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class="vector-toc-list"> </ul> </li> <li id="toc-Order" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Order"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Order</span> </div> </a> <ul id="toc-Order-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Numbers" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Numbers</span> </div> </a> <ul id="toc-Numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Arithmetics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Arithmetics"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Arithmetics</span> </div> </a> <button aria-controls="toc-Arithmetics-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Arithmetics subsection</span> </button> <ul id="toc-Arithmetics-sublist" class="vector-toc-list"> <li id="toc-Theory_of_arithmetic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Theory_of_arithmetic"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Theory of arithmetic</span> </div> </a> <ul id="toc-Theory_of_arithmetic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Exponentials" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exponentials"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Exponentials</span> </div> </a> <ul id="toc-Exponentials-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logarithms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logarithms"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Logarithms</span> </div> </a> <ul id="toc-Logarithms-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mathematics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Mathematics"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Mathematics</span> </div> </a> <ul id="toc-Mathematics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Changes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Changes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Changes</span> </div> </a> <ul id="toc-Changes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Revolutions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Revolutions"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Revolutions</span> </div> </a> <ul id="toc-Revolutions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Operations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Operations"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Operations</span> </div> </a> <ul id="toc-Operations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inclinations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Inclinations"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Inclinations</span> </div> </a> <ul id="toc-Inclinations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dimensional_analyses" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Dimensional_analyses"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Dimensional analyses</span> </div> </a> <ul id="toc-Dimensional_analyses-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Astronomical_units" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Astronomical_units"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>Astronomical units</span> </div> </a> <ul id="toc-Astronomical_units-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Regions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Regions"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</span> <span>Regions</span> </div> </a> <ul id="toc-Regions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Areas" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Areas"> <div class="vector-toc-text"> <span class="vector-toc-numb">16</span> <span>Areas</span> </div> </a> <ul id="toc-Areas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orbits" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Orbits"> <div class="vector-toc-text"> <span class="vector-toc-numb">17</span> <span>Orbits</span> </div> </a> <ul id="toc-Orbits-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Infinitesimals" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Infinitesimals"> <div class="vector-toc-text"> <span class="vector-toc-numb">18</span> <span>Infinitesimals</span> </div> </a> <ul id="toc-Infinitesimals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Distances" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Distances"> <div class="vector-toc-text"> <span class="vector-toc-numb">19</span> <span>Distances</span> </div> </a> <ul id="toc-Distances-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cosmic_distance_ladders" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Cosmic_distance_ladders"> <div class="vector-toc-text"> <span class="vector-toc-numb">20</span> <span>Cosmic distance ladders</span> </div> </a> <ul id="toc-Cosmic_distance_ladders-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Diameters" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Diameters"> <div class="vector-toc-text"> <span class="vector-toc-numb">21</span> <span>Diameters</span> </div> </a> <ul id="toc-Diameters-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Arithmetic_dimensional_analysis" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Arithmetic_dimensional_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">22</span> <span>Arithmetic dimensional analysis</span> </div> </a> <ul id="toc-Arithmetic_dimensional_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Obliquities" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Obliquities"> <div class="vector-toc-text"> <span class="vector-toc-numb">23</span> <span>Obliquities</span> </div> </a> <ul id="toc-Obliquities-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inverses" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Inverses"> <div class="vector-toc-text"> <span class="vector-toc-numb">24</span> <span>Inverses</span> </div> </a> <ul id="toc-Inverses-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Antapex" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Antapex"> <div class="vector-toc-text"> <span class="vector-toc-numb">25</span> <span>Antapex</span> </div> </a> <ul id="toc-Antapex-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebras" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Algebras"> <div class="vector-toc-text"> <span class="vector-toc-numb">26</span> <span>Algebras</span> </div> </a> <ul id="toc-Algebras-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometries" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Geometries"> <div class="vector-toc-text"> <span class="vector-toc-numb">27</span> <span>Geometries</span> </div> </a> <ul id="toc-Geometries-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Coordinates" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">28</span> <span>Coordinates</span> </div> </a> <button aria-controls="toc-Coordinates-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Coordinates subsection</span> </button> <ul id="toc-Coordinates-sublist" class="vector-toc-list"> <li id="toc-Triclinic_coordinates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Triclinic_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">28.1</span> <span>Triclinic coordinates</span> </div> </a> <ul id="toc-Triclinic_coordinates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Monoclinic_coordinates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Monoclinic_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">28.2</span> <span>Monoclinic coordinates</span> </div> </a> <ul id="toc-Monoclinic_coordinates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orthorhombic_coordinates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Orthorhombic_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">28.3</span> <span>Orthorhombic coordinates</span> </div> </a> <ul id="toc-Orthorhombic_coordinates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tetragonal_coordinates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tetragonal_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">28.4</span> <span>Tetragonal coordinates</span> </div> </a> <ul id="toc-Tetragonal_coordinates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rhombohedral_coordinates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rhombohedral_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">28.5</span> <span>Rhombohedral coordinates</span> </div> </a> <ul id="toc-Rhombohedral_coordinates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hexagonal_coordinates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hexagonal_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">28.6</span> <span>Hexagonal coordinates</span> </div> </a> <ul id="toc-Hexagonal_coordinates-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Triangles" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Triangles"> <div class="vector-toc-text"> <span class="vector-toc-numb">29</span> <span>Triangles</span> </div> </a> <ul id="toc-Triangles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Curvatures" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Curvatures"> <div class="vector-toc-text"> <span class="vector-toc-numb">30</span> <span>Curvatures</span> </div> </a> <ul id="toc-Curvatures-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conic_sections" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Conic_sections"> <div class="vector-toc-text"> <span class="vector-toc-numb">31</span> <span>Conic sections</span> </div> </a> <ul id="toc-Conic_sections-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Variations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Variations"> <div class="vector-toc-text"> <span class="vector-toc-numb">32</span> <span>Variations</span> </div> </a> <ul id="toc-Variations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Precessions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Precessions"> <div class="vector-toc-text"> <span class="vector-toc-numb">33</span> <span>Precessions</span> </div> </a> <ul id="toc-Precessions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rotations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Rotations"> <div class="vector-toc-text"> <span class="vector-toc-numb">34</span> <span>Rotations</span> </div> </a> <ul id="toc-Rotations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mirror_planes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Mirror_planes"> <div class="vector-toc-text"> <span class="vector-toc-numb">35</span> <span>Mirror planes</span> </div> </a> <ul id="toc-Mirror_planes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Resonances" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Resonances"> <div class="vector-toc-text"> <span class="vector-toc-numb">36</span> <span>Resonances</span> </div> </a> <ul id="toc-Resonances-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Eccentricities" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Eccentricities"> <div class="vector-toc-text"> <span class="vector-toc-numb">37</span> <span>Eccentricities</span> </div> </a> <ul id="toc-Eccentricities-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spherical_geometries" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Spherical_geometries"> <div class="vector-toc-text"> <span class="vector-toc-numb">38</span> <span>Spherical geometries</span> </div> </a> <ul id="toc-Spherical_geometries-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logical_laws" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Logical_laws"> <div class="vector-toc-text"> <span class="vector-toc-numb">39</span> <span>Logical laws</span> </div> </a> <ul id="toc-Logical_laws-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Horizontal_coordinate_system" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Horizontal_coordinate_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">40</span> <span>Horizontal coordinate system</span> </div> </a> <ul id="toc-Horizontal_coordinate_system-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fixed_point_in_the_sky" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Fixed_point_in_the_sky"> <div class="vector-toc-text"> <span class="vector-toc-numb">41</span> <span>Fixed point in the sky</span> </div> </a> <ul id="toc-Fixed_point_in_the_sky-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Trigonometries" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Trigonometries"> <div class="vector-toc-text"> <span class="vector-toc-numb">42</span> <span>Trigonometries</span> </div> </a> <ul id="toc-Trigonometries-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Angular_displacement" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Angular_displacement"> <div class="vector-toc-text"> <span class="vector-toc-numb">43</span> <span>Angular displacement</span> </div> </a> <ul id="toc-Angular_displacement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Radius_of_the_Earth" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Radius_of_the_Earth"> <div class="vector-toc-text"> <span class="vector-toc-numb">44</span> <span>Radius of the Earth</span> </div> </a> <ul id="toc-Radius_of_the_Earth-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Distance_computation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Distance_computation"> <div class="vector-toc-text"> <span class="vector-toc-numb">45</span> <span>Distance computation</span> </div> </a> <ul id="toc-Distance_computation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Distance_to_the_Moon" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Distance_to_the_Moon"> <div class="vector-toc-text"> <span class="vector-toc-numb">46</span> <span>Distance to the Moon</span> </div> </a> <ul id="toc-Distance_to_the_Moon-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Parallaxes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Parallaxes"> <div class="vector-toc-text"> <span class="vector-toc-numb">47</span> <span>Parallaxes</span> </div> </a> <ul id="toc-Parallaxes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Diurnal_parallax" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Diurnal_parallax"> <div class="vector-toc-text"> <span class="vector-toc-numb">48</span> <span>Diurnal parallax</span> </div> </a> <ul id="toc-Diurnal_parallax-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lunar_parallax" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Lunar_parallax"> <div class="vector-toc-text"> <span class="vector-toc-numb">49</span> <span>Lunar parallax</span> </div> </a> <ul id="toc-Lunar_parallax-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Calculuses" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Calculuses"> <div class="vector-toc-text"> <span class="vector-toc-numb">50</span> <span>Calculuses</span> </div> </a> <ul id="toc-Calculuses-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivatives" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Derivatives"> <div class="vector-toc-text"> <span class="vector-toc-numb">51</span> <span>Derivatives</span> </div> </a> <ul id="toc-Derivatives-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Partial_derivatives" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Partial_derivatives"> <div class="vector-toc-text"> <span class="vector-toc-numb">52</span> <span>Partial derivatives</span> </div> </a> <ul id="toc-Partial_derivatives-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gradients" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Gradients"> <div class="vector-toc-text"> <span class="vector-toc-numb">53</span> <span>Gradients</span> </div> </a> <ul id="toc-Gradients-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Area_under_a_curve" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Area_under_a_curve"> <div class="vector-toc-text"> <span class="vector-toc-numb">54</span> <span>Area under a curve</span> </div> </a> <ul id="toc-Area_under_a_curve-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integrals" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Integrals"> <div class="vector-toc-text"> <span class="vector-toc-numb">55</span> <span>Integrals</span> </div> </a> <ul id="toc-Integrals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Theoretical_calculus" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Theoretical_calculus"> <div class="vector-toc-text"> <span class="vector-toc-numb">56</span> <span>Theoretical calculus</span> </div> </a> <ul id="toc-Theoretical_calculus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Line_integrals" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Line_integrals"> <div class="vector-toc-text"> <span class="vector-toc-numb">57</span> <span>Line integrals</span> </div> </a> <ul id="toc-Line_integrals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vectors" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Vectors"> <div class="vector-toc-text"> <span class="vector-toc-numb">58</span> <span>Vectors</span> </div> </a> <ul id="toc-Vectors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tensors" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Tensors"> <div class="vector-toc-text"> <span class="vector-toc-numb">59</span> <span>Tensors</span> </div> </a> <ul id="toc-Tensors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Electronic_computers" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Electronic_computers"> <div class="vector-toc-text"> <span class="vector-toc-numb">60</span> <span>Electronic computers</span> </div> </a> <ul id="toc-Electronic_computers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Programmings" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Programmings"> <div class="vector-toc-text"> <span class="vector-toc-numb">61</span> <span>Programmings</span> </div> </a> <ul id="toc-Programmings-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Probabilities" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Probabilities"> <div class="vector-toc-text"> <span class="vector-toc-numb">62</span> <span>Probabilities</span> </div> </a> <ul id="toc-Probabilities-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Statistics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Statistics"> <div class="vector-toc-text"> <span class="vector-toc-numb">63</span> <span>Statistics</span> </div> </a> <ul id="toc-Statistics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hypotheses" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Hypotheses"> <div class="vector-toc-text"> 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<div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikiversity</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><div class="subpages">< <bdi dir="ltr"><a href="/wiki/Mathematics" title="Mathematics">Mathematics</a></bdi></div></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Superclusters_atlasoftheuniverse.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Superclusters_atlasoftheuniverse.gif/250px-Superclusters_atlasoftheuniverse.gif" decoding="async" width="250" height="234" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Superclusters_atlasoftheuniverse.gif/375px-Superclusters_atlasoftheuniverse.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/46/Superclusters_atlasoftheuniverse.gif/500px-Superclusters_atlasoftheuniverse.gif 2x" data-file-width="640" data-file-height="600" /></a><figcaption>Our universe within 1 billion light-years (307 Mpc) of Earth is shown to contain the local <a href="https://en.wikipedia.org/wiki/supercluster" class="extiw" title="w:supercluster">superclusters</a>, <a href="https://en.wikipedia.org/wiki/galaxy_filament" class="extiw" title="w:galaxy filament">galaxy filaments</a> and voids. Credit: Richard Powell.</figcaption></figure> <p>Although most of the mathematics needed to understand the information acquired through astronomical observation comes from physics, there are special needs from situations that intertwine mathematics with phenomena that may not yet have sufficient physics to explain the observations. These two uses of mathematics make <b>mathematical astronomy</b> a continuing challenge. </p><p>Astronomers use math all the time. One way it is used is when we look at objects in the sky with a telescope. The camera, specifically its charge-coupled device (CCD) detector, that is attached to the telescope basically converts or counts photons or electrons and records a series of numbers (the counts) - those numbers might correspond to how much light different objects in the sky are emitting, what type of light, etc. In order to be able to understand the information that these numbers contain, we need to use math and statistics to interpret them. </p><p>An initial use of mathematics in astronomy is counting <a href="/w/index.php?title=Radiation_astronomy/Entities&action=edit&redlink=1" class="new" title="Radiation astronomy/Entities (page does not exist)">entities</a>, sources, or objects in the sky. </p><p>Objects may be counted during the daytime or night. </p><p>One use of mathematics is the calculation of distance to an object in the sky. </p> <div style="clear:both;"></div> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Notations">Notations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=1" title="Edit section: Notations" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=1" title="Edit section's source code: Notations"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Notation</b>: let the symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oplus }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊕</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \oplus }"></span> indicate the <a href="/wiki/Earth" title="Earth">Earth</a>. </p><p><b>Notation</b>: let the symbol <b>ʘ</b> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \odot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊙</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \odot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e89e009eb8a8839c82aa5c76c15e9f2d67006276" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \odot }"></span> indicate the <a href="/wiki/Stars/Sun" title="Stars/Sun">Sun</a>. </p><p><b>Notation</b>: let the symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\odot }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊙</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\odot }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ba478a903e0a56d70246411ba707a43174b6f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.534ex; height:2.509ex;" alt="{\displaystyle I_{\odot }}"></span> indicate the total solar irradiance. </p><p><b>Notation</b>: let the symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84982d0f4b2453d70c502e13223a10e72b2f05b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.079ex; height:2.509ex;" alt="{\displaystyle L_{V}}"></span> indicate the solar visible luminosity. </p><p><b>Notation</b>: let the symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{\odot }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊙</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{\odot }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/373a5cff1f662ec2b7103301e2a6ed98c2b78498" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.094ex; height:2.509ex;" alt="{\displaystyle L_{\odot }}"></span> indicate the solar bolometric luminosity. </p><p><b>Notation</b>: let the symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{bol}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>o</mi> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{bol}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cee56f4c129219aeb0ac131b63f55d07a3d0b18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.808ex; height:2.509ex;" alt="{\displaystyle L_{bol}}"></span> indicate the solar bolometric luminosity. </p><p><b>Notation</b>: let the symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{bol}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>o</mi> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{bol}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/615effa1b0c8f030d2ba670ef44c4a7816575837" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.479ex; height:2.509ex;" alt="{\displaystyle M_{bol}}"></span> represent the <b>bolometric magnitude</b>, the total energy output. </p><p><b>Notation</b>: let the symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/982002375d43213f4fdbf6e63c65d4dc62a832ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.75ex; height:2.509ex;" alt="{\displaystyle M_{V}}"></span> represent the <b>visual magnitude</b>. </p><p><b>Notation</b>: let the symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\odot }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊙</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\odot }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fa96ed83d5db40d3350f1eb4f3e87754ef91793" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.765ex; height:2.509ex;" alt="{\displaystyle M_{\odot }}"></span> indicate the solar mass. </p><p><b>Notation</b>: let the symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{\odot }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊙</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{\odot }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b99cb7bd5053fbb9e5df11a7e1c0af5f89a6fed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.349ex; height:2.509ex;" alt="{\displaystyle Q_{\odot }}"></span> represent the net solar charge. </p><p><b>Notation</b>: let the symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\oplus }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊕</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\oplus }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98d8c196ed57b7c943fae8462bfc13c718e978ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.275ex; height:2.509ex;" alt="{\displaystyle R_{\oplus }}"></span> indicate the Earth's radius. </p><p><b>Notation</b>: let the symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{J}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{J}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fff22fd65de94cd43f50c1c009aa14acd295d41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.037ex; height:2.509ex;" alt="{\displaystyle R_{J}}"></span> indicate the radius of Jupiter. </p><p><b>Notation</b>: let the symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\odot }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊙</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\odot }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9117657b4cb9f470162cc84872f47307bab0c86a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.275ex; height:2.509ex;" alt="{\displaystyle R_{\odot }}"></span> indicate the solar radius. </p> <dl><dd><table border="1" cellpadding="5" cellspacing="0" align="none"> <tbody><tr> <th colspan="3"><b>Notational locations</b> </th></tr> <tr> <td><b>Weight</b> </td> <td><b>Oversymbol</b> </td> <td><b>Exponent</b> </td></tr> <tr> <td><b>Coefficient</b> </td> <td><b>Variable</b> </td> <td><b>Operation</b> </td></tr> <tr> <td><b>Number</b> </td> <td><b>Range</b> </td> <td><b>Index</b> </td></tr> </tbody></table></dd></dl> <p>For each of the notational locations around the central <b>Variable</b>, conventions are often set by consensus as to use. For example, <b>Exponent</b> is often used as an exponent to a number or variable: 2<sup>-2</sup> or x<sup>2</sup>. </p><p>In the <b>Notation</b>s at the top of this section, <b>Index</b> is replaced by symbols for the Sun (ʘ), <a href="/wiki/Earth" title="Earth">Earth</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\oplus }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊕</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\oplus }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98d8c196ed57b7c943fae8462bfc13c718e978ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.275ex; height:2.509ex;" alt="{\displaystyle R_{\oplus }}"></span>), or can be for Jupiter (J) such as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{J}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{J}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fff22fd65de94cd43f50c1c009aa14acd295d41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.037ex; height:2.509ex;" alt="{\displaystyle R_{J}}"></span>. </p><p>A common <b>Oversymbol</b> is one for the average <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {Variable}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>V</mi> <mi>a</mi> <mi>r</mi> <mi>i</mi> <mi>a</mi> <mi>b</mi> <mi>l</mi> <mi>e</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {Variable}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df4f8961f4d38509eb25f141312d85fd14d93e61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.987ex; height:3.009ex;" alt="{\displaystyle {\overline {Variable}}}"></span>. </p><p><b>Operation</b> may be replaced by a function, for example. </p><p>All notational locations could look something like </p> <dl><dd><table border="0" cellpadding="3" cellspacing="0" align="none"> <tbody><tr> <td>bx </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bd52ce670743d3b61bec928a7ec9f47309eb36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle -}"></span> </td> <td>x = n </td></tr> <tr> <td>a </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1d4e06539576633987e902f402ed46728d573b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.355ex; height:3.843ex;" alt="{\displaystyle \sum }"></span> </td> <td>f(x) </td></tr> <tr> <td>n </td> <td>→ </td> <td>∞ </td></tr> </tbody></table></dd></dl> <p>where the center line means "a x Σ f(x)" for all added up values of f(x) when x = n from say 0 to infinity with each term in the sum before summation multiplied by bn, then divided by n for an average whenever n is finite. </p> <div class="mw-heading mw-heading2"><h2 id="Abstractions">Abstractions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=2" title="Edit section: Abstractions" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=2" title="Edit section's source code: Abstractions"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r2661592">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}</style><div role="note" class="hatnote navigation-not-searchable">Main resource: <a href="/w/index.php?title=Abstractions&action=edit&redlink=1" class="new" title="Abstractions (page does not exist)">Abstractions</a></div> <p>A <b>nomy</b> (Latin <i>nomia</i>) is a "system of <a href="/w/index.php?title=Laws&action=edit&redlink=1" class="new" title="Laws (page does not exist)">laws</a> governing or [the] sum of <a href="/w/index.php?title=Knowledge&action=edit&redlink=1" class="new" title="Knowledge (page does not exist)">knowledge</a> regarding a (specified) field."<sup id="cite_ref-Gove_1-0" class="reference"><a href="#cite_note-Gove-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <i>Nomology</i> is the "<a href="/wiki/What_is_science%3F" title="What is science?">science</a> of physical and logical laws."<sup id="cite_ref-Gove_1-1" class="reference"><a href="#cite_note-Gove-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p><b>Def.</b> the quality of dealing with ideas rather than events is called <b>abstraction</b>. </p><p><b>Def.</b> the act of the theoretical way of looking at things; something that exists only in idealized form is called <b>abstraction</b>. </p> <div class="mw-heading mw-heading2"><h2 id="Relations">Relations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=3" title="Edit section: Relations" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=3" title="Edit section's source code: Relations"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resource: <a href="/w/index.php?title=Relations&action=edit&redlink=1" class="new" title="Relations (page does not exist)">Relations</a></div> <p><b>Notation</b>: let the relation symbol <b>≠</b> indicate that two expressions are different. </p><p>For example, 2 x 3 ≠ 5 x 7. </p><p><b>Notation</b>: let the relation symbol <b>~</b> represent <b>similar to</b>. </p><p>For example, depending on the scale involved, 7 ~ 8 on a scale of 10, 7/10 = 0.7 and 8/10 = 0.8. relative to numbers between 0.5 and 1.0, 0.7 ~ 0.8, but 0.2 ≁ 0.7. </p><p>Similarity may be close such as 0.7 ≈ 0.8, but 0.5 ~ 0.8. Or similarity may include equality, 5 ± 3 ≃ 4 ± 2. When the degree of equality is greater than the degree of similarity, the symbol ≅ is used. The reverse is represented by ≊. </p> <div class="mw-heading mw-heading2"><h2 id="Differences">Differences</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=4" title="Edit section: Differences" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=4" title="Edit section's source code: Differences"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resources: <a href="/w/index.php?title=Abstractions/Differences&action=edit&redlink=1" class="new" title="Abstractions/Differences (page does not exist)">Abstractions/Differences</a> and <a href="/w/index.php?title=Differences&action=edit&redlink=1" class="new" title="Differences (page does not exist)">Differences</a></div> <p>Here's a <a href="/wiki/Definitions/Theory#Theoretical_definition" title="Definitions/Theory">theoretical definition</a>: </p><p><b>Def.</b> an abstract relation between identity and sameness is called a <b>difference</b>. </p><p><b>Notation</b>: let the symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }"></span> represent <b>difference in</b>. </p> <div class="mw-heading mw-heading2"><h2 id="Order">Order</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=5" title="Edit section: Order" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=5" title="Edit section's source code: Order"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resource: <a href="/w/index.php?title=Order&action=edit&redlink=1" class="new" title="Order (page does not exist)">Order</a></div> <p>Ordering numbers may mean listing them in increasing value. For example, 2 is less than 3 so that in increasing order 2,3 is the list. </p><p><b>Notation</b>: let the symbol <b>></b> represent <b>greater than</b>. </p><p>For example, the integer five (5) is greater than the integer (2): 5 > 2. </p><p><b>Notation</b>: let the symbol <b><</b> represent <b>less than</b>. </p><p>For, example, 2 < 3. </p> <div class="mw-heading mw-heading2"><h2 id="Numbers">Numbers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=6" title="Edit section: Numbers" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=6" title="Edit section's source code: Numbers"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resource: <a href="/wiki/Numbers" title="Numbers">Numbers</a></div> <dl><dd><table style="border:1px solid #ddd; text-align:center; margin:auto" cellspacing="20"> <tbody><tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1,2,3,\ldots \!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1,2,3,\ldots \!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5961ff1e82de3e93621ebeb348d405c518df8436" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.204ex; width:9.129ex; height:2.509ex;" alt="{\displaystyle 1,2,3,\ldots \!}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ldots ,-2,-1,0,1,2\,\ldots \!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>…</mo> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mspace width="thinmathspace" /> <mo>…</mo> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ldots ,-2,-1,0,1,2\,\ldots \!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f33340d3b458b4a9782ff34b24b8c2d55abb4bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.204ex; width:21.023ex; height:2.509ex;" alt="{\displaystyle \ldots ,-2,-1,0,1,2\,\ldots \!}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -2,{\frac {2}{3}},1.21\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>,</mo> <mn>1.21</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -2,{\frac {2}{3}},1.21\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46f69bde2e37e56443b3c6839be8c010fdaad7a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:11.558ex; height:5.176ex;" alt="{\displaystyle -2,{\frac {2}{3}},1.21\,\!}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -e,{\sqrt {2}},3,\pi \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>e</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mi>π</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -e,{\sqrt {2}},3,\pi \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4344d8d6a980cb805d7b6cc29cd8770d8d8d61c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:11.973ex; height:3.009ex;" alt="{\displaystyle -e,{\sqrt {2}},3,\pi \,\!}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2,i,-2+3i,2e^{i{\frac {4\pi }{3}}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>,</mo> <mi>i</mi> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mi>i</mi> <mo>,</mo> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </mrow> </mrow> </msup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2,i,-2+3i,2e^{i{\frac {4\pi }{3}}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/694cccd4f4f8bf7e7159431b59477c67387b77ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:18.543ex; height:3.843ex;" alt="{\displaystyle 2,i,-2+3i,2e^{i{\frac {4\pi }{3}}}\,\!}"></span> </td></tr> <tr> <td><a href="/wiki/Natural_number" title="Natural number">Natural numbers</a></td> <td><a href="/wiki/Integer" title="Integer">Integers</a></td> <td><a href="https://en.wikipedia.org/wiki/Rational_number" class="extiw" title="w:Rational number">Rational numbers</a></td> <td><a href="/wiki/Real_Numbers" title="Real Numbers">Real numbers</a></td> <td><a href="/wiki/Complex_Numbers" title="Complex Numbers">Complex numbers</a> </td></tr></tbody></table></dd></dl> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Avogadro%27s_number_in_e_notation.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Avogadro%27s_number_in_e_notation.jpg/200px-Avogadro%27s_number_in_e_notation.jpg" decoding="async" width="200" height="43" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Avogadro%27s_number_in_e_notation.jpg/300px-Avogadro%27s_number_in_e_notation.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Avogadro%27s_number_in_e_notation.jpg/400px-Avogadro%27s_number_in_e_notation.jpg 2x" data-file-width="2955" data-file-height="631" /></a><figcaption>A calculator display showing an approximation to the <a href="https://en.wikipedia.org/wiki/Avogadro_constant" class="extiw" title="w:Avogadro constant">Avogadro constant</a> in E notation. Credit: <a href="https://commons.wikimedia.org/wiki/User:PRHaney" class="extiw" title="commons:User:PRHaney">PRHaney</a>.</figcaption></figure> <p><b>Scientific notation</b> (more commonly known as <b>standard form</b>) is a way of writing numbers that are too big or too small to be conveniently written in decimal form. Scientific notation has a number of useful properties and is commonly used in calculators and by scientists, mathematicians and engineers. </p> <table class="wikitable" style="float:left"> <tbody><tr> <th>Standard decimal notation </th> <th>Normalized scientific notation </th></tr> <tr> <td>2 </td> <td><span style="white-space:nowrap"><span style="white-space:nowrap">2</span><span style="white-space:nowrap;margin-left:0.25em;margin-right:0.15em">×</span>10<span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"><span style="white-space:nowrap">0</span></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span> </td></tr> <tr> <td>300 </td> <td><span style="white-space:nowrap"><span style="white-space:nowrap">3</span><span style="white-space:nowrap;margin-left:0.25em;margin-right:0.15em">×</span>10<span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"><span style="white-space:nowrap">2</span></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span> </td></tr> <tr> <td>4,321.768 </td> <td><span style="white-space:nowrap"><span style="white-space:nowrap">4.321<span style="margin-left:.25em">768</span></span><span style="white-space:nowrap;margin-left:0.25em;margin-right:0.15em">×</span>10<span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"><span style="white-space:nowrap">3</span></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span> </td></tr> <tr> <td>-53,000 </td> <td><span style="white-space:nowrap">−<span style="white-space:nowrap">5.3</span><span style="white-space:nowrap;margin-left:0.25em;margin-right:0.15em">×</span>10<span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"><span style="white-space:nowrap">4</span></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span> </td></tr> <tr> <td>6,720,000,000 </td> <td><span style="white-space:nowrap"><span style="white-space:nowrap">6.72</span><span style="white-space:nowrap;margin-left:0.25em;margin-right:0.15em">×</span>10<span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"><span style="white-space:nowrap">9</span></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span> </td></tr> <tr> <td>0.2 </td> <td><span style="white-space:nowrap"><span style="white-space:nowrap">2</span><span style="white-space:nowrap;margin-left:0.25em;margin-right:0.15em">×</span>10<span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">−<span style="white-space:nowrap">1</span></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span> </td></tr> <tr> <td>0.000 000 007 51 </td> <td><span style="white-space:nowrap"><span style="white-space:nowrap">7.51</span><span style="white-space:nowrap;margin-left:0.25em;margin-right:0.15em">×</span>10<span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">−<span style="white-space:nowrap">9</span></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span> </td></tr></tbody></table> <p>A <b>metric prefix</b> or <b>SI prefix</b> is a <a href="https://en.wikipedia.org/wiki/unit_prefix" class="extiw" title="w:unit prefix">unit prefix</a> that precedes a basic unit of measure to indicate a <a href="https://en.wikipedia.org/wiki/decimal" class="extiw" title="w:decimal">decadic</a> <a href="https://en.wikipedia.org/wiki/multiple_(mathematics)" class="extiw" title="w:multiple (mathematics)">multiple</a> or <a href="https://en.wikipedia.org/wiki/fraction_(mathematics)" class="extiw" title="w:fraction (mathematics)">fraction</a> of the unit. Each prefix has a unique symbol that is prepended to the unit symbol. </p> <table class="wikitable" style="padding: 0; text-align: center; width: 0"> <tbody><tr> <th colspan="2" style="background:#ccf;"><a href="https://en.wikipedia.org/wiki/Metric_prefix" class="extiw" title="w:Metric prefix">Metric prefixes</a> </th></tr> <tr> <td> <table style="border: 1px #aaa solid"> <tbody><tr> <th style="background:#edf; text-align: center">Prefix </th> <th style="background:#edf; text-align: center">Symbol </th> <th style="background:#edf; text-align: center">1000<sup><i>m</i></sup> </th> <th style="background:#edf; text-align: center">10<sup><i>n</i></sup> </th> <th style="background:#edf; text-align: center"><a href="https://en.wikipedia.org/wiki/Decimal" class="extiw" title="w:Decimal">Decimal</a> </th> <th style="background:#edf; text-align: center"><a href="https://en.wikipedia.org/wiki/Long_and_short_scales" class="extiw" title="w:Long and short scales">Short scale</a> </th> <th style="background:#edf; text-align: center"><a href="https://en.wikipedia.org/wiki/Long_and_short_scales" class="extiw" title="w:Long and short scales">Long scale</a> </th> <th style="background:#edf; text-align: center">Since<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>n 1<span class="cite-bracket">]</span></a></sup> </th></tr> <tr> <td><a href="https://en.wikipedia.org/wiki/yotta-" class="extiw" title="w:yotta-">yotta</a> </td> <td align="center">Y </td> <td>1000<sup>8</sup> </td> <td><a href="https://en.wikipedia.org/wiki/Orders_of_magnitude_(numbers)#1024" class="extiw" title="w:Orders of magnitude (numbers)">10<sup>24</sup></a> </td> <td align="right"><span style="white-space:nowrap">1<span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span></span> </td> <td>septillion </td> <td>quadrillion </td> <td>1991 </td></tr> <tr> <td><a href="https://en.wikipedia.org/wiki/zetta-" class="extiw" title="w:zetta-">zetta</a> </td> <td align="center">Z </td> <td>1000<sup>7</sup> </td> <td><a href="https://en.wikipedia.org/wiki/Orders_of_magnitude_(numbers)#1021" class="extiw" title="w:Orders of magnitude (numbers)">10<sup>21</sup></a> </td> <td align="right"><span style="white-space:nowrap">1<span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span></span> </td> <td>sextillion </td> <td>trilliard </td> <td>1991 </td></tr> <tr> <td><a href="https://en.wikipedia.org/wiki/exa-" class="extiw" title="w:exa-">exa</a> </td> <td align="center">E </td> <td>1000<sup>6</sup> </td> <td><a href="https://en.wikipedia.org/wiki/Orders_of_magnitude_(numbers)#1018" class="extiw" title="w:Orders of magnitude (numbers)">10<sup>18</sup></a> </td> <td align="right"><span style="white-space:nowrap">1<span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span></span> </td> <td>quintillion </td> <td>trillion </td> <td>1975 </td></tr> <tr> <td><a href="https://en.wikipedia.org/wiki/peta-" class="extiw" title="w:peta-">peta</a> </td> <td align="center">P </td> <td>1000<sup>5</sup> </td> <td><a href="https://en.wikipedia.org/wiki/Orders_of_magnitude_(numbers)#1015" class="extiw" title="w:Orders of magnitude (numbers)">10<sup>15</sup></a> </td> <td align="right"><span style="white-space:nowrap">1<span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span></span> </td> <td>quadrillion </td> <td>billiard </td> <td>1975 </td></tr> <tr> <td><a href="https://en.wikipedia.org/wiki/tera-" class="extiw" title="w:tera-">tera</a> </td> <td align="center">T </td> <td>1000<sup>4</sup> </td> <td><a href="https://en.wikipedia.org/wiki/Orders_of_magnitude_(numbers)#1012" class="extiw" title="w:Orders of magnitude (numbers)">10<sup>12</sup></a> </td> <td align="right"><span style="white-space:nowrap">1<span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span></span> </td> <td>trillion </td> <td>billion </td> <td>1960 </td></tr> <tr> <td><a href="https://en.wikipedia.org/wiki/giga-" class="extiw" title="w:giga-">giga</a> </td> <td align="center">G </td> <td>1000<sup>3</sup> </td> <td><a href="https://en.wikipedia.org/wiki/1000000000_(number)" class="extiw" title="w:1000000000 (number)">10<sup>9</sup></a> </td> <td align="right"><span style="white-space:nowrap">1<span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span></span> </td> <td>billion </td> <td>milliard </td> <td>1960 </td></tr> <tr> <td><a href="https://en.wikipedia.org/wiki/mega-" class="extiw" title="w:mega-">mega</a> </td> <td align="center">M </td> <td>1000<sup>2</sup> </td> <td><a href="https://en.wikipedia.org/wiki/1000000_(number)" class="extiw" title="w:1000000 (number)">10<sup>6</sup></a> </td> <td align="right"><span style="white-space:nowrap">1<span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span></span> </td> <td colspan="2" align="center">million </td> <td>1960 </td></tr> <tr> <td><a href="https://en.wikipedia.org/wiki/kilo-" class="extiw" title="w:kilo-">kilo</a> </td> <td align="center">k </td> <td>1000<sup>1</sup> </td> <td><a href="https://en.wikipedia.org/wiki/1000_(number)" class="extiw" title="w:1000 (number)">10<sup>3</sup></a> </td> <td align="right"><span style="white-space:nowrap">1<span style="margin-left:0.25em">000</span></span> </td> <td colspan="2" align="center">thousand </td> <td>1795 </td></tr> <tr> <td><a href="https://en.wikipedia.org/wiki/hecto-" class="extiw" title="w:hecto-">hecto</a> </td> <td align="center">h </td> <td>1000<sup>2/3</sup> </td> <td><a href="https://en.wikipedia.org/wiki/100_(number)" class="extiw" title="w:100 (number)">10<sup>2</sup></a> </td> <td align="right">100 </td> <td colspan="2" align="center">hundred </td> <td>1795 </td></tr> <tr> <td><a href="https://en.wikipedia.org/wiki/deca-" class="extiw" title="w:deca-">deca</a> </td> <td align="center">da </td> <td>1000<sup>1/3</sup> </td> <td><a href="https://en.wikipedia.org/wiki/10_(number)" class="extiw" title="w:10 (number)">10<sup>1</sup></a> </td> <td align="right">10 </td> <td colspan="2" align="center">ten </td> <td>1795 </td></tr> <tr style="background-color:#EEE"> <td colspan="2"> </td> <td>1000<sup>0</sup> </td> <td><a href="https://en.wikipedia.org/wiki/1_(number)" class="extiw" title="w:1 (number)">10<sup>0</sup></a> </td> <td align="center">1 </td> <td colspan="2" align="center">one </td> <td>– </td></tr> <tr> <td><a href="https://en.wikipedia.org/wiki/deci-" class="extiw" title="w:deci-">deci</a> </td> <td align="center">d </td> <td>1000<sup>−1/3</sup> </td> <td><a href="https://en.wikipedia.org/wiki/Orders_of_magnitude_(numbers)#10.E2.88.921" class="extiw" title="w:Orders of magnitude (numbers)">10<sup>−1</sup></a> </td> <td align="left">0.1 </td> <td colspan="2" align="center">tenth </td> <td>1795 </td></tr> <tr> <td><a href="https://en.wikipedia.org/wiki/centi-" class="extiw" title="w:centi-">centi</a> </td> <td align="center">c </td> <td>1000<sup>−2/3</sup> </td> <td><a href="https://en.wikipedia.org/wiki/Orders_of_magnitude_(numbers)#10.E2.88.922" class="extiw" title="w:Orders of magnitude (numbers)">10<sup>−2</sup></a> </td> <td align="left">0.01 </td> <td colspan="2" align="center">hundredth </td> <td>1795 </td></tr> <tr> <td><a href="https://en.wikipedia.org/wiki/milli-" class="extiw" title="w:milli-">milli</a> </td> <td align="center">m </td> <td>1000<sup>−1</sup> </td> <td><a href="https://en.wikipedia.org/wiki/Orders_of_magnitude_(numbers)#10.E2.88.923" class="extiw" title="w:Orders of magnitude (numbers)">10<sup>−3</sup></a> </td> <td align="left">0.001 </td> <td colspan="2" align="center">thousandth </td> <td>1795 </td></tr> <tr> <td><a href="https://en.wikipedia.org/wiki/micro-" class="extiw" title="w:micro-">micro</a> </td> <td align="center"><a href="https://en.wikipedia.org/wiki/Mu_(letter)" class="extiw" title="w:Mu (letter)">μ</a> </td> <td>1000<sup>−2</sup> </td> <td><a href="https://en.wikipedia.org/wiki/Orders_of_magnitude_(numbers)#10.E2.88.926" class="extiw" title="w:Orders of magnitude (numbers)">10<sup>−6</sup></a> </td> <td align="left"><span style="white-space:nowrap">0.000<span style="margin-left:0.25em">001</span></span> </td> <td colspan="2" align="center">millionth </td> <td>1960 </td></tr> <tr> <td><a href="https://en.wikipedia.org/wiki/nano-" class="extiw" title="w:nano-">nano</a> </td> <td align="center">n </td> <td>1000<sup>−3</sup> </td> <td><a href="https://en.wikipedia.org/wiki/Orders_of_magnitude_(numbers)#10.E2.88.929" class="extiw" title="w:Orders of magnitude (numbers)">10<sup>−9</sup></a> </td> <td align="left"><span style="white-space:nowrap">0.000<span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">001</span></span> </td> <td>billionth </td> <td>milliardth </td> <td>1960 </td></tr> <tr> <td><a href="https://en.wikipedia.org/wiki/pico-" class="extiw" title="w:pico-">pico</a> </td> <td align="center">p </td> <td>1000<sup>−4</sup> </td> <td><a href="https://en.wikipedia.org/wiki/Orders_of_magnitude_(numbers)#10.E2.88.9212" class="extiw" title="w:Orders of magnitude (numbers)">10<sup>−12</sup></a> </td> <td align="left"><span style="white-space:nowrap">0.000<span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">001</span></span> </td> <td>trillionth </td> <td>billionth </td> <td>1960 </td></tr> <tr> <td><a href="https://en.wikipedia.org/wiki/femto-" class="extiw" title="w:femto-">femto</a> </td> <td align="center">f </td> <td>1000<sup>−5</sup> </td> <td><a href="https://en.wikipedia.org/wiki/Orders_of_magnitude_(numbers)#10.E2.88.9215" class="extiw" title="w:Orders of magnitude (numbers)">10<sup>−15</sup></a> </td> <td align="left"><span style="white-space:nowrap">0.000<span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">001</span></span> </td> <td>quadrillionth </td> <td>billiardth </td> <td>1964 </td></tr> <tr> <td><a href="https://en.wikipedia.org/wiki/atto-" class="extiw" title="w:atto-">atto</a> </td> <td align="center">a </td> <td>1000<sup>−6</sup> </td> <td><a href="https://en.wikipedia.org/wiki/Orders_of_magnitude_(numbers)#10.E2.88.9218" class="extiw" title="w:Orders of magnitude (numbers)">10<sup>−18</sup></a> </td> <td align="left"><span style="white-space:nowrap">0.000<span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">001</span></span> </td> <td>quintillionth </td> <td>trillionth </td> <td>1964 </td></tr> <tr> <td><a href="https://en.wikipedia.org/wiki/zepto-" class="extiw" title="w:zepto-">zepto</a> </td> <td align="center">z </td> <td>1000<sup>−7</sup> </td> <td><a href="https://en.wikipedia.org/wiki/Orders_of_magnitude_(numbers)#10.E2.88.9221" class="extiw" title="w:Orders of magnitude (numbers)">10<sup>−21</sup></a> </td> <td align="left"><span style="white-space:nowrap">0.000<span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">001</span></span> </td> <td>sextillionth </td> <td>trilliardth </td> <td>1991 </td></tr> <tr> <td><a href="https://en.wikipedia.org/wiki/yocto-" class="extiw" title="w:yocto-">yocto</a> </td> <td align="center">y </td> <td>1000<sup>−8</sup> </td> <td><a href="https://en.wikipedia.org/wiki/Orders_of_magnitude_(numbers)#10.E2.88.9224" class="extiw" title="w:Orders of magnitude (numbers)">10<sup>−24</sup></a> </td> <td align="left"><span style="white-space:nowrap">0.000<span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">001</span></span> </td> <td>septillionth </td> <td>quadrillionth </td> <td>1991 </td></tr> <tr> <td colspan="8" style="font-size:xx-small;background:#EEE"> <ol class="references"> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">The metric system was introduced in 1795 with six prefixes. The other dates relate to recognition by a resolution of the <a href="https://en.wikipedia.org/wiki/General_Conference_on_Weights_and_Measures" class="extiw" title="w:General Conference on Weights and Measures">General Conference on Weights and Measures</a> (CGPM)]].</span> </li> </ol> </td></tr></tbody></table> </td></tr></tbody></table> <p>A significant figure is a digit in a number that adds to its precision. This includes all nonzero numbers, zeroes between significant digits, and zeroes <a href="https://en.wikipedia.org/wiki/Significant_figures#Identifying_significant_digits" class="extiw" title="w:Significant figures">indicated to be significant.</a> </p><p>Leading and trailing zeroes are not significant because they exist only to show the scale of the number. Therefore, 1,230,400 has five significant figures—1, 2, 3, 0, and 4; the two zeroes serve only as placeholders and add no precision to the original number. </p><p>When a number is converted into normalized scientific notation, it is scaled down to a number between 1 and 10. All of the significant digits remain, but all of the place holding zeroes are incorporated into the exponent. Following these rules, 1,230,400 becomes 1.2304 x 10<sup>6</sup>. </p><p>It is customary in scientific measurements to record all the significant digits from the measurements, for example, 1,230,400, but the measurement may have introduced an error which when calculated indicates the last significant digit has a range of values where the most likely one is the "4". The range may be 3-5 so that the last significant digit plus this error may be written as (4,1) meaning 4-1=3 and 4+1=5. </p><p>Another example of significant digits is the speed of all <a href="https://en.wikipedia.org/wiki/massless_particle" class="extiw" title="w:massless particle">massless particles</a> and associated <a href="https://en.wikipedia.org/wiki/field_(physics)" class="extiw" title="w:field (physics)">fields</a>—including <a href="https://en.wikipedia.org/wiki/electromagnetic_radiation" class="extiw" title="w:electromagnetic radiation">electromagnetic radiation</a> such as <a href="https://en.wikipedia.org/wiki/light" class="extiw" title="w:light">light</a>—in vacuum ... [The most accurate value is] 299792.4562±0.0011 [km/s].<sup id="cite_ref-NIST_heterodyne_3-0" class="reference"><a href="#cite_note-NIST_heterodyne-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The magnitude of the speed is 299792.4562 and the actual measured variation is ±0.0011 so that the last two significant digits "62" are most likely within a variation from "51" to "73". </p><p>Most <a href="https://en.wikipedia.org/wiki/calculator" class="extiw" title="w:calculator">calculators</a> and many <a href="https://en.wikipedia.org/wiki/computer_program" class="extiw" title="w:computer program">computer programs</a> present very large and very small results in E notation. The <a href="https://en.wikipedia.org/wiki/E" class="extiw" title="w:E">letter <i>E</i> or <i>e</i></a> is often used to represent <i>times ten raised to the power of</i> (which would be written as "x 10<sup><i>b</i></sup>") [where <i>b</i> represents a number] and is followed by the value of the exponent. </p><p><b>Def.</b> any real number that cannot be expressed as a ratio of two integers is called an <b>irrational number</b>. </p><p><b>Def.</b> incapable of being put into one-to-one correspondence with the natural numbers or any subset thereof is called <b>uncountable</b>. </p><p>An uncountable set of numbers such as the irrational numbers lies somewhere between a finite set of numbers, for example, the set of natural factors of 6: {1,2,3,6}, and an infinite set of numbers such as the <a href="https://en.wikipedia.org/wiki/Natural_number" class="extiw" title="w:Natural number">natural numbers</a>. </p><p><b>Def.</b> the branch of pure mathematics concerned with the properties of integers is called <b>number theory</b>. </p> <div class="mw-heading mw-heading2"><h2 id="Arithmetics">Arithmetics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=7" title="Edit section: Arithmetics" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=7" title="Edit section's source code: Arithmetics"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resource: <a href="/w/index.php?title=Arithmetics&action=edit&redlink=1" class="new" title="Arithmetics (page does not exist)">Arithmetics</a></div> <p>Arithmetic involves the manipulation of numbers. </p> <div class="mw-heading mw-heading3"><h3 id="Theory_of_arithmetic">Theory of arithmetic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=8" title="Edit section: Theory of arithmetic" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=8" title="Edit section's source code: Theory of arithmetic"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Def.</b> the <a href="https://en.wiktionary.org/wiki/mathematics" class="extiw" title="wikt:mathematics">mathematics</a> of <a href="https://en.wiktionary.org/wiki/number" class="extiw" title="wikt:number">numbers</a> (<a href="https://en.wiktionary.org/wiki/integer" class="extiw" title="wikt:integer">integers</a>, <a href="https://en.wiktionary.org/wiki/rational_number" class="extiw" title="wikt:rational number">rational numbers</a>, <a href="https://en.wiktionary.org/wiki/real_number" class="extiw" title="wikt:real number">real numbers</a>, or <a href="https://en.wiktionary.org/wiki/complex_number" class="extiw" title="wikt:complex number">complex numbers</a>) under the operations of <a href="https://en.wiktionary.org/wiki/addition" class="extiw" title="wikt:addition">addition</a>, <a href="https://en.wiktionary.org/wiki/subtraction" class="extiw" title="wikt:subtraction">subtraction</a>, <a href="https://en.wiktionary.org/wiki/multiplication" class="extiw" title="wikt:multiplication">multiplication</a>, and <a href="https://en.wiktionary.org/wiki/division" class="extiw" title="wikt:division">division</a> is called an <b>arithmetic</b>. </p><p><b>Def.</b> a symbol ( = ) used in mathematics to indicate that two values are the same is called <b>equals</b>, or <b>equal to</b>. </p><p>Consider the integers: 1 and 2. The statement, "1 + 2 = 3", contains the operation + (addition) and the relation = (equals). </p> <div class="mw-heading mw-heading3"><h3 id="Exponentials">Exponentials</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=9" title="Edit section: Exponentials" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=9" title="Edit section's source code: Exponentials"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resource: <a href="/w/index.php?title=Exponentials&action=edit&redlink=1" class="new" title="Exponentials (page does not exist)">Exponentials</a></div> <p>The exponential can require a different operator of arithmetic. </p><p>The number <b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span></b> is an important <a href="https://en.wikipedia.org/wiki/mathematical_constant" class="extiw" title="w:mathematical constant">mathematical constant</a>, approximately equal to 2.71828, that is the base of the <a href="https://en.wikipedia.org/wiki/natural_logarithm" class="extiw" title="w:natural logarithm">natural logarithm</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=1+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+{\frac {1}{1\cdot 2\cdot 3\cdot 4}}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>1</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> <mo>⋅</mo> <mn>4</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e=1+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+{\frac {1}{1\cdot 2\cdot 3\cdot 4}}+\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c4cb2aca2a6325796b15e05d48c4e8b121916f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:47.313ex; height:5.343ex;" alt="{\displaystyle e=1+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+{\frac {1}{1\cdot 2\cdot 3\cdot 4}}+\cdots }"></span></dd></dl> <dl><dd>e<sup>2</sup> + e<sup>3</sup> ≠ e<sup>5</sup>. Yet</dd> <dd>e<sup>2</sup> x e<sup>3</sup> = e<sup>(2 + 3)</sup> = e<sup>5</sup>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Logarithms">Logarithms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=10" title="Edit section: Logarithms" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=10" title="Edit section's source code: Logarithms"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resource: <a href="/w/index.php?title=Logarithms&action=edit&redlink=1" class="new" title="Logarithms (page does not exist)">Logarithms</a></div> <p><b>Def.</b> For a number [...], the [exponent or power] to which a given <i>base</i> number must be raised in order to obtain [the number] is called a <b>logarithm</b>. </p><p>Consider 10<sup>3</sup>, the <i>base</i> number is 10. The exponent is 3. The number itself is 1000. Using logarithm notation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{10}1000=3.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mn>1000</mn> <mo>=</mo> <mn>3.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{10}1000=3.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3011e145fee28065715bbb48e5232a35ee524c55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.793ex; height:2.676ex;" alt="{\displaystyle \log _{10}1000=3.}"></span></dd></dl> <p>For 2<sup>4</sup>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{2}16=4.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mn>16</mn> <mo>=</mo> <mn>4.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{2}16=4.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/950c935deac81eaee0436c30bbcda0404b8506de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.646ex; height:2.676ex;" alt="{\displaystyle \log _{2}16=4.}"></span></dd></dl> <p>For e<sup>4</sup>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln _{e}e^{4}=4.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ln</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>⁡</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mn>4.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln _{e}e^{4}=4.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/958c99cb3a34b5e26cc097359c4ec92ae18a492c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.37ex; height:3.009ex;" alt="{\displaystyle \ln _{e}e^{4}=4.}"></span></dd></dl> <p><b>Notation</b>: let the symbol <b>dex</b> represent the difference between powers of ten. </p><p>An order or factor of ten, dex is used both to refer to the function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dex(x)=10^{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>e</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dex(x)=10^{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a396fea19e6edf0866db3d63d8309aee7891013" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.364ex; height:2.843ex;" alt="{\displaystyle dex(x)=10^{x}}"></span> and the number of (possibly fractional) orders of magnitude separating two numbers. When dealing with <a href="https://en.wiktionary.org/wiki/log" class="extiw" title="wikt:log">log</a> to the base 10 transform of a number set, the transform of 10, 100, and 1 000 000 is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{10}(10)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>10</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{10}(10)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dafae4a380640ef0df2df2a956fa621134ca0aef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.243ex; height:2.843ex;" alt="{\displaystyle \log _{10}(10)=1}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{10}(100)=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>100</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{10}(100)=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9ebfab5c47d15fb80589f738fcaf8d9ee67db8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.406ex; height:2.843ex;" alt="{\displaystyle \log _{10}(100)=2}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{10}(1000000)=6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1000000</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{10}(1000000)=6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/072b8dc0d0447ed28b1fc59a2a13242203c62327" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.055ex; height:2.843ex;" alt="{\displaystyle \log _{10}(1000000)=6}"></span>, so the difference between 10 and 100 in base 10 is 1 dex and the difference between 1 and 1 000 000 is 6 dex. </p> <div class="mw-heading mw-heading2"><h2 id="Mathematics">Mathematics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=11" title="Edit section: Mathematics" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=11" title="Edit section's source code: Mathematics"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resource: <a href="/wiki/Mathematics" title="Mathematics">Mathematics</a></div> <p><b>Def.</b> an abstract representational system used in the study of <a href="https://en.wiktionary.org/wiki/number" class="extiw" title="wikt:number">numbers</a>, <a href="https://en.wiktionary.org/wiki/shape" class="extiw" title="wikt:shape">shapes</a>, <a href="https://en.wiktionary.org/wiki/structure" class="extiw" title="wikt:structure">structure</a> and <a href="https://en.wiktionary.org/wiki/change" class="extiw" title="wikt:change">change</a> and the relationships between these concepts is called <b>mathematics</b>. </p><p><b>Def.</b> the branches of mathematics used in the study of astronomy, <a href="/wiki/Astrophysics" title="Astrophysics">astrophysics</a> and cosmology is called <b>astromathematics</b>. </p> <div class="mw-heading mw-heading2"><h2 id="Changes">Changes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=12" title="Edit section: Changes" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=12" title="Edit section's source code: Changes"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resource: <a href="/w/index.php?title=Changes&action=edit&redlink=1" class="new" title="Changes (page does not exist)">Changes</a></div> <p><b>Def.</b> the act or instance of making, or the process of becoming different is called <b>change</b>. </p> <div class="mw-heading mw-heading2"><h2 id="Revolutions">Revolutions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=13" title="Edit section: Revolutions" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=13" title="Edit section's source code: Revolutions"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resources: <a href="/w/index.php?title=Changes/Revolutions&action=edit&redlink=1" class="new" title="Changes/Revolutions (page does not exist)">Changes/Revolutions</a> and <a href="/w/index.php?title=Revolutions&action=edit&redlink=1" class="new" title="Revolutions (page does not exist)">Revolutions</a></div> <p><b>Def.</b> the traversal of one body through an orbit around another body is called a <b>revolution</b>. </p> <div class="mw-heading mw-heading2"><h2 id="Operations">Operations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=14" title="Edit section: Operations" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=14" title="Edit section's source code: Operations"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resource: <a href="/w/index.php?title=Operations&action=edit&redlink=1" class="new" title="Operations (page does not exist)">Operations</a></div> <p>However, attempting to add 1 dome to 1 telescope may have little or no meaning. The operation of <b>addition</b> would be similar to the operation of <b>construction</b>. </p><p>If 1 G2V star is added to 1 M2V star the result may be a double star. The operation of addition here usually requires an explanation (a theory). Astronomers use math all the time. One way it is used is when we look at objects in the sky with a telescope. The camera that is attached to the telescope basically records a series of numbers - those numbers might correspond to how much light different objects in the sky are emitting, what type of light, etc. In order to be able to understand the information that these numbers contain, we need to use math and statistics to interpret them. </p> <div class="mw-heading mw-heading2"><h2 id="Inclinations">Inclinations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=15" title="Edit section: Inclinations" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=15" title="Edit section's source code: Inclinations"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resources: <a href="/w/index.php?title=Powers/Tendencies/Inclinations&action=edit&redlink=1" class="new" title="Powers/Tendencies/Inclinations (page does not exist)">Powers/Tendencies/Inclinations</a> and <a href="/w/index.php?title=Inclinations&action=edit&redlink=1" class="new" title="Inclinations (page does not exist)">Inclinations</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Orbit1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Orbit1.svg/200px-Orbit1.svg.png" decoding="async" width="200" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Orbit1.svg/300px-Orbit1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Orbit1.svg/400px-Orbit1.svg.png 2x" data-file-width="555" data-file-height="500" /></a><figcaption>The diagram describes the parameters associated with orbital inclination (<i>i</i>). Credit: <a href="https://en.wikipedia.org/wiki/User:Lasunncty" class="extiw" title="w:User:Lasunncty">Lasunncty</a>.</figcaption></figure> <p><b>Def.</b> the angle of intersection of a reference plane is called an <b>inclination</b>. </p><p>"The orbital inclination [<i>i</i>] [of Mercury] varies between 5° and 10° with a 10<sup>6</sup> yr period with smaller amplitude variations with a period of about 10<sup>5</sup> yr."<sup id="cite_ref-Peale_5-0" class="reference"><a href="#cite_note-Peale-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div style="clear:both;"></div> <div class="mw-heading mw-heading2"><h2 id="Dimensional_analyses">Dimensional analyses</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=16" title="Edit section: Dimensional analyses" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=16" title="Edit section's source code: Dimensional analyses"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resource: <a href="/w/index.php?title=Dimensional_analyses&action=edit&redlink=1" class="new" title="Dimensional analyses (page does not exist)">Dimensional analyses</a></div> <p><b>Def.</b> a single aspect of a given thing is called a <b>dimension</b>. </p><p>Usually, in astronomy, a number is associated with a dimension or aspect of an entity. For example, the Earth is 1.50 x 10<sup>8</sup> km on average from the Sun. Kilometer (km) is a dimension and 1.50 x 10<sup>8</sup> is a number. </p><p><b>Def.</b> the study of the dimensions of quantities; used to obtain information about large complex systems, and as a means of checking equations is called <b>dimensional analysis</b>. </p><p>Prefixed values cannot be multiplied or divided together, and they have to be converted into non-prefixed standard form for such calculations. For example, 5 mV × 5 mA ≠ 25 mW. The correct calculation is: 5 mV × 5 mA = 5 × 10<sup>−3</sup> V × 5 × 10<sup>−3</sup> A = 25 x 10<sup>−6</sup> W = 25 µW = 0.025 mW. </p><p>Prefixes corresponding to an exponent that is divisible by three are often recommended. Hence "100 m" rather than "1 hm" (hectometre) or "10 dam" (decametres). The "non-three" prefixes (hecto-, deca-, deci-, and centi-) are however more commonly used for everyday purposes than in science. </p><p>When units occur in exponentiation, for example, in square and cubic forms, any size prefix is considered part of the unit, and thus included in the exponentiation. </p> <ul><li><span style="white-space:nowrap">1<span style="margin-left:0.25em">km<sup>2</sup></span></span> means one square kilometre or the size of a square of 1000 m by 1000 m and not 1000 <a href="https://en.wikipedia.org/wiki/square_metre" class="extiw" title="w:square metre">square metres</a>.</li> <li><span style="white-space:nowrap">2<span style="margin-left:0.25em">Mm<sup>3</sup></span></span> means two cubic megametre or the size of two cubes of <span style="white-space:nowrap">1<span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">m</span></span> by <span style="white-space:nowrap">1<span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">m</span></span> by <span style="white-space:nowrap">1<span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">m</span></span> or <span style="white-space:nowrap">2<span style="margin-left:0.27em;margin-right:0.27em">×</span>10<span style="display:none">^</span><sup>18</sup> m<sup>3</sup></span>, and not <span style="white-space:nowrap">2<span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">cubic metres</span></span> (<span style="white-space:nowrap">2<span style="margin-left:0.27em;margin-right:0.27em">×</span>10<span style="display:none">^</span><sup>6</sup> m<sup>3</sup></span>).</li></ul> <dl><dt>Examples</dt></dl> <ul><li><span style="white-space:nowrap">5 cm</span> = <span style="white-space:nowrap">5<span style="margin-left:0.27em;margin-right:0.27em">×</span>10<span style="display:none">^</span><sup>−2</sup> m</span> = <span style="white-space:nowrap">5<span style="margin-left:0.25em">×</span><span style="margin-left:0.25em">0.01</span><span style="margin-left:0.25em">m</span></span> = <span style="white-space:nowrap">0.05<span style="margin-left:0.25em">m</span></span></li> <li><span style="white-space:nowrap">3 MW</span> = <span style="white-space:nowrap">3<span style="margin-left:0.27em;margin-right:0.27em">×</span>10<span style="display:none">^</span><sup>6</sup> W</span> = <span style="white-space:nowrap">3<span style="margin-left:0.25em">×</span><span style="margin-left:0.25em">1</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">W</span></span> = <span style="white-space:nowrap">3<span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">W</span></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Astronomical_units">Astronomical units</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=17" title="Edit section: Astronomical units" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=17" title="Edit section's source code: Astronomical units"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resources: <a href="/wiki/Astronomy/Units" class="mw-redirect" title="Astronomy/Units">Astronomy/Units</a> and <a href="/w/index.php?title=Astronomical_units&action=edit&redlink=1" class="new" title="Astronomical units (page does not exist)">Astronomical units</a></div> <p><b>Def.</b> "1 day (d)" is called the <b>astronomical unit of time</b>.<sup id="cite_ref-Seidelmann_6-0" class="reference"><a href="#cite_note-Seidelmann-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p><b>Def.</b> "the distance from the centre of the Sun at which a particle of negligible mass, in an unperturbed circular orbit, would have an orbital period of 365.2568983 days" is called the <b>Astronomical Unit</b> (AU).<sup id="cite_ref-Seidelmann_6-1" class="reference"><a href="#cite_note-Seidelmann-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p><b>Def.</b> "the distance at which one Astronomical Unit subtends an angle of one arcsecond" is called the <b>parsec</b> (pc).<sup id="cite_ref-Seidelmann_6-2" class="reference"><a href="#cite_note-Seidelmann-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p><b>Def.</b> "365.25 days" is called a <b>Julian year</b>.<sup id="cite_ref-Seidelmann_6-3" class="reference"><a href="#cite_note-Seidelmann-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p><b>Def.</b> "36,525 days" is called a <b>Julian century</b>.<sup id="cite_ref-Seidelmann_6-4" class="reference"><a href="#cite_note-Seidelmann-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable" style="float:center"> <caption>Units of Physics and Astronomy </caption> <tbody><tr> <th>Dimension</th> <th>Astronomy</th> <th>Symbol</th> <th>Physics</th> <th>Symbol</th> <th>Conversion </th></tr> <tr> <td>time</td> <td>1 day</td> <td>d</td> <td>1 second</td> <td>s</td> <td>1 d = 86,400 s<sup id="cite_ref-Seidelmann_6-5" class="reference"><a href="#cite_note-Seidelmann-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td>time</td> <td>1 "Julian year"<sup id="cite_ref-Wilkins_7-0" class="reference"><a href="#cite_note-Wilkins-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></td> <td>J</td> <td>1 second</td> <td>s</td> <td>1 J = 31,557,600 s </td></tr> <tr> <td>distance</td> <td>1 astronomical unit</td> <td>AU</td> <td>1 meter</td> <td>m</td> <td>1 AU = 149,597,870.691 km<sup id="cite_ref-Seidelmann_6-6" class="reference"><a href="#cite_note-Seidelmann-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td>mass</td> <td>1 Sun</td> <td>M<sub>ʘ</sub></td> <td>1 kilogram</td> <td>kg</td> <td>1 M<sub>ʘ</sub> = 1.9891 x 10<sup>30</sup> kg<sup id="cite_ref-Seidelmann_6-7" class="reference"><a href="#cite_note-Seidelmann-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td>luminosity</td> <td>1 Sun</td> <td>L<sub>ʘ</sub></td> <td>1 watt</td> <td>W</td> <td>1 L<sub>ʘ</sub> = 3.846 x 10<sup>26</sup> W<sup id="cite_ref-Williams2004_8-0" class="reference"><a href="#cite_note-Williams2004-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td>angular distance</td> <td>1 parsec</td> <td>pc</td> <td>1 meter</td> <td>m</td> <td>1 pc ~ 30.857 x 10<sup>12</sup> km<sup id="cite_ref-Seidelmann_6-8" class="reference"><a href="#cite_note-Seidelmann-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </td></tr> </tbody></table> <div class="mw-heading mw-heading2"><h2 id="Regions">Regions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=18" title="Edit section: Regions" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=18" title="Edit section's source code: Regions"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resources: <a href="/w/index.php?title=Spaces/Regions&action=edit&redlink=1" class="new" title="Spaces/Regions (page does not exist)">Spaces/Regions</a> and <a href="/w/index.php?title=Regions&action=edit&redlink=1" class="new" title="Regions (page does not exist)">Regions</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Center_of_the_Milky_Way_Galaxy_IV_%E2%80%93_Composite.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Center_of_the_Milky_Way_Galaxy_IV_%E2%80%93_Composite.jpg/250px-Center_of_the_Milky_Way_Galaxy_IV_%E2%80%93_Composite.jpg" decoding="async" width="250" height="125" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Center_of_the_Milky_Way_Galaxy_IV_%E2%80%93_Composite.jpg/375px-Center_of_the_Milky_Way_Galaxy_IV_%E2%80%93_Composite.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/00/Center_of_the_Milky_Way_Galaxy_IV_%E2%80%93_Composite.jpg/500px-Center_of_the_Milky_Way_Galaxy_IV_%E2%80%93_Composite.jpg 2x" data-file-width="9725" data-file-height="4862" /></a><figcaption>This is a composite image of the central region of our Milky Way galaxy. Credit: NASA/JPL-Caltech/ESA/CXC/STScI.</figcaption></figure> <p>A <b>region</b> is a division or part of a space having definable characteristics but not always fixed boundaries. </p><p><b>Def.</b> any <a href="https://en.wiktionary.org/wiki/considerable" class="extiw" title="wikt:considerable">considerable</a> and <a href="https://en.wiktionary.org/wiki/connected" class="extiw" title="wikt:connected">connected</a> <a href="https://en.wiktionary.org/wiki/part" class="extiw" title="wikt:part">part</a> of a <a href="https://en.wiktionary.org/wiki/space" class="extiw" title="wikt:space">space</a> or <a href="https://en.wiktionary.org/wiki/surface" class="extiw" title="wikt:surface">surface</a>; specifically, a <a href="https://en.wiktionary.org/wiki/tract" class="extiw" title="wikt:tract">tract</a> of <a href="https://en.wiktionary.org/wiki/land" class="extiw" title="wikt:land">land</a> or <a href="https://en.wiktionary.org/wiki/sea" class="extiw" title="wikt:sea">sea</a> of considerable but <a href="https://en.wiktionary.org/wiki/indefinite" class="extiw" title="wikt:indefinite">indefinite</a> <a href="https://en.wiktionary.org/wiki/extent" class="extiw" title="wikt:extent">extent</a>; a <a href="https://en.wiktionary.org/wiki/country" class="extiw" title="wikt:country">country</a>; a <a href="https://en.wiktionary.org/wiki/district" class="extiw" title="wikt:district">district</a>; in a broad sense, a place without special <a href="https://en.wiktionary.org/wiki/reference" class="extiw" title="wikt:reference">reference</a> to <a href="https://en.wiktionary.org/wiki/location" class="extiw" title="wikt:location">location</a> or extent but viewed as an <a href="https://en.wiktionary.org/wiki/entity" class="extiw" title="wikt:entity">entity</a> for <a href="https://en.wiktionary.org/wiki/geographical" class="extiw" title="wikt:geographical">geographical</a>, <a href="https://en.wiktionary.org/wiki/social" class="extiw" title="wikt:social">social</a> or <a href="https://en.wiktionary.org/wiki/cultural" class="extiw" title="wikt:cultural">cultural</a> reasons is called a <b>region</b>. </p><p><b>Region</b> is most commonly found as a term used in terrestrial and astrophysics sciences also an area, notably among the different sub-disciplines of <a href="/wiki/Geography" title="Geography">geography</a>, studied by <a href="https://en.wikipedia.org/wiki/Regional_geography" class="extiw" title="w:Regional geography">regional geographers</a>. Regions consist of <a href="https://en.wikipedia.org/wiki/subregions" class="extiw" title="w:subregions">subregions</a> that contain clusters of like areas that are distinctive by their uniformity of description based on a range of statistical data, for example demographic, and locales. </p><p><b>Def.</b> a subset of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> that is <a href="https://en.wikipedia.org/wiki/open_set" class="extiw" title="w:open set">open</a> (in the standard <a href="https://en.wikipedia.org/wiki/Euclidean_topology" class="extiw" title="w:Euclidean topology">Euclidean topology</a>), <a href="https://en.wikipedia.org/wiki/connected_set" class="extiw" title="w:connected set">connected</a> and <a href="https://en.wikipedia.org/wiki/empty_set" class="extiw" title="w:empty set">non-empty</a> is called a <b>region</b>, or <b>region of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> is the n-dimensional real number system.</b> </p> <div style="clear:both;"></div> <div class="mw-heading mw-heading2"><h2 id="Areas">Areas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=19" title="Edit section: Areas" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=19" title="Edit section's source code: Areas"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resources: <a href="/w/index.php?title=Spaces/Areas&action=edit&redlink=1" class="new" title="Spaces/Areas (page does not exist)">Spaces/Areas</a> and <a href="/wiki/Areas" class="mw-redirect" title="Areas">Areas</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:RandLintegrals.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/RandLintegrals.png/250px-RandLintegrals.png" decoding="async" width="250" height="153" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/RandLintegrals.png/375px-RandLintegrals.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1b/RandLintegrals.png/500px-RandLintegrals.png 2x" data-file-width="569" data-file-height="349" /></a><figcaption>This diagram shows an approximation to an area under a curve. Credit: <a href="https://commons.wikimedia.org/wiki/User:Dubhe" class="extiw" title="commons:User:Dubhe">Dubhe</a>.</figcaption></figure> <p>In the figure on the right, an area is the difference in the x-direction times the difference in the y-direction. </p><p>This rectangle cornered at the origin of the curvature represents an area for the curve. </p> <div style="clear:both;"></div> <div class="mw-heading mw-heading2"><h2 id="Orbits">Orbits</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=20" title="Edit section: Orbits" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=20" title="Edit section's source code: Orbits"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resources: <a href="/w/index.php?title=Spaces/Regions/Spheres/Orbits&action=edit&redlink=1" class="new" title="Spaces/Regions/Spheres/Orbits (page does not exist)">Spaces/Regions/Spheres/Orbits</a> and <a href="/w/index.php?title=Orbits&action=edit&redlink=1" class="new" title="Orbits (page does not exist)">Orbits</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Kepler_orbits.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Kepler_orbits.svg/250px-Kepler_orbits.svg.png" decoding="async" width="250" height="304" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Kepler_orbits.svg/375px-Kepler_orbits.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Kepler_orbits.svg/500px-Kepler_orbits.svg.png 2x" data-file-width="494" data-file-height="600" /></a><figcaption>A hyperbolic pass is indicated by the blue line with an eccentricity of 1.3. A parabolic pass is the green line. The elliptical orbit in red has an eccentricity (<i>e</i>) of 0.7. Credit: <a href="https://commons.wikimedia.org/wiki/User:Stamcose" class="extiw" title="commons:User:Stamcose">Stamcose</a>.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Orbit3.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/59/Orbit3.gif" decoding="async" width="200" height="200" class="mw-file-element" data-file-width="200" data-file-height="200" /></a><figcaption>Two bodies orbiting around a common barycenter (red cross) with circular orbits. Credit: <a href="https://commons.wikimedia.org/wiki/User:Zhatt" class="extiw" title="commons:User:Zhatt">Zhatt</a>.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Orbit5.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Orbit5.gif/250px-Orbit5.gif" decoding="async" width="250" height="125" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Orbit5.gif/375px-Orbit5.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/0/0e/Orbit5.gif 2x" data-file-width="400" data-file-height="200" /></a><figcaption>Two bodies orbiting around a common barycenter (red cross) with elliptic orbits. Credit: <a href="https://commons.wikimedia.org/wiki/User:Zhatt" class="extiw" title="commons:User:Zhatt">Zhatt</a>.</figcaption></figure> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:ISEE3-ICE-trajectory.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/ISEE3-ICE-trajectory.gif/250px-ISEE3-ICE-trajectory.gif" decoding="async" width="250" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/ISEE3-ICE-trajectory.gif/375px-ISEE3-ICE-trajectory.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d7/ISEE3-ICE-trajectory.gif/500px-ISEE3-ICE-trajectory.gif 2x" data-file-width="710" data-file-height="557" /></a><figcaption>ISEE-3 is inserted into a "halo" orbit on June 10, 1982. Credit: NASA.</figcaption></figure> <p><b>Def.</b> a circular or elliptical path of one object around another object is called an <b>orbit</b>. </p><p>Historically, the apparent motions of the planets were first understood geometrically (and without regard to gravity) in terms of <a href="https://en.wikipedia.org/wiki/epicycles" class="extiw" title="w:epicycles">epicycles</a>, which are the sums of numerous circular motions.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Theories of this kind predicted paths of the planets moderately well, until <a href="https://en.wikipedia.org/wiki/Johannes_Kepler" class="extiw" title="w:Johannes Kepler">Johannes Kepler</a> was able to show that the motions of planets were in fact (at least approximately) elliptical motions.<sup id="cite_ref-Caspar_10-0" class="reference"><a href="#cite_note-Caspar-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>In the <a href="https://en.wikipedia.org/wiki/geocentric_model" class="extiw" title="w:geocentric model">geocentric model</a> of the solar system, the <a href="https://en.wikipedia.org/wiki/celestial_spheres" class="extiw" title="w:celestial spheres">celestial spheres</a> model was originally used to explain the apparent motion of the planets in the sky in terms of perfect spheres or rings, but after the planets' motions were more accurately measured, theoretical mechanisms such as <a href="https://en.wikipedia.org/wiki/deferent_and_epicycle" class="extiw" title="w:deferent and epicycle">deferent and epicycles</a> were added. Although it was capable of accurately predicting the planets' position in the sky, more and more epicycles were required over time, and the model became more and more unwieldy. </p><p>In <a href="/wiki/Astronomy/Theory" class="mw-redirect" title="Astronomy/Theory">theoretical astronomy</a>, whether the Earth moves or not, serving as a fixed point with which to measure movements by objects or entities, or there is a <a href="https://en.wikipedia.org/wiki/solar_system" class="extiw" title="w:solar system">solar system</a> with the <a href="/wiki/Stars/Sun" title="Stars/Sun">Sun</a> near its center, is a matter of simplicity and calculational accuracy. Copernicus's theory provided a strikingly simple explanation for the apparent retrograde motions of the planets—namely as <a href="https://en.wikipedia.org/wiki/parallax" class="extiw" title="w:parallax">parallactic</a> displacements resulting from the Earth's motion around the Sun—an important consideration in <a href="https://en.wikipedia.org/wiki/Johannes_Kepler" class="extiw" title="w:Johannes Kepler">Johannes Kepler</a>'s conviction that the theory was substantially correct.<sup id="cite_ref-Linton_11-0" class="reference"><a href="#cite_note-Linton-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> "[Kepler] knew that the tables constructed from the heliocentric theory were more accurate than those of Ptolemy"<sup id="cite_ref-Linton_11-1" class="reference"><a href="#cite_note-Linton-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> with the Earth at the center. Using a computer, this means that for competing programs, one written for each theory, the heliocentric program finishes first (for a mutually specified high degree of accuracy). </p><p>Orbits come in many shapes and motions. The simplest forms are a circle or an ellipse. </p> <div style="clear:both;"></div> <div class="mw-heading mw-heading2"><h2 id="Infinitesimals">Infinitesimals</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=21" title="Edit section: Infinitesimals" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=21" title="Edit section's source code: Infinitesimals"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resource: <a href="/w/index.php?title=Infinitesimals&action=edit&redlink=1" class="new" title="Infinitesimals (page does not exist)">Infinitesimals</a></div> <p><b>Notation</b>: let the symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> represent an <b>infinitesimal difference in</b>. </p><p><b>Notation</b>: let the symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62b4e7c1cedb9564609aefd2aa2309972f455c24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.318ex; height:2.176ex;" alt="{\displaystyle \partial }"></span> represent an <b>infinitesimal difference in</b> one of more than one. </p> <div class="mw-heading mw-heading2"><h2 id="Distances">Distances</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=22" title="Edit section: Distances" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=22" title="Edit section's source code: Distances"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resource: <a href="/w/index.php?title=Distances&action=edit&redlink=1" class="new" title="Distances (page does not exist)">Distances</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Distancedisplacement.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Distancedisplacement.svg/250px-Distancedisplacement.svg.png" decoding="async" width="250" height="155" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Distancedisplacement.svg/375px-Distancedisplacement.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/40/Distancedisplacement.svg/500px-Distancedisplacement.svg.png 2x" data-file-width="323" data-file-height="200" /></a><figcaption>Distance along a path is compared in this diagram with displacement. Credit: .</figcaption></figure> <p><b>Def.</b> the amount of space between two points, usually geographical points, usually (but not necessarily) measured along a straight line is called a <b>distance</b>. </p><p><b>Distance</b> (or <b>farness</b>) is a numerical description of how far apart objects are. In <a href="/wiki/Physics" title="Physics">physics</a> or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria (e.g. "two counties over"). In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a distance function or <a href="https://en.wikipedia.org/wiki/Metric_(mathematics)" class="extiw" title="w:Metric (mathematics)">metric</a> is a generalization of the concept of physical distance. A metric is a function that behaves according to a specific set of rules, and provides a concrete way of describing what it means for elements of some space to be "close to" or "far away from" each other. </p><p><b>Def.</b> </p> <ol><li>a series of interconnected rings or links usually made of metal,</li> <li>a series of interconnected links of known length, used as a measuring device,</li> <li>a long measuring tape,</li> <li>a unit of length equal to 22 yards. The length of a Gunter's surveying chain. The length of a cricket pitch. Equal to 20.12 metres. Equal to 4 rods. Equal to 100 links.</li> <li>a totally ordered set, especially a totally ordered subset of a poset,</li> <li>iron links bolted to the side of a vessel to bold the dead-eyes connected with the shrouds; also, the channels, or</li> <li>the warp threads of a web</li></ol> <p>is called a <b>chain</b>. </p><p><b>Def.</b> a unit of length equal to 220 yards or exactly 201.168 meters, now only used in measuring distances in horse racing is called a <b>furlong</b>. </p><p><b>Def.</b> </p> <ol><li>a trench cut in the soil, as when plowed in order to plant a crop or</li> <li>any trench, channel, or groove, as in wood or metal</li></ol> <p>is called a <b>furrow</b>. </p><p><b>Def.</b> the distance that a person can walk in one hour, commonly taken to be approximately three English miles (about five kilometers) is called a <b>league</b>. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\odot }(equatorial)=696,342km}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊙</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>e</mi> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>t</mi> <mi>o</mi> <mi>r</mi> <mi>i</mi> <mi>a</mi> <mi>l</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>696</mn> <mo>,</mo> <mn>342</mn> <mi>k</mi> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\odot }(equatorial)=696,342km}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83d3cdfed71ecbabc658f176d41b41b8f8b042c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.897ex; height:2.843ex;" alt="{\displaystyle R_{\odot }(equatorial)=696,342km}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{J}(equatorial)=71,492km}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>e</mi> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>t</mi> <mi>o</mi> <mi>r</mi> <mi>i</mi> <mi>a</mi> <mi>l</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>71</mn> <mo>,</mo> <mn>492</mn> <mi>k</mi> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{J}(equatorial)=71,492km}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8802d0b3b5e2d1afec94142013e5c3f1bcb4edb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.496ex; height:2.843ex;" alt="{\displaystyle R_{J}(equatorial)=71,492km}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{S}(equatorial)=60,268km}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>e</mi> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>t</mi> <mi>o</mi> <mi>r</mi> <mi>i</mi> <mi>a</mi> <mi>l</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>60</mn> <mo>,</mo> <mn>268</mn> <mi>k</mi> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{S}(equatorial)=60,268km}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3df5dac6800cde6660a273bbf8a34326835c58b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.516ex; height:2.843ex;" alt="{\displaystyle R_{S}(equatorial)=60,268km}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{U}(equatorial)=25,559km}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>e</mi> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>t</mi> <mi>o</mi> <mi>r</mi> <mi>i</mi> <mi>a</mi> <mi>l</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>25</mn> <mo>,</mo> <mn>559</mn> <mi>k</mi> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{U}(equatorial)=25,559km}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7befb48588a704c5fff33fdb2cd0021e6a29db58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.716ex; height:2.843ex;" alt="{\displaystyle R_{U}(equatorial)=25,559km}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{N}(equatorial)=24,764km}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>e</mi> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>t</mi> <mi>o</mi> <mi>r</mi> <mi>i</mi> <mi>a</mi> <mi>l</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>24</mn> <mo>,</mo> <mn>764</mn> <mi>k</mi> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{N}(equatorial)=24,764km}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57a3a342a0f3387593cb2df9ff6bab4c2e58827d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.915ex; height:2.843ex;" alt="{\displaystyle R_{N}(equatorial)=24,764km}"></span></dd></dl> <p>Then, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{J}(equatorial)=F_{J}*R_{\odot }(equatorial),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>e</mi> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>t</mi> <mi>o</mi> <mi>r</mi> <mi>i</mi> <mi>a</mi> <mi>l</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </msub> <mo>∗</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊙</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>e</mi> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>t</mi> <mi>o</mi> <mi>r</mi> <mi>i</mi> <mi>a</mi> <mi>l</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{J}(equatorial)=F_{J}*R_{\odot }(equatorial),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42293a76dd1d8677797fcec138934ffeaa0616f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.545ex; height:2.843ex;" alt="{\displaystyle R_{J}(equatorial)=F_{J}*R_{\odot }(equatorial),}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{S}(equatorial)=F_{S}*R_{\odot }(equatorial),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>e</mi> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>t</mi> <mi>o</mi> <mi>r</mi> <mi>i</mi> <mi>a</mi> <mi>l</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>∗</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊙</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>e</mi> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>t</mi> <mi>o</mi> <mi>r</mi> <mi>i</mi> <mi>a</mi> <mi>l</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{S}(equatorial)=F_{S}*R_{\odot }(equatorial),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02e9c38216e76fd81550d1d029a0e30c20479d56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.585ex; height:2.843ex;" alt="{\displaystyle R_{S}(equatorial)=F_{S}*R_{\odot }(equatorial),}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{U}(equatorial)=F_{U}*R_{\odot }(equatorial),and}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>e</mi> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>t</mi> <mi>o</mi> <mi>r</mi> <mi>i</mi> <mi>a</mi> <mi>l</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo>∗</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊙</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>e</mi> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>t</mi> <mi>o</mi> <mi>r</mi> <mi>i</mi> <mi>a</mi> <mi>l</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>a</mi> <mi>n</mi> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{U}(equatorial)=F_{U}*R_{\odot }(equatorial),and}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c4017fc92a4746aa0187a37b60bc4651622dfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.213ex; height:2.843ex;" alt="{\displaystyle R_{U}(equatorial)=F_{U}*R_{\odot }(equatorial),and}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{N}(equatorial)=F_{N}*R_{\odot }(equatorial).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>e</mi> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>t</mi> <mi>o</mi> <mi>r</mi> <mi>i</mi> <mi>a</mi> <mi>l</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>∗</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊙</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>e</mi> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>t</mi> <mi>o</mi> <mi>r</mi> <mi>i</mi> <mi>a</mi> <mi>l</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{N}(equatorial)=F_{N}*R_{\odot }(equatorial).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bef84e59ac1cc455264a9ec9979767a09479ac27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.383ex; height:2.843ex;" alt="{\displaystyle R_{N}(equatorial)=F_{N}*R_{\odot }(equatorial).}"></span></dd></dl> <div style="clear:both;"></div> <div class="mw-heading mw-heading2"><h2 id="Cosmic_distance_ladders">Cosmic distance ladders</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=23" title="Edit section: Cosmic distance ladders" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=23" title="Edit section's source code: Cosmic distance ladders"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resources: <a href="/w/index.php?title=Distances/Extragalactics&action=edit&redlink=1" class="new" title="Distances/Extragalactics (page does not exist)">Distances/Extragalactics</a> and <a href="/w/index.php?title=Cosmic_distance_ladders&action=edit&redlink=1" class="new" title="Cosmic distance ladders (page does not exist)">Cosmic distance ladders</a></div> <p>The <a href="https://en.wikipedia.org/wiki/apparent_magnitude" class="extiw" title="w:apparent magnitude">apparent magnitude</a>, or the magnitude as seen by the observer, can be used to determine the distance <i>D</i> to the object in kiloparsecs (where 1 kpc equals 1000 parsecs) as follows: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{smallmatrix}5\cdot \log _{10}{\frac {D}{\mathrm {kpc} }}\ =\ m\ -\ M\ -\ 10,\end{smallmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mn>5</mn> <mo>⋅</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">c</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mi>m</mi> <mtext> </mtext> <mo>−</mo> <mtext> </mtext> <mi>M</mi> <mtext> </mtext> <mo>−</mo> <mtext> </mtext> <mn>10</mn> <mo>,</mo> </mtd> </mtr> </mtable> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{smallmatrix}5\cdot \log _{10}{\frac {D}{\mathrm {kpc} }}\ =\ m\ -\ M\ -\ 10,\end{smallmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28cb4aee74a8734d11a2e2041cfb19bdebf19d89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:21.484ex; height:3.509ex;" alt="{\displaystyle {\begin{smallmatrix}5\cdot \log _{10}{\frac {D}{\mathrm {kpc} }}\ =\ m\ -\ M\ -\ 10,\end{smallmatrix}}}"></span></dd></dl> <p>where <i>m</i> the apparent magnitude and <i>M</i> the absolute magnitude. </p> <div class="mw-heading mw-heading2"><h2 id="Diameters">Diameters</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=24" title="Edit section: Diameters" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=24" title="Edit section's source code: Diameters"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resources: <a href="/w/index.php?title=Dimensions/Breadths/Diameters&action=edit&redlink=1" class="new" title="Dimensions/Breadths/Diameters (page does not exist)">Dimensions/Breadths/Diameters</a> and <a href="/w/index.php?title=Diameters&action=edit&redlink=1" class="new" title="Diameters (page does not exist)">Diameters</a></div> <p><b>Def.</b> the length of any straight line between two points on the circumference of a circle that passes through the centre/center of the circle is called a <b>diameter</b>. </p> <div class="mw-heading mw-heading2"><h2 id="Arithmetic_dimensional_analysis">Arithmetic dimensional analysis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=25" title="Edit section: Arithmetic dimensional analysis" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=25" title="Edit section's source code: Arithmetic dimensional analysis"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Usually, pure <a href="/w/index.php?title=Arithmetic&action=edit&redlink=1" class="new" title="Arithmetic (page does not exist)">arithmetic</a> only involves numbers. But, when arithmetic is used in a science such as <a href="/w/index.php?title=Radiation_astronomy&action=edit&redlink=1" class="new" title="Radiation astronomy (page does not exist)">radiation astronomy</a>, dimensional analysis is also applicable. </p><p>To build an observatory usually requires adding components together. </p><p>For example: 1 dome + 1 telescope + 1 outbuilding + 1 control room + 1 laboratory + 1 observation room may = 1 observatory. </p><p>Yet, </p> <dl><dd>1 + 1 + 1 + 1 + 1 + 1 = 6 components in 1 simple observatory.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Obliquities">Obliquities</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=26" title="Edit section: Obliquities" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=26" title="Edit section's source code: Obliquities"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resources: <a href="/w/index.php?title=Spaces/Obliquities&action=edit&redlink=1" class="new" title="Spaces/Obliquities (page does not exist)">Spaces/Obliquities</a> and <a href="/w/index.php?title=Obliquities&action=edit&redlink=1" class="new" title="Obliquities (page does not exist)">Obliquities</a></div> <p><b>Def.</b> the quality of being oblique in direction, deviating from the horizontal or vertical; or the angle created by such a deviation is called <b>obliquity</b>. </p><p><b>Axial tilt</b> (also called <b>obliquity</b>) is the angle between an object's <a href="https://en.wikipedia.org/wiki/Axis_of_rotation" class="extiw" title="w:Axis of rotation">rotational axis</a>, and a line <a href="https://en.wikipedia.org/wiki/Perpendicular" class="extiw" title="w:Perpendicular">perpendicular</a> to its <a href="https://en.wikipedia.org/wiki/Orbital_plane_(astronomy)" class="extiw" title="w:Orbital plane (astronomy)">orbital plane</a>. The planet <a href="https://en.wikipedia.org/wiki/Venus" class="extiw" title="w:Venus">Venus</a> has an axial tilt of 177.3° because it is rotating in retrograde direction, opposite to other planets like <a href="/wiki/Earth" title="Earth">Earth</a>. The planet <a href="https://en.wikipedia.org/wiki/Uranus" class="extiw" title="w:Uranus">Uranus</a> is rotating on its side in such a way that its rotational axis, and hence its north pole, is pointed almost in the direction of its orbit around the <a href="/wiki/Stars/Sun" title="Stars/Sun">Sun</a>. Hence the axial tilt of Uranus is 97°.<sup id="cite_ref-Williams_12-0" class="reference"><a href="#cite_note-Williams-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>The obliquity of the Earth's axis has a period of about 41,000 years.<sup id="cite_ref-Hays_13-0" class="reference"><a href="#cite_note-Hays-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Inverses">Inverses</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=27" title="Edit section: Inverses" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=27" title="Edit section's source code: Inverses"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resource: <a href="/w/index.php?title=Inverses&action=edit&redlink=1" class="new" title="Inverses (page does not exist)">Inverses</a></div> <p><b>Def.</b> the set of points that map to a given point (or set of points) under a specified function is called an <b>inverse image</b>. </p><p>Under the function given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84ddac4ae10b1aa4a11741c79771a583419fb1fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.9ex; height:3.176ex;" alt="{\displaystyle f(x)=x^{2}}"></span>, the <b>inverse image</b> of 4 is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{-2,2\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{-2,2\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98c59c335b2e3313e277b47503c6478b9cf6b0c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.492ex; height:2.843ex;" alt="{\displaystyle \{-2,2\}}"></span>, as is the <b>inverse image</b> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{4\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>4</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{4\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1733e4be0b2f30707f38027dec7815ff18c7258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{4\}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Antapex">Antapex</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=28" title="Edit section: Antapex" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=28" title="Edit section's source code: Antapex"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Def.</b> the point to which the Sun appears to be moving with respect to the local stars is called the <b>solar apex</b>. </p><p>An antapex is a point that an astronomical object's total motion is directed away from. It is opposite to the apex. </p><p>The <b>local standard of rest</b> or <b>LSR</b> follows the mean motion of material in the <a href="https://en.wikipedia.org/wiki/Milky_Way" class="extiw" title="w:Milky Way">Milky Way</a> in the neighborhood of the <a href="/wiki/Stars/Sun" title="Stars/Sun">Sun</a>.<sup id="cite_ref-Shu_14-0" class="reference"><a href="#cite_note-Shu-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> The path of this material is not precisely circular.<sup id="cite_ref-Binney_15-0" class="reference"><a href="#cite_note-Binney-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> The Sun follows the <b>solar circle</b> (<a href="https://en.wikipedia.org/wiki/Ellipse#Eccentricity" class="extiw" title="w:Ellipse">eccentricity</a> <i>e</i> < 0.1 ) at a speed of about 220 km/s in a clockwise direction when viewed from the <a href="https://en.wikipedia.org/wiki/galactic_coordinates" class="extiw" title="w:galactic coordinates">galactic north pole</a> at a radius of ≈ 8 <a href="https://en.wikipedia.org/wiki/kiloparsec" class="extiw" title="w:kiloparsec">kpc</a> about the center of the galaxy near <a href="https://en.wikipedia.org/wiki/Sgr_A*" class="extiw" title="w:Sgr A*">Sgr A*</a>, and has only a slight motion, towards the <a href="https://en.wikipedia.org/wiki/Solar_apex" class="extiw" title="w:Solar apex">solar apex</a>, relative to the LSR.<sup id="cite_ref-Reid_16-0" class="reference"><a href="#cite_note-Reid-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> The Sun's <a href="https://en.wikipedia.org/wiki/peculiar_motion" class="extiw" title="w:peculiar motion">peculiar motion</a> relative to the LSR is 13.4 km/s.<sup id="cite_ref-Binney1_17-0" class="reference"><a href="#cite_note-Binney1-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Mamajek_18-0" class="reference"><a href="#cite_note-Mamajek-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> The LSR velocity is anywhere from 202–241 km/s.<sup id="cite_ref-Majewski_19-0" class="reference"><a href="#cite_note-Majewski-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Algebras">Algebras</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=29" title="Edit section: Algebras" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=29" title="Edit section's source code: Algebras"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resource: <a href="/w/index.php?title=Algebras&action=edit&redlink=1" class="new" title="Algebras (page does not exist)">Algebras</a></div> <p><b>Notation</b>: let the symbol <b>*</b> designate an as yet unspecified operation. </p><p><b>Notation</b>: let the symbol <b>R</b> designate an as yet unspecified relation. </p><p><b>Def.</b> a system for computation using letters or other symbols to represent numbers, with rules for manipulating these symbols is called an <b>algebra</b>. </p><p>Fundamentally, <a href="/wiki/Portal:Algebra" class="mw-redirect" title="Portal:Algebra">algebra</a> uses letters to represent as yet unspecified <a href="/wiki/Numbers" title="Numbers">numbers</a>. The numbers may be <a href="/wiki/Numbers/Integers" title="Numbers/Integers">integers</a>, <a href="/wiki/The_Number_System#Rational_Numbers" title="The Number System">rational numbers</a>, <a href="/wiki/The_Number_System#Irrational_Numbers" title="The Number System">irrational numbers</a>, or any <a href="/wiki/Real_Numbers" title="Real Numbers">real number</a> or <a href="/wiki/Complex_Numbers" title="Complex Numbers">complex number</a>. As an <a href="https://en.wikipedia.org/wiki/Experimentalist" class="extiw" title="w:Experimentalist">experimentalist</a>, eventually you must find a way to change unspecified numbers into specified ones. But, as a theoretician, first you are free to leave the numbers in some algebraic form, then to have your theory tested by any experimentalist you need to relate the algebraic terms of your theory to real or complex numbers. </p><p>Consider the lower case letters of the English alphabet: a and n. The statement, "a * n R an", contains the operation * (followed by) and the relation R (spells the word). </p><p>The manipulations of these symbols are performed using operations. </p><p><b>Def.</b> a <a href="https://en.wiktionary.org/wiki/procedure" class="extiw" title="wikt:procedure">procedure</a> for generating a <a href="https://en.wiktionary.org/wiki/value" class="extiw" title="wikt:value">value</a> from one or more other values (the <a href="https://en.wiktionary.org/wiki/operand" class="extiw" title="wikt:operand">operands</a>; the value for any particular [operand] is unique) is called an <b>operation</b>. </p><p><b>Notation</b>: let the symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1d4e06539576633987e902f402ed46728d573b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.355ex; height:3.843ex;" alt="{\displaystyle \sum }"></span> represent the summation of many terms. </p><p><b>Notation</b>: let the symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eed3e3db6cc2028a183af948212ed2551d25c954" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \Pi }"></span> represent the product of many terms. </p><p>The results are recorded using statements of relation. </p><p><b>Def.</b> a relation in which each element of the domain is associated with exactly one element of the codomain is called a <b>function</b>. </p> <div class="mw-heading mw-heading2"><h2 id="Geometries">Geometries</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=30" title="Edit section: Geometries" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=30" title="Edit section's source code: Geometries"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resource: <a href="/w/index.php?title=Geometries&action=edit&redlink=1" class="new" title="Geometries (page does not exist)">Geometries</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Similar-geometric-shapes.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Similar-geometric-shapes.svg/250px-Similar-geometric-shapes.svg.png" decoding="async" width="250" height="173" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Similar-geometric-shapes.svg/375px-Similar-geometric-shapes.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Similar-geometric-shapes.svg/500px-Similar-geometric-shapes.svg.png 2x" data-file-width="936" data-file-height="648" /></a><figcaption>Mathematics: Figures shown in the same color are similar. Credit: <a href="https://commons.wikimedia.org/wiki/User:Amada44" class="extiw" title="commons:User:Amada44">Amada44</a>.</figcaption></figure> <p><b>Def.</b> of geometrical figures including triangles, squares, ellipses, arcs and more complex figures, having the same shape but possibly different size, rotational orientation, and position; in particular, having corresponding angles equal and corresponding line segments proportional; such that one can be had from the other using a sequence of operations of rotation, translation and scaling is called <b>similar</b>. </p><p><b>Def.</b> a branch of mathematics that studies solutions of systems of algebraic equations using both algebra and geometry is called <b>algebraic geometry</b>. </p><p><b>Def.</b> a branch of mathematics that investigates properties of figures through the coordinates of their points is called <b>analytic geometry</b>. </p><p><b>Def.</b> a branch of mathematics that investigates those properties of figures that are invariant when projected from a point to a line or plane is called <b>projective geometry</b>. </p><p><b>Def.</b> a set along with a collection of finitary functions and relations is called a <b>structure</b>. </p><p><b>Def.</b> </p> <ol><li>a set of points with some added structure,</li> <li>distance between things,</li> <li>physical extent a range of values or locations across two or three dimensions,</li> <li>physical extent in all directions, seen as an attribute of the universe,</li> <li>a set of points, each of which is uniquely specified by a number (the dimensionality) of coordinates,</li> <li>a generalized construct or set whose members have some property in common; typically there will be a geometric metaphor allowing these members to be viewed as "points",</li> <li>a gap; an empty place,</li> <li>a (chiefly empty) area or volume with set limits or boundaries,</li></ol> <p>is called a <b>space</b>. </p><p>The universe as often perceived may be described spatially, sometimes with plane <a href="/wiki/Geometry" title="Geometry">geometry</a>, other occasions with <a href="/wiki/Ideas_in_Geometry/Spherical_Geometry" title="Ideas in Geometry/Spherical Geometry">spherical geometry</a>. </p> <div style="clear:both;"></div> <div class="mw-heading mw-heading2"><h2 id="Coordinates">Coordinates</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=31" title="Edit section: Coordinates" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=31" title="Edit section's source code: Coordinates"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resources: <a href="/w/index.php?title=Measurements/Coordinates&action=edit&redlink=1" class="new" title="Measurements/Coordinates (page does not exist)">Measurements/Coordinates</a> and <a href="/w/index.php?title=Coordinates&action=edit&redlink=1" class="new" title="Coordinates (page does not exist)">Coordinates</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Cartesian-coordinate-system-with-circle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Cartesian-coordinate-system-with-circle.svg/250px-Cartesian-coordinate-system-with-circle.svg.png" decoding="async" width="250" height="257" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Cartesian-coordinate-system-with-circle.svg/375px-Cartesian-coordinate-system-with-circle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Cartesian-coordinate-system-with-circle.svg/500px-Cartesian-coordinate-system-with-circle.svg.png 2x" data-file-width="768" data-file-height="790" /></a><figcaption>Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red. The equation of a circle is (<i>x</i> - <i>a</i>)<sup>2</sup> + (<i>y</i> - <i>b</i>)<sup>2</sup> = <i>r</i><sup>2</sup> where <i>a</i> and <i>b</i> are the coordinates of the center (<i>a</i>, <i>b</i>) and <i>r</i> is the radius. Credit: <a href="https://en.wikipedia.org/wiki/User:345Kai" class="extiw" title="w:User:345Kai">345Kai</a>.</figcaption></figure> <p>A <b>Cartesian coordinate system</b> specifies each point uniquely in a plane by a pair of numerical <b>coordinates</b>, which are the [positive and negative numbers] signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a <i>coordinate axis</i> or just <i>axis</i> of the system, and the point where they meet is its <i>origin</i>, usually at ordered pair (0,0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin. </p> <div style="clear:both;"></div> <div class="mw-heading mw-heading3"><h3 id="Triclinic_coordinates">Triclinic coordinates</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=32" title="Edit section: Triclinic coordinates" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=32" title="Edit section's source code: Triclinic coordinates"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Reseaux_3D_aP.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Reseaux_3D_aP.png/100px-Reseaux_3D_aP.png" decoding="async" width="100" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Reseaux_3D_aP.png/150px-Reseaux_3D_aP.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Reseaux_3D_aP.png/200px-Reseaux_3D_aP.png 2x" data-file-width="230" data-file-height="173" /></a><figcaption></figcaption></figure><p> A triclinic coordinate system has coordinates of different lengths (a ≠ b ≠ c) along x, y, and z axes, respectively, with interaxial angles that are not 90°. The interaxial angles α, β, and γ vary such that (α ≠ β ≠ γ). These interaxial angles are α = y⋀z, β = z⋀x, and γ = x⋀y, where the symbol "⋀" means "angle between". </p><div style="clear:both;"></div> <div class="mw-heading mw-heading3"><h3 id="Monoclinic_coordinates">Monoclinic coordinates</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=33" title="Edit section: Monoclinic coordinates" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=33" title="Edit section's source code: Monoclinic coordinates"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Monoclinic_cell.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Monoclinic_cell.svg/100px-Monoclinic_cell.svg.png" decoding="async" width="100" height="137" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Monoclinic_cell.svg/150px-Monoclinic_cell.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/Monoclinic_cell.svg/200px-Monoclinic_cell.svg.png 2x" data-file-width="512" data-file-height="701" /></a><figcaption></figcaption></figure><p> In a monoclinic coordinate system, a ≠ b ≠ c, and depending on setting α = β = 90° ≠ γ, α = γ = 90° ≠ β, α = 90° ≠ β ≠ γ, or α = β ≠ γ ≠ 90°. </p><div style="clear:both;"></div> <div class="mw-heading mw-heading3"><h3 id="Orthorhombic_coordinates">Orthorhombic coordinates</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=34" title="Edit section: Orthorhombic coordinates" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=34" title="Edit section's source code: Orthorhombic coordinates"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Reseaux_3D_oP.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Reseaux_3D_oP.png/100px-Reseaux_3D_oP.png" decoding="async" width="100" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Reseaux_3D_oP.png/150px-Reseaux_3D_oP.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Reseaux_3D_oP.png/200px-Reseaux_3D_oP.png 2x" data-file-width="242" data-file-height="181" /></a><figcaption></figcaption></figure><p> In an orthorhombic coordinate system α = β = γ = 90° and a ≠ b ≠ c. </p><div style="clear:both;"></div> <div class="mw-heading mw-heading3"><h3 id="Tetragonal_coordinates">Tetragonal coordinates</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=35" title="Edit section: Tetragonal coordinates" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=35" title="Edit section's source code: Tetragonal coordinates"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Reseaux_3D_tP-2011-03-12.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Reseaux_3D_tP-2011-03-12.png/100px-Reseaux_3D_tP-2011-03-12.png" decoding="async" width="100" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Reseaux_3D_tP-2011-03-12.png/150px-Reseaux_3D_tP-2011-03-12.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Reseaux_3D_tP-2011-03-12.png/200px-Reseaux_3D_tP-2011-03-12.png 2x" data-file-width="212" data-file-height="192" /></a><figcaption></figcaption></figure><p> A tetragonal coordinate system has α = β = γ = 90°, and a = b ≠ c. </p><div style="clear:both;"></div> <div class="mw-heading mw-heading3"><h3 id="Rhombohedral_coordinates">Rhombohedral coordinates</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=36" title="Edit section: Rhombohedral coordinates" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=36" title="Edit section's source code: Rhombohedral coordinates"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Rhombohedral.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Rhombohedral.svg/100px-Rhombohedral.svg.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Rhombohedral.svg/150px-Rhombohedral.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Rhombohedral.svg/200px-Rhombohedral.svg.png 2x" data-file-width="139" data-file-height="141" /></a><figcaption></figcaption></figure><p> A rhombohedral system has a = b = c and α = β = γ < 120°, ≠ 90°. </p><div style="clear:both;"></div> <div class="mw-heading mw-heading3"><h3 id="Hexagonal_coordinates">Hexagonal coordinates</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=37" title="Edit section: Hexagonal coordinates" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=37" title="Edit section's source code: Hexagonal coordinates"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Reseaux_3D_hP.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Reseaux_3D_hP.png/100px-Reseaux_3D_hP.png" decoding="async" width="100" height="90" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Reseaux_3D_hP.png/150px-Reseaux_3D_hP.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Reseaux_3D_hP.png/200px-Reseaux_3D_hP.png 2x" data-file-width="270" data-file-height="244" /></a><figcaption></figcaption></figure><p> A hexagonal system has a = b ≠ c and α = β = 90°, γ = 120°. </p><div style="clear:both;"></div> <div class="mw-heading mw-heading2"><h2 id="Triangles">Triangles</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=38" title="Edit section: Triangles" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=38" title="Edit section's source code: Triangles"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resources: <a href="/w/index.php?title=Spaces/Angularities/Triangles&action=edit&redlink=1" class="new" title="Spaces/Angularities/Triangles (page does not exist)">Spaces/Angularities/Triangles</a> and <a href="/w/index.php?title=Triangles&action=edit&redlink=1" class="new" title="Triangles (page does not exist)">Triangles</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Simple_triangle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Simple_triangle.svg/200px-Simple_triangle.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Simple_triangle.svg/300px-Simple_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Simple_triangle.svg/400px-Simple_triangle.svg.png 2x" data-file-width="100" data-file-height="100" /></a><figcaption>This diagram shows a regular <b>triangle</b>, the geometric shape. Credit: <a href="https://commons.wikimedia.org/wiki/User:Dbc334" class="extiw" title="commons:User:Dbc334">Dbc334</a>.</figcaption></figure> <p><b>Def.</b> a polygon with three sides and three angles is called a <b>triangle</b>. </p> <div style="clear:both;"></div> <div class="mw-heading mw-heading2"><h2 id="Curvatures">Curvatures</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=39" title="Edit section: Curvatures" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=39" title="Edit section's source code: Curvatures"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resources: <a href="/w/index.php?title=Spaces/Curvatures&action=edit&redlink=1" class="new" title="Spaces/Curvatures (page does not exist)">Spaces/Curvatures</a> and <a href="/w/index.php?title=Curvatures&action=edit&redlink=1" class="new" title="Curvatures (page does not exist)">Curvatures</a></div> <p>The graph at the top of <a href="/wiki/Astronomy/Mathematics#Areas" class="mw-redirect" title="Astronomy/Mathematics">areas</a> shows a curve or curvature. </p> <div class="mw-heading mw-heading2"><h2 id="Conic_sections">Conic sections</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=40" title="Edit section: Conic sections" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=40" title="Edit section's source code: Conic sections"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resources: <a href="/w/index.php?title=Forms/Rotundity/Conics/Sections&action=edit&redlink=1" class="new" title="Forms/Rotundity/Conics/Sections (page does not exist)">Forms/Rotundity/Conics/Sections</a> and <a href="/wiki/Conic_sections" title="Conic sections">Conic sections</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Conic_Sections.svg" class="mw-file-description"><img alt="Diagram of conic sections" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Conic_Sections.svg/250px-Conic_Sections.svg.png" decoding="async" width="250" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Conic_Sections.svg/375px-Conic_Sections.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/Conic_Sections.svg/500px-Conic_Sections.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>Conics are of three types: parabolas , ellipses, including circles, and hyperbolas. Credit: .</figcaption></figure> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Ellipse_parameters_2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Ellipse_parameters_2.svg/300px-Ellipse_parameters_2.svg.png" decoding="async" width="300" height="178" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Ellipse_parameters_2.svg/450px-Ellipse_parameters_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/76/Ellipse_parameters_2.svg/600px-Ellipse_parameters_2.svg.png 2x" data-file-width="424" data-file-height="251" /></a><figcaption>Conic parameters are shown in the case of an ellipse. Credit: .</figcaption></figure> <p><b>Def.</b> any of the four distinct shapes that are the intersections of a cone with a plane, namely the circle, ellipse, parabola and hyperbola is called a <b>conic section</b>. </p><p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>conic section</b> (or just <b>conic</b>) is a <a href="https://en.wikipedia.org/wiki/curve" class="extiw" title="w:curve">curve</a> obtained as the intersection of a <a href="https://en.wikipedia.org/wiki/cone_(geometry)" class="extiw" title="w:cone (geometry)">cone</a> (more precisely, a right circular <a href="https://en.wikipedia.org/wiki/conical_surface" class="extiw" title="w:conical surface">conical surface</a>) with a <a href="https://en.wikipedia.org/wiki/plane_(mathematics)" class="extiw" title="w:plane (mathematics)">plane</a>. </p><p>Various parameters are associated with a conic section, as shown in the following table. (For the ellipse, the table gives the case of <i>a</i>><i>b</i>, for which the major axis is horizontal; for the reverse case, interchange the symbols <i>a</i> and <i>b</i>. For the hyperbola the east-west opening case is given. In all cases, <i>a</i> and <i>b</i> are positive.) </p> <table class="wikitable"> <tbody><tr> <th>conic section </th> <th>equation </th> <th>eccentricity (<i>e</i>) </th> <th>linear eccentricity (<i>c</i>) </th> <th>semi-latus rectum (<i>ℓ</i>) </th> <th>focal parameter (<i>p</i>) </th></tr> <tr> <td><a href="https://en.wikipedia.org/wiki/circle" class="extiw" title="w:circle">circle</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}=a^{2}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}=a^{2}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e380520745a14d3cecb320b0075e4e75290de026" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.209ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}=a^{2}\,}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db4b06f9315849466a0502680377e30a9da8a1b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle 0\,}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db4b06f9315849466a0502680377e30a9da8a1b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle 0\,}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d73aa5354c24942dab5316be466465a9d171510" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.617ex; height:1.676ex;" alt="{\displaystyle a\,}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }"></span> </td></tr> <tr> <td><a href="https://en.wikipedia.org/wiki/ellipse" class="extiw" title="w:ellipse">ellipse</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d7eb067b1ac196e718e5003ed60a0ea37577483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.372ex; height:6.009ex;" alt="{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {1-{\frac {b^{2}}{a^{2}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {1-{\frac {b^{2}}{a^{2}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62a566ff429a7cfd4a4ae93b0491bc9b1e283540" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:9.447ex; height:7.509ex;" alt="{\displaystyle {\sqrt {1-{\frac {b^{2}}{a^{2}}}}}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {a^{2}-b^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {a^{2}-b^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15343dec20e53eac7195118d549040cf7dc6cb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.5ex; height:3.509ex;" alt="{\displaystyle {\sqrt {a^{2}-b^{2}}}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {b^{2}}{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>a</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {b^{2}}{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69a6ebdcebcf1718d50e2366e75c8e463841429a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.888ex; height:5.676ex;" alt="{\displaystyle {\frac {b^{2}}{a}}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {b^{2}}{\sqrt {a^{2}-b^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {b^{2}}{\sqrt {a^{2}-b^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/972b8a91b7411575da35dbf752a6d6a03cb0fb74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:10.336ex; height:7.009ex;" alt="{\displaystyle {\frac {b^{2}}{\sqrt {a^{2}-b^{2}}}}}"></span> </td></tr> <tr> <td><a href="https://en.wikipedia.org/wiki/parabola" class="extiw" title="w:parabola">parabola</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{2}=4ax\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>4</mn> <mi>a</mi> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}=4ax\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2fce9233a405ac73f53aed4004a29b84f176b92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.422ex; height:3.009ex;" alt="{\displaystyle y^{2}=4ax\,}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfd1e7984fe6e1b79a26404a8138a6c6ee41a476" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle 1\,}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d73aa5354c24942dab5316be466465a9d171510" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.617ex; height:1.676ex;" alt="{\displaystyle a\,}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2a\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>a</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2a\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/137dc8413b1aa650a8f05418298059e45bfb20b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.779ex; height:2.176ex;" alt="{\displaystyle 2a\,}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2a\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>a</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2a\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/137dc8413b1aa650a8f05418298059e45bfb20b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.779ex; height:2.176ex;" alt="{\displaystyle 2a\,}"></span> </td></tr> <tr> <td><a href="/wiki/Conic_sections" title="Conic sections">hyperbola</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a24e3784b3cc27be20faa8b06c0c64e08dcabf7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.372ex; height:6.009ex;" alt="{\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {1+{\frac {b^{2}}{a^{2}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {1+{\frac {b^{2}}{a^{2}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8859cddd06d015b4695e5c8febfaaf78ba558d88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:9.447ex; height:7.509ex;" alt="{\displaystyle {\sqrt {1+{\frac {b^{2}}{a^{2}}}}}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {a^{2}+b^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {a^{2}+b^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/460372bc2a2886a1a99b9280394eb32ec5c4fea4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.5ex; height:3.509ex;" alt="{\displaystyle {\sqrt {a^{2}+b^{2}}}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {b^{2}}{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>a</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {b^{2}}{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69a6ebdcebcf1718d50e2366e75c8e463841429a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.888ex; height:5.676ex;" alt="{\displaystyle {\frac {b^{2}}{a}}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {b^{2}}{\sqrt {a^{2}+b^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {b^{2}}{\sqrt {a^{2}+b^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dde459be2d91ef203b790d4f707ddc78ab66897e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:10.336ex; height:7.009ex;" alt="{\displaystyle {\frac {b^{2}}{\sqrt {a^{2}+b^{2}}}}}"></span> </td></tr></tbody></table> <p>The general parabola equation with a vertical axis </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ax^{2}+bx+c=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax^{2}+bx+c=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23e70cfa003f402d108ec04d97983fb62f69536e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.89ex; height:2.843ex;" alt="{\displaystyle ax^{2}+bx+c=0}"></span></dd></dl> <p>is solved in terms of the constants <i>a</i>, <i>b</i>, and <i>c</i> for x by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>b</mi> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mi>a</mi> <mi>c</mi> </msqrt> </mrow> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a9804ca8ce019507e3199ca8fced800fb5b7d7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.172ex; height:6.176ex;" alt="{\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.}"></span></dd></dl> <p>The general conic equation in a Cartesian plane (x,y) is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>B</mi> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mi>C</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>D</mi> <mi>x</mi> <mo>+</mo> <mi>E</mi> <mi>y</mi> <mo>+</mo> <mi>F</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6ea68a2e3fd5ef356e93aa3567acd751fdb1183" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:39.78ex; height:3.009ex;" alt="{\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0\,,}"></span></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (AC-B^{2}/4)F+BED/4-CD^{2}/4-AE^{2}/4\neq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mi>C</mi> <mo>−</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo stretchy="false">)</mo> <mi>F</mi> <mo>+</mo> <mi>B</mi> <mi>E</mi> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo>−</mo> <mi>C</mi> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo>−</mo> <mi>A</mi> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo>≠</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (AC-B^{2}/4)F+BED/4-CD^{2}/4-AE^{2}/4\neq 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/053debe8c6379a821da0452bd2f2f7b1c792a4f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.247ex; height:3.176ex;" alt="{\displaystyle (AC-B^{2}/4)F+BED/4-CD^{2}/4-AE^{2}/4\neq 0.}"></span></dd></dl> <p>For parabolas, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B^{2}=4AC,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>4</mn> <mi>A</mi> <mi>C</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B^{2}=4AC,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e224327ccb8ed7d02edc3e6869657badb4b864a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.235ex; height:3.009ex;" alt="{\displaystyle B^{2}=4AC,}"></span></dd></dl> <p>where <i>A</i> and <i>C</i> are not both zero. </p> <div style="clear:both;"></div> <div class="mw-heading mw-heading2"><h2 id="Variations">Variations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=41" title="Edit section: Variations" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=41" title="Edit section's source code: Variations"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resources: <a href="/w/index.php?title=Spaces/Variations&action=edit&redlink=1" class="new" title="Spaces/Variations (page does not exist)">Spaces/Variations</a> and <a href="/w/index.php?title=Variations&action=edit&redlink=1" class="new" title="Variations (page does not exist)">Variations</a></div> <p><b>Def.</b> a partial change in the form, position, state, or qualities of a thing or a related but distinct thing is called a <b>variation</b>. </p> <div class="mw-heading mw-heading2"><h2 id="Precessions">Precessions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=42" title="Edit section: Precessions" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=42" title="Edit section's source code: Precessions"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resources: <a href="/w/index.php?title=Motions/Precessions&action=edit&redlink=1" class="new" title="Motions/Precessions (page does not exist)">Motions/Precessions</a> and <a href="/w/index.php?title=Precessions&action=edit&redlink=1" class="new" title="Precessions (page does not exist)">Precessions</a></div> <p><b>Def.</b> any of several slow changes in an astronomical body's rotational or orbital parameters such as the slow gyration of the <a href="/wiki/Earth" title="Earth">Earth</a>’s axis around the pole of the ecliptic is called a <b>precession</b>. </p><p><b>Def.</b> the slow westward shift of the equinoxes along the plane of the ecliptic, resulting from precession of an object's axis of rotation, and causing the equinoxes to occur earlier each year is called the <b>precession of the equinoxes</b>. </p><p>The equinoxes of Earth precess with a period of about 21,000 years.<sup id="cite_ref-Hays_13-1" class="reference"><a href="#cite_note-Hays-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Rotations">Rotations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=43" title="Edit section: Rotations" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=43" title="Edit section's source code: Rotations"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resource: <a href="/w/index.php?title=Rotations&action=edit&redlink=1" class="new" title="Rotations (page does not exist)">Rotations</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Rotating_Sphere.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/0/02/Rotating_Sphere.gif" decoding="async" width="100" height="121" class="mw-file-element" data-file-width="100" data-file-height="121" /></a><figcaption>A <a href="https://en.wikipedia.org/wiki/polyhedron" class="extiw" title="w:polyhedron">polyhedron</a> resembling a sphere rotating around an axis. Credit: <a href="https://en.wikipedia.org/wiki/User:BorisFromStockdale" class="extiw" title="w:User:BorisFromStockdale">BorisFromStockdale</a>.</figcaption></figure> <p><b>Def.</b> the act of turning around a centre or an axis is called a <b>rotation</b>. </p><p>A <b>rotation</b> is a <a href="https://en.wikipedia.org/wiki/circular_motion" class="extiw" title="w:circular motion">circular</a> movement of an object around a <i>center</i> (or <i><a href="https://en.wikipedia.org/wiki/point_(geometry)" class="extiw" title="w:point (geometry)">point</a></i>) <i>of rotation</i>. A <a href="https://en.wikipedia.org/wiki/Three-dimensional_space" class="extiw" title="w:Three-dimensional space">three-dimensional</a> object rotates always around an imaginary <a href="https://en.wikipedia.org/wiki/Line_(geometry)" class="extiw" title="w:Line (geometry)">line</a> called a <i>rotation axis</i>. If the axis is within the body, and passes through its <a href="https://en.wikipedia.org/wiki/center_of_mass" class="extiw" title="w:center of mass">center of mass</a> the body is said to rotate upon itself, or <i><a href="https://en.wiktionary.org/wiki/spin" class="extiw" title="wikt:spin">spin</a></i>. A <b>rotation</b> about an external point, e.g. the <a href="/wiki/Earth" title="Earth">Earth</a> about the <a href="/wiki/Stars/Sun" title="Stars/Sun">Sun</a>, is called a revolution or <i><a href="https://en.wikipedia.org/wiki/orbit" class="extiw" title="w:orbit">orbital revolution</a></i>. </p><p>Axes of rotation can be multiple: </p> <ol><li>one-fold - ⨀, ⦺, ⧀</li> <li>two-fold - ⨸,</li> <li>three-fold - ▲,</li> <li>four-fold - ◈,</li> <li>five-fold - ✪, or</li> <li>six-fold - ✱.</li></ol> <p>Higher-fold axes of rotation are possible. As the number-fold of axes increases, the polyhedron approaches a circle. Or, in three dimensions, a sphere. </p> <div style="clear:both;"></div> <div class="mw-heading mw-heading2"><h2 id="Mirror_planes">Mirror planes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=44" title="Edit section: Mirror planes" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=44" title="Edit section's source code: Mirror planes"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resource: <a href="/w/index.php?title=Mirror_planes&action=edit&redlink=1" class="new" title="Mirror planes (page does not exist)">Mirror planes</a></div> <p>A mirror plane reflects on the other side the handedness that is on the initial side: </p> <ol><li>⨴ | ⨵, the plane between is the mirror so that on either side is the reflection of the other, here an axis of rotation out of the plane of the paper could place the reflection on top of the object on the other side of the mirror plane,</li> <li>∀ | ∀, here such an axis of rotation would not work,</li> <li>⊆ | ⊇, this one is like number two,</li> <li>⊕ | ⊕, here rotational symmetry is preserved, and</li> <li>⨫ | ⨬, here rotation axes exist in the plane of the paper.</li></ol> <div class="mw-heading mw-heading2"><h2 id="Resonances">Resonances</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=45" title="Edit section: Resonances" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=45" title="Edit section's source code: Resonances"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resource: <a href="/w/index.php?title=Resonances&action=edit&redlink=1" class="new" title="Resonances (page does not exist)">Resonances</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Galilean_moon_Laplace_resonance_animation.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/8/83/Galilean_moon_Laplace_resonance_animation.gif" decoding="async" width="365" height="245" class="mw-file-element" data-file-width="365" data-file-height="245" /></a><figcaption>The <a href="https://en.wikipedia.org/wiki/Laplace_resonance" class="extiw" title="w:Laplace resonance">Laplace resonances</a> of Ganymede, <a href="/w/index.php?title=Rocks/Ice_sheets/Europa&action=edit&redlink=1" class="new" title="Rocks/Ice sheets/Europa (page does not exist)">Europa</a>, and <a href="/w/index.php?title=Io&action=edit&redlink=1" class="new" title="Io (page does not exist)">Io</a> is illustrated. Credit: User:Matma Rex.</figcaption></figure> <p>An orbital resonance occurs when two orbiting bodies exert a regular, periodic gravitational influence on each other, usually due to their orbital periods being related by a ratio of two small integers. The physics principle behind orbital resonance is similar in concept to pushing a child on a swing, where the orbit and the swing both have a natural frequency, and the other body doing the "pushing" will act in periodic repetition to have a cumulative effect on the motion. Orbital resonances greatly enhance the mutual gravitational influence of the bodies, i.e., their ability to alter or constrain each other's orbits. In most cases, this results in an unstable interaction, in which the bodies exchange momentum and shift orbits until the resonance no longer exists. Under some circumstances, a resonant system can be stable and self-correcting, so that the bodies remain in resonance. Examples are the 1:2:4 resonance of <a href="/wiki/Jupiter" title="Jupiter">Jupiter</a>'s moons <a href="/w/index.php?title=Rocks/Rocky_objects/Ganymede&action=edit&redlink=1" class="new" title="Rocks/Rocky objects/Ganymede (page does not exist)">Ganymede</a>, <a href="/w/index.php?title=Rocks/Ice_sheets/Europa&action=edit&redlink=1" class="new" title="Rocks/Ice sheets/Europa (page does not exist)">Europa</a> and <a href="/w/index.php?title=Io&action=edit&redlink=1" class="new" title="Io (page does not exist)">Io</a>, and the 2:3 resonance between <a href="/w/index.php?title=Pluto&action=edit&redlink=1" class="new" title="Pluto (page does not exist)">Pluto</a> and <a href="/wiki/Neptune" class="mw-redirect" title="Neptune">Neptune</a>. Unstable resonances with <a href="/w/index.php?title=Saturn&action=edit&redlink=1" class="new" title="Saturn (page does not exist)">Saturn</a>'s inner moons give rise to gaps in the rings of Saturn. The special case of 1:1 resonance (between bodies with similar orbital radii) causes large <a href="/wiki/Solar_System" class="mw-disambig" title="Solar System">Solar System</a> bodies to eject most other bodies sharing their orbits; this is part of the much more extensive process of clearing the neighbourhood, an effect that is used in the current definition of a planet. </p> <div style="clear:both;"></div> <div class="mw-heading mw-heading2"><h2 id="Eccentricities">Eccentricities</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=46" title="Edit section: Eccentricities" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=46" title="Edit section's source code: Eccentricities"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resource: <a href="/w/index.php?title=Eccentricities&action=edit&redlink=1" class="new" title="Eccentricities (page does not exist)">Eccentricities</a></div> <p><b>Def.</b> the ratio, constant for any particular conic section, of the distance of a point from the focus to its distance from the <a href="https://en.wiktionary.org/wiki/directrix" class="extiw" title="wikt:directrix">directrix</a> is called the <b>eccentricity</b>. </p><p>For an ellipse, the eccentricity is the ratio of the distance from the center to a focus divided by the length of the semi-major axis. </p><p>"Mercury's orbit eccentricity [<i>e</i>] varies between about 0.11 and 0.24 with the shortest time lapse between the extremes being about 4 x 10<sup>5</sup> yr".<sup id="cite_ref-Peale_5-1" class="reference"><a href="#cite_note-Peale-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> "Smaller amplitude variations occur with about a 10<sup>5</sup> yr period."<sup id="cite_ref-Peale_5-2" class="reference"><a href="#cite_note-Peale-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Spherical_geometries">Spherical geometries</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=47" title="Edit section: Spherical geometries" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=47" title="Edit section's source code: Spherical geometries"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resources: <a href="/w/index.php?title=Geometries/Spheres&action=edit&redlink=1" class="new" title="Geometries/Spheres (page does not exist)">Geometries/Spheres</a> and <a href="/w/index.php?title=Spherical_geometries&action=edit&redlink=1" class="new" title="Spherical geometries (page does not exist)">Spherical geometries</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Triangles_(spherical_geometry).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Triangles_%28spherical_geometry%29.jpg/250px-Triangles_%28spherical_geometry%29.jpg" decoding="async" width="250" height="206" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Triangles_%28spherical_geometry%29.jpg/375px-Triangles_%28spherical_geometry%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/Triangles_%28spherical_geometry%29.jpg/500px-Triangles_%28spherical_geometry%29.jpg 2x" data-file-width="2489" data-file-height="2048" /></a><figcaption>On a sphere, the sum of the angles of a triangle is not equal to 180°. Credit: .</figcaption></figure> <p><b>Def.</b> the <a href="https://en.wiktionary.org/wiki/non-Euclidean_geometry" class="extiw" title="wikt:non-Euclidean geometry">non-Euclidean geometry</a> on the surface of a <a href="https://en.wiktionary.org/wiki/sphere" class="extiw" title="wikt:sphere">sphere</a> is called <b>spherical geometry</b>. </p><p><b>Spherical geometry</b> is the <a href="/wiki/Geometry" title="Geometry">geometry</a> of the two-<a href="https://en.wikipedia.org/wiki/dimension" class="extiw" title="w:dimension">dimensional</a> surface of a <a href="/w/index.php?title=Sphere&action=edit&redlink=1" class="new" title="Sphere (page does not exist)">sphere</a>. It is an example of a <a href="/wiki/Geometry" title="Geometry">geometry</a> which is not Euclidean. Two practical applications of the principles of spherical geometry are to <a href="https://en.wikipedia.org/wiki/navigation" class="extiw" title="w:navigation">navigation</a> and astronomy. </p><p>A sphere [suggested by the image of the Earth at right] is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is very nearly 180. The surface of a sphere can be represented by a collection of two dimensional maps. Therefore it is a two dimensional <a href="https://en.wikipedia.org/wiki/manifold" class="extiw" title="w:manifold">manifold</a>. </p><p>The <b>great-circle</b> or <b><a href="https://en.wikipedia.org/wiki/Great_Circle" class="extiw" title="w:Great Circle">orthodromic</a> distance</b> is the shortest <a href="https://en.wikipedia.org/wiki/distance" class="extiw" title="w:distance">distance</a> between any two <a href="https://en.wikipedia.org/wiki/Point_(geometry)" class="extiw" title="w:Point (geometry)">points</a> on the surface of a <a href="https://en.wikipedia.org/wiki/sphere" class="extiw" title="w:sphere">sphere</a> measured along a path on the surface of the sphere (as opposed to going through the sphere's interior). Because spherical geometry is different from ordinary <a href="https://en.wikipedia.org/wiki/Euclidean_geometry" class="extiw" title="w:Euclidean geometry">Euclidean geometry</a>, the equations for distance take on a different form. The distance between two points in <a href="https://en.wikipedia.org/wiki/Euclidean_space" class="extiw" title="w:Euclidean space">Euclidean space</a> is the length of a straight line from one point to the other. On the sphere, however, there are no straight lines. In <a href="https://en.wikipedia.org/wiki/non-Euclidean_geometry" class="extiw" title="w:non-Euclidean geometry">non-Euclidean geometry</a>, straight lines are replaced with <a href="https://en.wikipedia.org/wiki/geodesic" class="extiw" title="w:geodesic">geodesics</a>. Geodesics on the sphere are the <i><a href="https://en.wikipedia.org/wiki/great_circle" class="extiw" title="w:great circle">great circles</a></i> (circles on the sphere whose centers are coincident with the center of the sphere). </p><p>Through any two points on a sphere which are not directly opposite each other, there is a unique great circle. The two points separate the great circle into two arcs. The length of the shorter arc is the great-circle distance between the points. A great circle endowed with such a distance is the <a href="https://en.wikipedia.org/wiki/Riemannian_circle" class="extiw" title="w:Riemannian circle">Riemannian circle</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Logical_laws">Logical laws</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=48" title="Edit section: Logical laws" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=48" title="Edit section's source code: Logical laws"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resources: <a href="/w/index.php?title=Maxims/Axioms/Logical_laws&action=edit&redlink=1" class="new" title="Maxims/Axioms/Logical laws (page does not exist)">Maxims/Axioms/Logical laws</a> and <a href="/w/index.php?title=Logical_laws&action=edit&redlink=1" class="new" title="Logical laws (page does not exist)">Logical laws</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Kepler_laws_diagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/Kepler_laws_diagram.svg/300px-Kepler_laws_diagram.svg.png" decoding="async" width="300" height="322" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/Kepler_laws_diagram.svg/450px-Kepler_laws_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/98/Kepler_laws_diagram.svg/600px-Kepler_laws_diagram.svg.png 2x" data-file-width="400" data-file-height="429" /></a><figcaption>The diagram illustrates Kepler's three laws using two planetary orbits. Credit: <a href="https://commons.wikimedia.org/wiki/User:Hankwang" class="extiw" title="commons:User:Hankwang">Hankwang</a>.</figcaption></figure> <p>Kepler's laws of planetary motion: </p> <ol><li>The orbit of every planet is an <a href="https://en.wikipedia.org/wiki/ellipse" class="extiw" title="w:ellipse">ellipse</a> with the Sun at one of the two <a href="https://en.wikipedia.org/wiki/Focus_(geometry)" class="extiw" title="w:Focus (geometry)">foci</a>.</li> <li>A <a href="https://en.wikipedia.org/wiki/line_(geometry)" class="extiw" title="w:line (geometry)">line</a> joining a planet and the Sun sweeps out equal <a href="https://en.wikipedia.org/wiki/area" class="extiw" title="w:area">areas</a> during equal intervals of time.<sup id="cite_ref-Wolfram2nd_20-0" class="reference"><a href="#cite_note-Wolfram2nd-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup></li> <li>The <a href="https://en.wikipedia.org/wiki/square_(algebra)" class="extiw" title="w:square (algebra)">square</a> of the <a href="https://en.wikipedia.org/wiki/orbital_period" class="extiw" title="w:orbital period">orbital period</a> of a planet is directly <a href="https://en.wikipedia.org/wiki/Proportionality_(mathematics)" class="extiw" title="w:Proportionality (mathematics)">proportional</a> to the <a href="https://en.wikipedia.org/wiki/cube_(arithmetic)" class="extiw" title="w:cube (arithmetic)">cube</a> of the <a href="https://en.wikipedia.org/wiki/semi-major_axis" class="extiw" title="w:semi-major axis">semi-major axis</a> of its orbit.</li></ol> <p>The diagram at the right illustrates Kepler's three laws of planetary orbits: (1) The orbits are ellipses, with focal points <i>ƒ</i><sub>1</sub> and <i>ƒ</i><sub>2</sub> for the first planet and <i>ƒ</i><sub>1</sub> and <i>ƒ</i><sub>3</sub> for the second planet. The Sun is placed in focal point <i>ƒ</i><sub>1</sub>. (2) The two shaded sectors <i>A</i><sub>1</sub> and <i>A</i><sub>2</sub> have the same surface area and the time for planet 1 to cover segment <i>A</i><sub>1</sub> is equal to the time to cover segment <i>A</i><sub>2</sub>. (3) The total orbit times for planet 1 and planet 2 have a ratio <i>a</i><sub>1</sub><sup>3/2</sup> : <i>a</i><sub>2</sub><sup>3/2</sup>. </p><p>The simplest description of the paths astronomical objects may take when passing each other includes a hyperbolic and parabolic pass. When capture occurs it usually produces an elliptical orbit. </p> <div class="mw-heading mw-heading2"><h2 id="Horizontal_coordinate_system">Horizontal coordinate system</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=49" title="Edit section: Horizontal coordinate system" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=49" title="Edit section's source code: Horizontal coordinate system"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resources: <a href="/w/index.php?title=Coordinates/Horizontals&action=edit&redlink=1" class="new" title="Coordinates/Horizontals (page does not exist)">Coordinates/Horizontals</a> and <a href="/w/index.php?title=Horizontal_coordinate_systems&action=edit&redlink=1" class="new" title="Horizontal coordinate systems (page does not exist)">Horizontal coordinate systems</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Horizontal_coordinate_system_2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Horizontal_coordinate_system_2.svg/250px-Horizontal_coordinate_system_2.svg.png" decoding="async" width="250" height="175" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Horizontal_coordinate_system_2.svg/375px-Horizontal_coordinate_system_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Horizontal_coordinate_system_2.svg/500px-Horizontal_coordinate_system_2.svg.png 2x" data-file-width="590" data-file-height="412" /></a><figcaption>This diagram describes altitude and azimuth. Credit: Francisco Javier Blanco González.</figcaption></figure> <p>The altitude of an entity in the sky is given by the angle of the arc from the local horizon to the entity. </p><p>The horizontal coordinate system is a <a href="https://en.wikipedia.org/wiki/celestial_coordinate_system" class="extiw" title="w:celestial coordinate system">celestial coordinate system</a> that uses the observer's local <a href="https://en.wikipedia.org/wiki/horizon" class="extiw" title="w:horizon">horizon</a> as the <a href="https://en.wikipedia.org/wiki/Fundamental_plane_(spherical_coordinates)" class="extiw" title="w:Fundamental plane (spherical coordinates)">fundamental plane</a>. This coordinate system divides the sky into the upper <a href="https://en.wikipedia.org/wiki/sphere" class="extiw" title="w:sphere">hemisphere</a> where objects are visible, and the lower hemisphere where objects cannot be seen since the earth is in the way. The <a href="https://en.wikipedia.org/wiki/Great_circle" class="extiw" title="w:Great circle">great circle</a> separating hemispheres [is] called [the] celestial horizon or rational horizon. The pole of the upper hemisphere is called the <a href="https://en.wikipedia.org/wiki/Zenith" class="extiw" title="w:Zenith">zenith</a>. The pole of the lower hemisphere is called the <a href="https://en.wikipedia.org/wiki/Nadir" class="extiw" title="w:Nadir">nadir</a>.<sup id="cite_ref-Schombert_21-0" class="reference"><a href="#cite_note-Schombert-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>The horizontal coordinates are: </p> <ul><li><b>Altitude (Alt)</b>, sometimes referred to as <a href="https://en.wikipedia.org/wiki/elevation_(disambiguation)" class="extiw" title="w:elevation (disambiguation)"> elevation</a>, is the angle between the object and the observer's local horizon. It is expressed as an angle between 0 degrees to 90 degrees.</li> <li><b><a href="https://en.wikipedia.org/wiki/Azimuth" class="extiw" title="w:Azimuth">Azimuth</a> (Az)</b>, that is the angle of the object around the horizon, usually measured from the north increasing towards the east.</li> <li><b>Zenith distance</b>, the distance from directly overhead (i.e. the zenith) is sometimes used instead of altitude in some calculations using these coordinates. The zenith distance is the <a href="https://en.wikipedia.org/wiki/complementary_angles" class="extiw" title="w:complementary angles">complement</a> of altitude (i.e. 90°-altitude).</li></ul> <div class="mw-heading mw-heading2"><h2 id="Fixed_point_in_the_sky">Fixed point in the sky</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=50" title="Edit section: Fixed point in the sky" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=50" title="Edit section's source code: Fixed point in the sky"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:EquatorialDecRA.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/EquatorialDecRA.png/100px-EquatorialDecRA.png" decoding="async" width="100" height="117" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/EquatorialDecRA.png/150px-EquatorialDecRA.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/EquatorialDecRA.png/200px-EquatorialDecRA.png 2x" data-file-width="458" data-file-height="538" /></a><figcaption>By choosing an equal day/night position among the fixed objects in the night sky, the observer can measure <a href="https://en.wikipedia.org/wiki/equatorial_coordinates" class="extiw" title="w:equatorial coordinates">equatorial coordinates</a>: <a href="https://en.wikipedia.org/wiki/Declination" class="extiw" title="w:Declination">declination</a> (Dec) and <a href="https://en.wikipedia.org/wiki/Right_ascension" class="extiw" title="w:Right ascension">right ascension</a> (RA). Credit: .</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:AxialTiltObliquity.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/AxialTiltObliquity.png/250px-AxialTiltObliquity.png" decoding="async" width="250" height="194" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/AxialTiltObliquity.png/375px-AxialTiltObliquity.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/61/AxialTiltObliquity.png/500px-AxialTiltObliquity.png 2x" data-file-width="760" data-file-height="590" /></a><figcaption>Earth is shown as viewed from the Sun; the orbit direction is counter-clockwise (to the left). Description of the relations between axial tilt (or obliquity), rotation axis, plane of orbit, celestial equator and ecliptic. Credit: .</figcaption></figure> <p>The observations require precise <a href="https://en.wikipedia.org/wiki/measurement" class="extiw" title="w:measurement">measurement</a> and adaptations to the movements of the Earth, especially when and where, for a time, an object or entity is available. </p><p>With the creation of a geographical grid, an observer needs to be able to fix a point in the sky. From many observations within a period of stability, an observer notices that patterns of visual objects or entities in the night sky repeat. Further, a choice is available: is the Earth moving or are the star patterns moving? Depending on latitude, the observer may have noticed that the days vary in length and the pattern of variation repeats after some number of days and nights. By choosing an equal day/night position among the fixed objects in the night sky, the observer can measure <a href="https://en.wikipedia.org/wiki/equatorial_coordinates" class="extiw" title="w:equatorial coordinates">equatorial coordinates</a>: <a href="https://en.wikipedia.org/wiki/Declination" class="extiw" title="w:Declination">declination</a> (Dec) and <a href="https://en.wikipedia.org/wiki/Right_ascension" class="extiw" title="w:Right ascension">right ascension</a> (RA). </p><p>Once these can be determined, the apparent absolute positions of objects or entities are available in a communicable form. The repeat pattern of (day/night)s allows the observer to calculate the RA and Dec at any point during the cycle for a new object, or approximations are made using RA and Dec for recognized objects. </p><p>Independent of the choice made (Earth moves or not), the pattern of objects is the same for days or nights of the repeating length once a year. The <b><a href="https://en.wikipedia.org/wiki/Equinox" class="extiw" title="w:Equinox">vernal equinox</a></b> is a day/night of equal length and the same pattern of objects in the night sky. The <b>autumnal equinox</b> is the other equal length day/night with its own pattern of objects in the night sky. </p><p>The projection of the Earth's equator and poles of rotation, or if the observer hasn't concluded as yet that it's the Earth that's rotating, the circulating pattern of stars in ever smaller circles heading in specific directions, is the celestial sphere. </p> <div style="clear:both;"></div> <div class="mw-heading mw-heading2"><h2 id="Trigonometries">Trigonometries</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=51" title="Edit section: Trigonometries" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=51" title="Edit section's source code: Trigonometries"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resource: <a href="/w/index.php?title=Trigonometries&action=edit&redlink=1" class="new" title="Trigonometries (page does not exist)">Trigonometries</a></div> <p><b>Def.</b> the relationships between the <a href="https://en.wiktionary.org/wiki/side" class="extiw" title="wikt:side">sides</a> and the <a href="https://en.wiktionary.org/wiki/angle" class="extiw" title="wikt:angle">angles</a> of <a href="https://en.wiktionary.org/wiki/triangle" class="extiw" title="wikt:triangle">triangles</a> and the <a href="https://en.wiktionary.org/wiki/calculation" class="extiw" title="wikt:calculation">calculations</a> based on them is called <b>trigonometry</b>. </p><p><b>Trigonometry</b> ... studies <a href="https://en.wikipedia.org/wiki/triangle" class="extiw" title="w:triangle">triangles</a> and the relationships between their sides and the angles between these sides. Trigonometry defines the <a href="https://en.wikipedia.org/wiki/trigonometric_functions" class="extiw" title="w:trigonometric functions">trigonometric functions</a>, which describe those relationships and have applicability to cyclical phenomena, such as waves. </p> <div class="mw-heading mw-heading2"><h2 id="Angular_displacement">Angular displacement</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=52" title="Edit section: Angular displacement" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=52" title="Edit section's source code: Angular displacement"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For the speeds in units of <i>c</i>, <i>β</i> = <i>v</i>/<i>c</i>, "[i]n the usual interpretation of superluminal motion, the apparent velocity is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{app}={\beta _{jet}\sin \phi \over 1-\beta _{jet}\cos \phi },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>p</mi> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>e</mi> <mi>t</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>e</mi> <mi>t</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{app}={\beta _{jet}\sin \phi \over 1-\beta _{jet}\cos \phi },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/514d8367a96cf8792453bc5fe9e678d126c314b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.512ex; height:6.509ex;" alt="{\displaystyle \beta _{app}={\beta _{jet}\sin \phi \over 1-\beta _{jet}\cos \phi },}"></span></dd></dl> <p>where <i>β</i><sub>jet</sub><i>c</i> is the jet velocity, and the jet makes an angle <i>Φ</i> to the line of sight."<sup id="cite_ref-Gabuzda_22-0" class="reference"><a href="#cite_note-Gabuzda-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Radius_of_the_Earth">Radius of the Earth</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=53" title="Edit section: Radius of the Earth" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=53" title="Edit section's source code: Radius of the Earth"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Because the <a href="/wiki/Earth" title="Earth">Earth</a> is not perfectly spherical, no single value serves as its natural <a href="https://en.wiktionary.org/wiki/radius" class="extiw" title="wikt:radius">radius</a>. <i>Earth radius</i> is used as a unit of distance, especially in astronomy and <a href="/w/index.php?title=Geology&action=edit&redlink=1" class="new" title="Geology (page does not exist)">geology</a>. Any radius a distance from a point on the surface to the center falls between the polar minimum of about 6,357 km and the equatorial maximum of about 6,378 km (≈3,950 – 3,963 mi). </p><p>Equations for great-circle distance can be used to roughly calculate the shortest distance between points on the surface of the Earth (<i>as the crow flies</i>), and so have applications in <a href="https://en.wikipedia.org/wiki/navigation" class="extiw" title="w:navigation">navigation</a>. </p><p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{s},\lambda _{s};\ \phi _{f},\lambda _{f}\;\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>;</mo> <mtext> </mtext> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mspace width="thickmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{s},\lambda _{s};\ \phi _{f},\lambda _{f}\;\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bb5db7537186e1f6dfc268b6a4b746116d02292" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-right: -0.387ex; width:14.088ex; height:2.843ex;" alt="{\displaystyle \phi _{s},\lambda _{s};\ \phi _{f},\lambda _{f}\;\!}"></span> be the geographical <a href="https://en.wikipedia.org/wiki/latitude" class="extiw" title="w:latitude">latitude</a> and <a href="https://en.wikipedia.org/wiki/longitude" class="extiw" title="w:longitude">longitude</a> of two points (a base "standpoint" and the destination "forepoint"), respectively, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \phi ,\Delta \lambda \;\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>ϕ</mi> <mo>,</mo> <mi mathvariant="normal">Δ</mi> <mi>λ</mi> <mspace width="thickmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \phi ,\Delta \lambda \;\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e95dac011503403b232574195bd8bc308d7351d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:8.292ex; height:2.509ex;" alt="{\displaystyle \Delta \phi ,\Delta \lambda \;\!}"></span> their absolute differences; then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta {\widehat {\sigma }}\;\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta {\widehat {\sigma }}\;\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e094e03934ef4fce492d6450ff71d5b105576da3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:3.911ex; height:2.343ex;" alt="{\displaystyle \Delta {\widehat {\sigma }}\;\!}"></span>, the <a href="https://en.wikipedia.org/wiki/central_angle" class="extiw" title="w:central angle">central angle</a> between them, is given by the <a href="https://en.wikipedia.org/wiki/spherical_law_of_cosines" class="extiw" title="w:spherical law of cosines">spherical law of cosines</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\color {white}{\Big |}}\Delta {\widehat {\sigma }}=\arccos {\big (}\sin \phi _{s}\sin \phi _{f}+\cos \phi _{s}\cos \phi _{f}\cos \Delta \lambda {\big )}.\;\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="white"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">|</mo> </mrow> </mrow> </mstyle> </mrow> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>arccos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">Δ</mi> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>.</mo> <mspace width="thickmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\color {white}{\Big |}}\Delta {\widehat {\sigma }}=\arccos {\big (}\sin \phi _{s}\sin \phi _{f}+\cos \phi _{s}\cos \phi _{f}\cos \Delta \lambda {\big )}.\;\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/659f673cc78ae3ca131ced79500c5907b2ced035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; margin-right: -0.387ex; width:51.518ex; height:4.176ex;" alt="{\displaystyle {\color {white}{\Big |}}\Delta {\widehat {\sigma }}=\arccos {\big (}\sin \phi _{s}\sin \phi _{f}+\cos \phi _{s}\cos \phi _{f}\cos \Delta \lambda {\big )}.\;\!}"></span></dd></dl> <p>The distance <i>d</i>, i.e. the <a href="https://en.wikipedia.org/wiki/arc_length" class="extiw" title="w:arc length">arc length</a>, for a sphere of radius <i>r</i> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta {\widehat {\sigma }}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta {\widehat {\sigma }}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/183c26b79e73268a43f19ca1287a578c56b93d2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.389ex; width:3.267ex; height:2.343ex;" alt="{\displaystyle \Delta {\widehat {\sigma }}\!}"></span> given in </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=r\,\Delta {\widehat {\sigma }}.\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mi>r</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>.</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d=r\,\Delta {\widehat {\sigma }}.\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa0661b0420275e570db90b059bd78171d492ccd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:10.05ex; height:2.343ex;" alt="{\displaystyle d=r\,\Delta {\widehat {\sigma }}.\,\!}"></span></dd></dl> <p>This arccosine formula above can have large <a href="https://en.wikipedia.org/wiki/rounding_error" class="extiw" title="w:rounding error">rounding errors</a> if the distance is small (if the two points are a kilometer apart the cosine of the central angle comes out 0.99999999). An equation known as the <a href="https://en.wikipedia.org/wiki/haversine_formula" class="extiw" title="w:haversine formula">haversine formula</a> is <a href="https://en.wikipedia.org/wiki/Condition_number" class="extiw" title="w:Condition number">numerically better-conditioned</a> for small distances:<sup id="cite_ref-Sinnott_23-0" class="reference"><a href="#cite_note-Sinnott-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>A formula that is accurate for all distances is the following special case (a sphere, which is an ellipsoid with equal major and minor axes) of the <a href="https://en.wikipedia.org/wiki/Vincenty%27s_formulae" class="extiw" title="w:Vincenty's formulae">Vincenty formula</a> (which more generally is a method to compute distances on ellipsoids):<sup id="cite_ref-Vincenty_24-0" class="reference"><a href="#cite_note-Vincenty-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\color {white}{\frac {\bigg |}{|}}|}\Delta {\widehat {\sigma }}=\arctan \left({\frac {\sqrt {\left(\cos \phi _{f}\sin \Delta \lambda \right)^{2}+\left(\cos \phi _{s}\sin \phi _{f}-\sin \phi _{s}\cos \phi _{f}\cos \Delta \lambda \right)^{2}}}{\sin \phi _{s}\sin \phi _{f}+\cos \phi _{s}\cos \phi _{f}\cos \Delta \lambda }}\right).\;\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="white"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <msup> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mi mathvariant="normal">Δ</mi> <mi>λ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">Δ</mi> <mi>λ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> <mrow> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">Δ</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> <mspace width="thickmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\color {white}{\frac {\bigg |}{|}}|}\Delta {\widehat {\sigma }}=\arctan \left({\frac {\sqrt {\left(\cos \phi _{f}\sin \Delta \lambda \right)^{2}+\left(\cos \phi _{s}\sin \phi _{f}-\sin \phi _{s}\cos \phi _{f}\cos \Delta \lambda \right)^{2}}}{\sin \phi _{s}\sin \phi _{f}+\cos \phi _{s}\cos \phi _{f}\cos \Delta \lambda }}\right).\;\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0202001e058fab43c1e6843ad528865415a3c295" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; margin-right: -0.387ex; width:79.081ex; height:10.843ex;" alt="{\displaystyle {\color {white}{\frac {\bigg |}{|}}|}\Delta {\widehat {\sigma }}=\arctan \left({\frac {\sqrt {\left(\cos \phi _{f}\sin \Delta \lambda \right)^{2}+\left(\cos \phi _{s}\sin \phi _{f}-\sin \phi _{s}\cos \phi _{f}\cos \Delta \lambda \right)^{2}}}{\sin \phi _{s}\sin \phi _{f}+\cos \phi _{s}\cos \phi _{f}\cos \Delta \lambda }}\right).\;\!}"></span></dd></dl> <p>When programming a computer, one should use the <code><a href="https://en.wikipedia.org/wiki/atan2" class="extiw" title="w:atan2">atan2</a>()</code> function rather than the ordinary arctangent function (<code>atan()</code>), in order to simplify handling of the case where the denominator is zero, and to compute <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta {\widehat {\sigma }}\;\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta {\widehat {\sigma }}\;\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e094e03934ef4fce492d6450ff71d5b105576da3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:3.911ex; height:2.343ex;" alt="{\displaystyle \Delta {\widehat {\sigma }}\;\!}"></span> unambiguously in all quadrants. Also, make sure that all latitudes and longitudes are in radians (rather than degrees) if that is what your programming language's sin(), cos() and atan2() functions expect (1 radian = 180 / π degrees, 1 degree = π / 180 radians). </p> <div class="mw-heading mw-heading2"><h2 id="Distance_computation">Distance computation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=54" title="Edit section: Distance computation" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=54" title="Edit section's source code: Distance computation"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Stellarparallax2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Stellarparallax2.svg/125px-Stellarparallax2.svg.png" decoding="async" width="125" height="255" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Stellarparallax2.svg/188px-Stellarparallax2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Stellarparallax2.svg/250px-Stellarparallax2.svg.png 2x" data-file-width="385" data-file-height="784" /></a><figcaption>The diagram describes <a href="https://en.wikipedia.org/wiki/Stellar_parallax" class="extiw" title="w:Stellar parallax">Stellar parallax</a> motion.</figcaption></figure> <p>Distance measurement by parallax is a special case of the principle of <a href="https://en.wikipedia.org/wiki/triangulation" class="extiw" title="w:triangulation">triangulation</a>, which states that one can solve for all the sides and angles in a network of triangles if, in addition to all the angles in the network, the length of at least one side has been measured. Thus, the careful measurement of the length of one baseline can fix the scale of an entire triangulation network. In parallax, the triangle is extremely long and narrow, and by measuring both its shortest side (the motion of the observer) and the small top angle (always less than 1 <a href="https://en.wikipedia.org/wiki/arcsecond" class="extiw" title="w:arcsecond">arcsecond</a>,<sup id="cite_ref-ZG44_25-0" class="reference"><a href="#cite_note-ZG44-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> leaving the other two close to 90 degrees), the length of the long sides (in practice considered to be equal) can be determined. </p><p>Assuming the angle is small (see <a href="https://en.wikipedia.org/wiki/Parallax#Derivation" class="extiw" title="w:Parallax">derivation</a> below), the distance to an object (measured in <a href="https://en.wikipedia.org/wiki/parsec" class="extiw" title="w:parsec">parsecs</a>) is the <a href="https://en.wikipedia.org/wiki/Reciprocal_(mathematics)" class="extiw" title="w:Reciprocal (mathematics)">reciprocal</a> of the parallax (measured in <a href="https://en.wikipedia.org/wiki/arcsecond" class="extiw" title="w:arcsecond">arcseconds</a>): <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(\mathrm {pc} )=1/p(\mathrm {arcsec} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">c</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(\mathrm {pc} )=1/p(\mathrm {arcsec} ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9668dddf26ddb56d02b3fbb01608df8db720c081" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.487ex; height:2.843ex;" alt="{\displaystyle d(\mathrm {pc} )=1/p(\mathrm {arcsec} ).}"></span> For example, the distance to <a href="https://en.wikipedia.org/wiki/Proxima_Centauri" class="extiw" title="w:Proxima Centauri">Proxima Centauri</a> is 1/0.7687=1.3009 parsecs (4.243 ly).<sup id="cite_ref-apj118_26-0" class="reference"><a href="#cite_note-apj118-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p> <div style="clear:both;"></div> <div class="mw-heading mw-heading2"><h2 id="Distance_to_the_Moon">Distance to the Moon</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=55" title="Edit section: Distance to the Moon" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=55" title="Edit section's source code: Distance to the Moon"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Any distance to the Moon is often initially calculated as a multiple of the Earth radius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\oplus }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊕</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\oplus }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98d8c196ed57b7c943fae8462bfc13c718e978ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.275ex; height:2.509ex;" alt="{\displaystyle R_{\oplus }}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Parallaxes">Parallaxes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=56" title="Edit section: Parallaxes" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=56" title="Edit section's source code: Parallaxes"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Parallax</b> is a displacement or difference in the <a href="https://en.wikipedia.org/wiki/apparent_position" class="extiw" title="w:apparent position">apparent position</a> of an object viewed along two different lines of sight, and is measured by the angle or semi-angle of inclination between those two lines."<sup id="cite_ref-Shorter_27-0" class="reference"><a href="#cite_note-Shorter-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup>"<i>Astron.</i> Apparent displacement, or difference in the apparent position, of an object, caused by actual change (or difference) of position of the point of observation; spec. the angular amount of such displacement or difference of position, being the angle contained between the two straight lines drawn to the object from the two different points of view, and constituting a measure of the distance of the object."<sup id="cite_ref-Oxford_28-0" class="reference"><a href="#cite_note-Oxford-28"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p><p>Nearby objects have a larger parallax than more distant objects when observed from different positions, so parallax can be used to determine distances. </p><p>Astronomers use the principle of parallax to measure distances to celestial objects including to the <a href="/wiki/Moon" title="Moon">Moon</a>, the <a href="/wiki/Sun" class="mw-redirect" title="Sun">Sun</a>, and to <a href="https://en.wikipedia.org/wiki/star" class="extiw" title="w:star">stars</a> beyond the <a href="/wiki/Solar_System" class="mw-disambig" title="Solar System">Solar System</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Diurnal_parallax">Diurnal parallax</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=57" title="Edit section: Diurnal parallax" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=57" title="Edit section's source code: Diurnal parallax"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Diurnal parallax</i> is a parallax that varies with rotation of the Earth or with difference of location on the Earth. The Moon and to a smaller extent the <a href="https://en.wikipedia.org/wiki/terrestrial_planet" class="extiw" title="w:terrestrial planet">terrestrial planets</a> or <a href="https://en.wikipedia.org/wiki/asteroid" class="extiw" title="w:asteroid">asteroids</a> seen from different viewing positions on the Earth (at one given moment) can appear differently placed against the background of fixed stars."<sup id="cite_ref-Seidelmann2005_29-0" class="reference"><a href="#cite_note-Seidelmann2005-29"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Barbieri_30-0" class="reference"><a href="#cite_note-Barbieri-30"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Lunar_parallax">Lunar parallax</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=58" title="Edit section: Lunar parallax" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=58" title="Edit section's source code: Lunar parallax"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Lunaparallax.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Lunaparallax.png/250px-Lunaparallax.png" decoding="async" width="250" height="302" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Lunaparallax.png/375px-Lunaparallax.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Lunaparallax.png/500px-Lunaparallax.png 2x" data-file-width="1582" data-file-height="1913" /></a><figcaption>Diagram of daily lunar parallax. Credit: .</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Lunarparallax_22_3_1988.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Lunarparallax_22_3_1988.png/250px-Lunarparallax_22_3_1988.png" decoding="async" width="250" height="229" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Lunarparallax_22_3_1988.png/375px-Lunarparallax_22_3_1988.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/13/Lunarparallax_22_3_1988.png/500px-Lunarparallax_22_3_1988.png 2x" data-file-width="554" data-file-height="508" /></a><figcaption>Example of lunar parallax: Occultation of Pleiades by the Moon. Credit: .</figcaption></figure> <p><i>Lunar parallax</i> (often short for <i>lunar horizontal parallax</i> or <i>lunar equatorial horizontal parallax</i>), is a special case of (diurnal) parallax: the Moon, being the nearest celestial body, has by far the largest maximum parallax of any celestial body, it can exceed 1 degree.<sup id="cite_ref-aa1981_31-0" class="reference"><a href="#cite_note-aa1981-31"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p><p>The diagram (above) for stellar parallax can illustrate lunar parallax as well, if the diagram is taken to be scaled right down and slightly modified. Instead of 'near star', read 'Moon', and instead of taking the circle at the bottom of the diagram to represent the size of the Earth's orbit around the Sun, take it to be the size of the Earth's globe, and of a circle around the Earth's surface. Then, the lunar (horizontal) parallax amounts to the difference in angular position, relative to the background of distant stars, of the Moon as seen from two different viewing positions on the Earth:- one of the viewing positions is the place from which the Moon can be seen directly overhead at a given moment (that is, viewed along the vertical line in the diagram); and the other viewing position is a place from which the Moon can be seen on the horizon at the same moment (that is, viewed along one of the diagonal lines, from an Earth-surface position corresponding roughly to one of the blue dots on the modified diagram). </p><p>The lunar (horizontal) parallax can alternatively be defined as the angle subtended at the distance of the Moon by the radius of the Earth<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> -- equal to angle p in the diagram when scaled-down and modified as mentioned above. </p><p>The lunar horizontal parallax at any time depends on the linear distance of the Moon from the Earth. The Earth-Moon linear distance varies continuously as the Moon follows its <a href="https://en.wikipedia.org/wiki/orbit_of_the_moon" class="extiw" title="w:orbit of the moon">perturbed and approximately elliptical orbit</a> around the Earth. The range of the variation in linear distance is from about 56 to 63.7 earth-radii, corresponding to horizontal parallax of about a degree of arc, but ranging from about 61.4' to about 54'.<sup id="cite_ref-aa1981_31-1" class="reference"><a href="#cite_note-aa1981-31"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> The <a href="https://en.wikipedia.org/wiki/Astronomical_Almanac" class="extiw" title="w:Astronomical Almanac">Astronomical Almanac</a> and similar publications tabulate the lunar horizontal parallax and/or the linear distance of the Moon from the Earth on a periodical e.g. daily basis for the convenience of astronomers (and formerly, of navigators), and the study of the way in which this coordinate varies with time forms part of <a href="https://en.wikipedia.org/wiki/lunar_theory" class="extiw" title="w:lunar theory">lunar theory</a>. </p><p>Parallax can also be used to determine the distance to the <a href="/wiki/Moon" title="Moon">Moon</a>. </p><p>One way to determine the lunar parallax from one location is by using a lunar eclipse. A full shadow of the Earth on the Moon has an apparent radius of curvature equal to the difference between the apparent radii of the Earth and the Sun as seen from the Moon. This radius can be seen to be equal to 0.75 degree, from which (with the solar apparent radius 0.25 degree) we get an Earth apparent radius of 1 degree. This yields for the Earth-Moon distance 60 Earth radii or 384,000 km. This procedure was first used by <a href="https://en.wikipedia.org/wiki/Aristarchus_of_Samos" class="extiw" title="w:Aristarchus of Samos">Aristarchus of Samos</a><sup id="cite_ref-Gutzwiller_33-0" class="reference"><a href="#cite_note-Gutzwiller-33"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> and <a href="https://en.wikipedia.org/wiki/Hipparchus" class="extiw" title="w:Hipparchus">Hipparchus</a>, and later found its way into the work of <a href="https://en.wikipedia.org/wiki/Ptolemy" class="extiw" title="w:Ptolemy">Ptolemy</a>. The diagram at right shows how daily lunar parallax arises on the geocentric and geostatic planetary model in which the Earth is at the centre of the planetary system and does not rotate. It also illustrates the important point that parallax need not be caused by any motion of the observer, contrary to some definitions of parallax that say it is, but may arise purely from motion of the observed. </p><p>Another method is to take two pictures of the Moon at exactly the same time from two locations on Earth and compare the positions of the Moon relative to the stars. Using the orientation of the Earth, those two position measurements, and the distance between the two locations on the Earth, the distance to the Moon can be triangulated: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {distance} _{\textrm {moon}}={\frac {\mathrm {distance} _{\mathrm {observerbase} }}{\tan(\mathrm {angle} )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>moon</mtext> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mrow> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {distance} _{\textrm {moon}}={\frac {\mathrm {distance} _{\mathrm {observerbase} }}{\tan(\mathrm {angle} )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5704d2c3c888b402b7178011565952385fa99ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:34.001ex; height:6.176ex;" alt="{\displaystyle \mathrm {distance} _{\textrm {moon}}={\frac {\mathrm {distance} _{\mathrm {observerbase} }}{\tan(\mathrm {angle} )}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Calculuses">Calculuses</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=59" title="Edit section: Calculuses" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=59" title="Edit section's source code: Calculuses"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resources: <a href="/w/index.php?title=Calculuses&action=edit&redlink=1" class="new" title="Calculuses (page does not exist)">Calculuses</a> and <a href="/wiki/Calculus" title="Calculus">Calculus</a></div> <p><b>Calculus</b> uses methods originally based on the summation of infinitesimal differences. </p><p>It includes the examination of changes in an expression by smaller and smaller differences. </p> <div class="mw-heading mw-heading2"><h2 id="Derivatives">Derivatives</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=60" title="Edit section: Derivatives" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=60" title="Edit section's source code: Derivatives"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resources: <a href="/w/index.php?title=Effects/Derivatives&action=edit&redlink=1" class="new" title="Effects/Derivatives (page does not exist)">Effects/Derivatives</a> and <a href="/wiki/Derivatives" title="Derivatives">Derivatives</a></div> <p><b>Def.</b> a result of an operation of deducing one function from another according to some fixed law is called a <b>derivative</b>. </p><p>Let </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2311a6a75c54b0ea085a381ba472c31d59321514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle y=f(x)}"></span></dd></dl> <p>be a function where values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> may be any real number and values resulting in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> are also any real number. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3890eb866b6258d7a304fc34c70ee3fb3a81a70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.266ex; height:2.176ex;" alt="{\displaystyle \Delta x}"></span> is a small finite difference in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> which when put into the function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> produces a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7caae142d915be8ef4d8c423bf91d1f6ea10e8e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.091ex; height:2.509ex;" alt="{\displaystyle \Delta y}"></span>.</dd></dl> <p>These small differences can be manipulated with the operations of arithmetic: addition (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"></span>), subtraction (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bd52ce670743d3b61bec928a7ec9f47309eb36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle -}"></span>), multiplication (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}"></span>), and division (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle /}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle /}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da0c4de1fba637d9799f6c64a6c77bf016d0ce1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.162ex; height:2.843ex;" alt="{\displaystyle /}"></span>). </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta y=f(x+\Delta x)-f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta y=f(x+\Delta x)-f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce43d41f207f5cdc2737e774b438b6cee1aa7401" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.971ex; height:2.843ex;" alt="{\displaystyle \Delta y=f(x+\Delta x)-f(x)}"></span></dd></dl> <p>Dividing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7caae142d915be8ef4d8c423bf91d1f6ea10e8e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.091ex; height:2.509ex;" alt="{\displaystyle \Delta y}"></span> by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3890eb866b6258d7a304fc34c70ee3fb3a81a70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.266ex; height:2.176ex;" alt="{\displaystyle \Delta x}"></span> and taking the limit as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3890eb866b6258d7a304fc34c70ee3fb3a81a70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.266ex; height:2.176ex;" alt="{\displaystyle \Delta x}"></span> → 0, produces the slope of a line tangent to f(x) at the point x. </p><p>For example, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84ddac4ae10b1aa4a11741c79771a583419fb1fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.9ex; height:3.176ex;" alt="{\displaystyle f(x)=x^{2}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x+\Delta x)=(x+\Delta x)^{2}=x^{2}+2x\Delta x+\Delta x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x+\Delta x)=(x+\Delta x)^{2}=x^{2}+2x\Delta x+\Delta x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d2aaa071fe456d344f460ddef7fe4f76c3e13f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.162ex; height:3.176ex;" alt="{\displaystyle f(x+\Delta x)=(x+\Delta x)^{2}=x^{2}+2x\Delta x+\Delta x^{2}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta y/\Delta x=(x^{2}+2x\Delta x+\Delta x^{2}-x^{2})/\Delta x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta y/\Delta x=(x^{2}+2x\Delta x+\Delta x^{2}-x^{2})/\Delta x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1deac27348045e5779deeb265728a4287cba05f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.222ex; height:3.176ex;" alt="{\displaystyle \Delta y/\Delta x=(x^{2}+2x\Delta x+\Delta x^{2}-x^{2})/\Delta x}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta y/\Delta x=2x+\Delta x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta y/\Delta x=2x+\Delta x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb46d47298eb738168b134a8c11c35cc001bf469" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.216ex; height:2.843ex;" alt="{\displaystyle \Delta y/\Delta x=2x+\Delta x}"></span></dd></dl> <p>as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3890eb866b6258d7a304fc34c70ee3fb3a81a70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.266ex; height:2.176ex;" alt="{\displaystyle \Delta x}"></span> and<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7caae142d915be8ef4d8c423bf91d1f6ea10e8e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.091ex; height:2.509ex;" alt="{\displaystyle \Delta y}"></span> go towards zero, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dy/dx=2x+dx=limit_{\Delta x\to 0}{f(x+\Delta x)-f(x) \over \Delta x}=2x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>l</mi> <mi>i</mi> <mi>m</mi> <mi>i</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dy/dx=2x+dx=limit_{\Delta x\to 0}{f(x+\Delta x)-f(x) \over \Delta x}=2x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f2289aa53c1df99960d8519af0f22b4ba531d2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:55.194ex; height:5.843ex;" alt="{\displaystyle dy/dx=2x+dx=limit_{\Delta x\to 0}{f(x+\Delta x)-f(x) \over \Delta x}=2x.}"></span></dd></dl> <p>This ratio is called the derivative. </p> <div class="mw-heading mw-heading2"><h2 id="Partial_derivatives">Partial derivatives</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=61" title="Edit section: Partial derivatives" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=61" title="Edit section's source code: Partial derivatives"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resources: <a href="/w/index.php?title=Effects/Derivatives/Partials&action=edit&redlink=1" class="new" title="Effects/Derivatives/Partials (page does not exist)">Effects/Derivatives/Partials</a> and <a href="/w/index.php?title=Partial_derivatives&action=edit&redlink=1" class="new" title="Partial derivatives (page does not exist)">Partial derivatives</a></div> <p>Let </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(x,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(x,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f00993c5c87552467aabf1e5d30180e8462768a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.794ex; height:2.843ex;" alt="{\displaystyle y=f(x,z)}"></span></dd></dl> <p>then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial y=\partial f(x,z)=\partial f(x,z)\partial x+\partial f(x,z)\partial z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>y</mi> <mo>=</mo> <mi mathvariant="normal">∂</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∂</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">∂</mi> <mi>x</mi> <mo>+</mo> <mi mathvariant="normal">∂</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial y=\partial f(x,z)=\partial f(x,z)\partial x+\partial f(x,z)\partial z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00daf6974402d01757e4c0c1616559ece6a248ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.138ex; height:2.843ex;" alt="{\displaystyle \partial y=\partial f(x,z)=\partial f(x,z)\partial x+\partial f(x,z)\partial z}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial y/\partial x=\partial f(x,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> <mo>=</mo> <mi mathvariant="normal">∂</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial y/\partial x=\partial f(x,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/869ac81a689d3b55e49f47813f2dca2dfb197937" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.24ex; height:2.843ex;" alt="{\displaystyle \partial y/\partial x=\partial f(x,z)}"></span></dd></dl> <p>where z is held constant and </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial y/\partial z=\partial f(x,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> <mo>=</mo> <mi mathvariant="normal">∂</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial y/\partial z=\partial f(x,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe178f4a9d3f690fb36eb901f37677008d3566bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.998ex; height:2.843ex;" alt="{\displaystyle \partial y/\partial z=\partial f(x,z)}"></span></dd></dl> <p>where x is held constant. </p> <div class="mw-heading mw-heading2"><h2 id="Gradients">Gradients</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=62" title="Edit section: Gradients" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=62" title="Edit section's source code: Gradients"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resources: <a href="/w/index.php?title=Spaces/Gradients&action=edit&redlink=1" class="new" title="Spaces/Gradients (page does not exist)">Spaces/Gradients</a> and <a href="/w/index.php?title=Gradients&action=edit&redlink=1" class="new" title="Gradients (page does not exist)">Gradients</a></div> <p><b>Notation</b>: let the symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3d0e93b78c50237f9ea83d027e4ebbdaef354b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \nabla }"></span> be the gradient, i.e., derivatives for multivariable functions. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla f(x,z)=\partial y=\partial f(x,z)=\partial f(x,z)\partial x+\partial f(x,z)\partial z.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∂</mi> <mi>y</mi> <mo>=</mo> <mi mathvariant="normal">∂</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∂</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">∂</mi> <mi>x</mi> <mo>+</mo> <mi mathvariant="normal">∂</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">∂</mi> <mi>z</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla f(x,z)=\partial y=\partial f(x,z)=\partial f(x,z)\partial x+\partial f(x,z)\partial z.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/073c43eecaef60705a3972ac5dcadc7537d37367" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.359ex; height:2.843ex;" alt="{\displaystyle \nabla f(x,z)=\partial y=\partial f(x,z)=\partial f(x,z)\partial x+\partial f(x,z)\partial z.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Area_under_a_curve">Area under a curve</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=63" title="Edit section: Area under a curve" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=63" title="Edit section's source code: Area under a curve"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider the curve in the graph in the section about <a href="/wiki/Astronomy/Mathematics#Areas" class="mw-redirect" title="Astronomy/Mathematics">areas</a>. The x-direction is left and right, the y-direction is vertical. </p><p>For </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x*\Delta y=[f(x+\Delta x)-f(x)]*\Delta x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>∗</mo> <mi mathvariant="normal">Δ</mi> <mi>y</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>∗</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x*\Delta y=[f(x+\Delta x)-f(x)]*\Delta x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c87941451bfff5016af8c18139a939d5f95302e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.186ex; height:2.843ex;" alt="{\displaystyle \Delta x*\Delta y=[f(x+\Delta x)-f(x)]*\Delta x}"></span></dd></dl> <p>the area under the curve shown in the diagram at right is the light purple rectangle plus the dark purple rectangle in the top figure </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x*\Delta y+f(x)*\Delta x=f(x+\Delta x)*\Delta x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>∗</mo> <mi mathvariant="normal">Δ</mi> <mi>y</mi> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x*\Delta y+f(x)*\Delta x=f(x+\Delta x)*\Delta x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8cc416ac49c27fa0908217edf033eaf127847f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.999ex; height:2.843ex;" alt="{\displaystyle \Delta x*\Delta y+f(x)*\Delta x=f(x+\Delta x)*\Delta x.}"></span></dd></dl> <p>Any particular individual rectangle for a sum of rectangular areas is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x_{i}+\Delta x_{i})*\Delta x_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∗</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x_{i}+\Delta x_{i})*\Delta x_{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d96ba19600335775e2d867b337c3b684237fea9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.03ex; height:2.843ex;" alt="{\displaystyle f(x_{i}+\Delta x_{i})*\Delta x_{i}.}"></span></dd></dl> <p>The approximate area under the curve is the sum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1d4e06539576633987e902f402ed46728d573b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.355ex; height:3.843ex;" alt="{\displaystyle \sum }"></span> of all the individual (i) areas from i = 0 to as many as the area needed (n): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=0}^{n}f(x_{i}+\Delta x_{i})*\Delta x_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∗</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=0}^{n}f(x_{i}+\Delta x_{i})*\Delta x_{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1649a5642fa44b0f5e17d7fe573d9199e485618" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.772ex; height:6.843ex;" alt="{\displaystyle \sum _{i=0}^{n}f(x_{i}+\Delta x_{i})*\Delta x_{i}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Integrals">Integrals</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=64" title="Edit section: Integrals" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=64" title="Edit section's source code: Integrals"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Def.</b> a number, the limit of the sums computed in a process in which the domain of a function is divided into small subsets and a possibly nominal value of the function on each subset is multiplied by the measure of that subset, all these products then being summed is called an <b>integral</b>. </p><p><b>Notation</b>: let the symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca732e5519cd2210bd59f1ab281b4e8f5a6a4413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:2.194ex; height:5.676ex;" alt="{\displaystyle \int }"></span> represent the <b>integral</b>. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle limit_{\Delta x\to 0}\sum _{i=0}^{n}f(x_{i}+\Delta x_{i})*\Delta x_{i}=\int f(x)dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mi>i</mi> <mi>m</mi> <mi>i</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </msub> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∗</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo>∫</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle limit_{\Delta x\to 0}\sum _{i=0}^{n}f(x_{i}+\Delta x_{i})*\Delta x_{i}=\int f(x)dx.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c6a3f7d503d1193a9a6586ce91b2fe3eecc44c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:45.986ex; height:6.843ex;" alt="{\displaystyle limit_{\Delta x\to 0}\sum _{i=0}^{n}f(x_{i}+\Delta x_{i})*\Delta x_{i}=\int f(x)dx.}"></span></dd></dl> <p>This can be within a finite interval [a,b] </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f(x)\;dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\;dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa07118298885ff05f58f8524ca4d62e25c363cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.397ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\;dx}"></span></dd></dl> <p>when i = 0 the integral is evaluated at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> and i = n the integral is evaluated at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>. Or, an indefinite integral (without notation on the integral symbol) as n goes to infinity and i = 0 is the integral evaluated at x = 0. </p> <div class="mw-heading mw-heading2"><h2 id="Theoretical_calculus">Theoretical calculus</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=65" title="Edit section: Theoretical calculus" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=65" title="Edit section's source code: Theoretical calculus"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Def.</b> a branch of mathematics that deals with the finding and properties ... of infinitesimal differences [or changes] is called a <b>calculus</b>. </p><p><b>Calculus</b> focuses on <a href="https://en.wikipedia.org/wiki/limit_(mathematics)" class="extiw" title="w:limit (mathematics)">limits</a>, <a href="https://en.wikipedia.org/wiki/function_(mathematics)" class="extiw" title="w:function (mathematics)">functions</a>, <a href="https://en.wikipedia.org/wiki/derivative" class="extiw" title="w:derivative">derivatives</a>, <a href="https://en.wikipedia.org/wiki/integral" class="extiw" title="w:integral">integrals</a>, and <a href="https://en.wikipedia.org/wiki/Series_(mathematics)" class="extiw" title="w:Series (mathematics)">infinite series</a>. </p><p>Although <i>calculus</i> (in the sense of analysis) is usually synonymous with infinitesimal calculus, not all historical formulations have relied on infinitesimals (infinitely small numbers that are nevertheless not zero). </p> <div class="mw-heading mw-heading2"><h2 id="Line_integrals">Line integrals</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=66" title="Edit section: Line integrals" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=66" title="Edit section's source code: Line integrals"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Def.</b> an integral the domain of whose integrand is a curve is called a <b>line integral</b>. </p><p>"The pulsar dispersion measures [(DM)] provide directly the value of </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle DM=\int _{0}^{\infty }n_{e}\,ds}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mi>M</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle DM=\int _{0}^{\infty }n_{e}\,ds}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bf0a83759bb5db5c98e6b4ff2d7644b0b42eced" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.278ex; height:5.843ex;" alt="{\displaystyle DM=\int _{0}^{\infty }n_{e}\,ds}"></span></dd></dl> <p>along the line of sight to the pulsar, while the interstellar Hα intensity (at high Galactic latitudes where optical extinction is minimal) is proportional to the emission measure"<sup id="cite_ref-Reynolds_34-0" class="reference"><a href="#cite_note-Reynolds-34"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle EM=\int _{0}^{\infty }n_{e}^{2}ds.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mi>M</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msubsup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>d</mi> <mi>s</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle EM=\int _{0}^{\infty }n_{e}^{2}ds.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e1e2c59062c7814e9211dc740d3cf5572d7785f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.445ex; height:5.843ex;" alt="{\displaystyle EM=\int _{0}^{\infty }n_{e}^{2}ds.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Vectors">Vectors</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=67" title="Edit section: Vectors" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=67" title="Edit section's source code: Vectors"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resource: <a href="/w/index.php?title=Vectors&action=edit&redlink=1" class="new" title="Vectors (page does not exist)">Vectors</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:3D_Vector.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/3D_Vector.svg/100px-3D_Vector.svg.png" decoding="async" width="100" height="95" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/3D_Vector.svg/150px-3D_Vector.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fd/3D_Vector.svg/200px-3D_Vector.svg.png 2x" data-file-width="555" data-file-height="525" /></a><figcaption></figcaption></figure><p> For standard basis, or unit, vectors (<b>i</b>, <b>j</b>, <b>k</b>) there may be vector components of <b>a</b> (<b>a</b><sub>x</sub>, <b>a</b><sub>y</sub>, <b>a</b><sub>z</sub>). </p><p><b>Def.</b> a directed quantity, one with both magnitude and direction; the signed difference between two points is called a <b>vector</b>. </p><p>"An observed time series consists of <i>N</i> data values x(t<sub>α</sub>) taken at a set of <i>N</i> discrete times {t<sub>α</sub>}. Hence it defines an <i>N</i>-dimensional <i>contravariant vector</i> in <i>sampling space</i>, by taking as the α<sup>th</sup> component of the vector, the value of the data at time t<sub>α</sub>, i.e., </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\alpha }=[x({t_{1}}),x({t_{2}}),...,x({t_{N}})].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\alpha }=[x({t_{1}}),x({t_{2}}),...,x({t_{N}})].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1add046244106341cafc36c0974056652731c1d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.592ex; height:2.843ex;" alt="{\displaystyle x^{\alpha }=[x({t_{1}}),x({t_{2}}),...,x({t_{N}})].}"></span></dd></dl> <p>This representation is the <i>canonical basis</i> for sampling space."<sup id="cite_ref-Foster_35-0" class="reference"><a href="#cite_note-Foster-35"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p> <div style="clear:both;"></div> <div class="mw-heading mw-heading2"><h2 id="Tensors">Tensors</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=68" title="Edit section: Tensors" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=68" title="Edit section's source code: Tensors"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resource: <a href="/wiki/Tensors" title="Tensors">Tensors</a></div> <p><b>Def.</b> a mathematical object consisting of a set of components with n indices each of which range from 1 to m where n is the rank and m is the dimension is called a <b>tensor</b>. </p><p>"An impressive array of time series analysis methods are equivalent to treating the data as a vector in function space, then projecting the data vector onto a subspace of low dimension. A geometric approach isolates and exposes many of the important features of time series techniques, directly adapts to irregular time spacing, and easily accommodates variable statistical weights. Tensor notation provides an ideal formalism for these techniques. It is quite convenient for distinguishing a variety of different vector spaces, and is the most compact notation for all the sums which arise in the analysis."<sup id="cite_ref-Foster_35-1" class="reference"><a href="#cite_note-Foster-35"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p><p>"[T]he generally invariant line element </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds^{2}=g_{\mu \nu }dx^{\mu }dx^{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}=g_{\mu \nu }dx^{\mu }dx^{\nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3dc481422966d8add7666f0b500fc120a49c30f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.081ex; height:3.343ex;" alt="{\displaystyle ds^{2}=g_{\mu \nu }dx^{\mu }dx^{\nu }}"></span></dd></dl> <p>[contains] the spacetime metric tensor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\mu \nu }(x^{\rho }),\mu ,\nu ,\rho =0,1,2,3,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> <mi>μ</mi> <mo>,</mo> <mi>ν</mi> <mo>,</mo> <mi>ρ</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\mu \nu }(x^{\rho }),\mu ,\nu ,\rho =0,1,2,3,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a54b42e79a604f1de82920586d984d6ea2e05ffc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.859ex; height:3.009ex;" alt="{\displaystyle g_{\mu \nu }(x^{\rho }),\mu ,\nu ,\rho =0,1,2,3,}"></span> [which] plays a dual role: on the one hand it determines the spacetime geometry, on the other it represents the (ten components of the) gravitational potential, and is thus a dynamical variable."<sup id="cite_ref-Bicak_36-0" class="reference"><a href="#cite_note-Bicak-36"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Electronic_computers">Electronic computers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=69" title="Edit section: Electronic computers" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=69" title="Edit section's source code: Electronic computers"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resource: <a href="/w/index.php?title=Electronic_computers&action=edit&redlink=1" class="new" title="Electronic computers (page does not exist)">Electronic computers</a></div> <p><b>Def.</b> a programmable electronic device that performs mathematical calculations and logical operations, especially one that can process, store and retrieve large amounts of data very quickly; now especially, a small one for personal or home use employed for manipulating text or graphics, accessing the Internet, or playing games or media is called a <b>computer</b>. </p><p>A <b>computer</b> is a general purpose device that can be <a href="https://en.wikipedia.org/wiki/Computer_program" class="extiw" title="w:Computer program">programmed</a> to carry out a finite set of arithmetic or logical operations. Since a sequence of operations can be readily changed, the computer can solve more than one kind of problem. </p> <div class="mw-heading mw-heading2"><h2 id="Programmings">Programmings</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=70" title="Edit section: Programmings" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=70" title="Edit section's source code: Programmings"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resource: <a href="/w/index.php?title=Programmings&action=edit&redlink=1" class="new" title="Programmings (page does not exist)">Programmings</a></div> <p>A <b>computer program</b> (also <b><a href="https://en.wikipedia.org/wiki/computer_software" class="extiw" title="w:computer software">software</a></b>, or just a <b>program</b>) is a sequence of <a href="https://en.wikipedia.org/wiki/instruction_(computer_science)" class="extiw" title="w:instruction (computer science)">instructions</a> written to perform a specified task with a computer.<sup id="cite_ref-pis-ch4-p132_37-0" class="reference"><a href="#cite_note-pis-ch4-p132-37"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> A computer requires programs to function, typically <a href="https://en.wikipedia.org/wiki/execution_(computing)" class="extiw" title="w:execution (computing)">executing</a> the program's instructions in a <a href="https://en.wikipedia.org/wiki/central_processing_unit" class="extiw" title="w:central processing unit">central processor</a>.<sup id="cite_ref-osc-ch3-p58_38-0" class="reference"><a href="#cite_note-osc-ch3-p58-38"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p><p><b>Computer programming</b> (often shortened to <b>programming</b> or <b>coding</b>) is the process of <a href="https://en.wikipedia.org/wiki/Software_design" class="extiw" title="w:Software design">designing</a>, writing, <a href="https://en.wikipedia.org/wiki/Software_testing" class="extiw" title="w:Software testing">testing</a>, <a href="https://en.wikipedia.org/wiki/debugging" class="extiw" title="w:debugging">debugging</a>, and maintaining the <a href="https://en.wikipedia.org/wiki/source_code" class="extiw" title="w:source code">source code</a> of <a href="https://en.wikipedia.org/wiki/computer_program" class="extiw" title="w:computer program">computer programs</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Probabilities">Probabilities</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=71" title="Edit section: Probabilities" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=71" title="Edit section's source code: Probabilities"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resource: <a href="/w/index.php?title=Probabilities&action=edit&redlink=1" class="new" title="Probabilities (page does not exist)">Probabilities</a></div> <p><b>Def.</b> a number, between 0 and 1, expressing the precise likelihood of an event happening is called a <b>probability</b>. </p><p><b>Probability</b> is a measure of the expectation that an event will occur or a statement is true. Probabilities are given a value between 0 (will not occur) and 1 (will occur).<sup id="cite_ref-Feller_39-0" class="reference"><a href="#cite_note-Feller-39"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> The higher the probability of an event, the more certain we are that the event will occur. </p> <div class="mw-heading mw-heading2"><h2 id="Statistics">Statistics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=72" title="Edit section: Statistics" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=72" title="Edit section's source code: Statistics"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resource: <a href="/wiki/Statistics" title="Statistics">Statistics</a></div> <p><b>Def.</b> a mathematical science concerned with data collection, presentation, analysis, and interpretation is called <b>statistics</b>. </p><p><b><a href="/wiki/Statistics" title="Statistics">Statistics</a></b> is the study of the collection, organization, analysis, interpretation, and presentation of <a href="https://en.wikipedia.org/wiki/data" class="extiw" title="w:data">data</a>.<sup id="cite_ref-Dodge_40-0" class="reference"><a href="#cite_note-Dodge-40"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> It deals with all aspects of this, including the planning of data collection in terms of the design of <a href="https://en.wikipedia.org/wiki/statistical_survey" class="extiw" title="w:statistical survey">surveys</a> and <a href="https://en.wikipedia.org/wiki/experimental_design" class="extiw" title="w:experimental design">experiments</a>.<sup id="cite_ref-Dodge_40-1" class="reference"><a href="#cite_note-Dodge-40"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> </p><p>"Statistics of projections are derived under a number of different null hypotheses."<sup id="cite_ref-Foster_35-2" class="reference"><a href="#cite_note-Foster-35"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Hypotheses">Hypotheses</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=73" title="Edit section: Hypotheses" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=73" title="Edit section's source code: Hypotheses"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2661592"><div role="note" class="hatnote navigation-not-searchable">Main resource: <a href="/wiki/Hypotheses" class="mw-redirect" title="Hypotheses">Hypotheses</a></div> <ol><li>Each mathematical approach requires a proof of concept.</li></ol> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=74" title="Edit section: See also" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=74" title="Edit section's source code: See also"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r2670040">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 12em;"> <ul><li><a href="https://en.wikipedia.org/wiki/Cosmic_distance_ladder" class="extiw" title="w:Cosmic distance ladder">Cosmic distance ladder</a></li> <li><a href="/wiki/Portal:Euclidean_geometry" title="Portal:Euclidean geometry">Euclidean geometry</a></li> <li><a href="/w/index.php?title=Topic:Mathematical_physics&action=edit&redlink=1" class="new" title="Topic:Mathematical physics (page does not exist)">Mathematical physics</a></li> <li><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics</a></li> <li><a href="/w/index.php?title=Radiation_astronomy/Courses/Principles&action=edit&redlink=1" class="new" title="Radiation astronomy/Courses/Principles (page does not exist)">Principles of Radiation Astronomy</a></li> <li><a href="/wiki/Probability" title="Probability">Probability</a></li> <li><a href="/w/index.php?title=Radiation_astronomy/Mathematics&action=edit&redlink=1" class="new" title="Radiation astronomy/Mathematics (page does not exist)">Radiation mathematics</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Astronomy/Theory" class="mw-redirect" title="Astronomy/Theory">Theoretical astronomy</a></li> <li><a href="/w/index.php?title=Radiation_astronomy/Theory&action=edit&redlink=1" class="new" title="Radiation astronomy/Theory (page does not exist)">Theoretical radiation astronomy</a></li> <li><a href="/w/index.php?title=Topic:Trigonometry&action=edit&redlink=1" class="new" title="Topic:Trigonometry (page does not exist)">Trigonometry</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=75" title="Edit section: References" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=75" title="Edit section's source code: References"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r2661605">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width reflist-columns-2"> <ol class="references"> <li id="cite_note-Gove-1"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Gove_1-0">1.0</a></sup> <sup><a href="#cite_ref-Gove_1-1">1.1</a></sup></span> <span class="reference-text"><span class="citation book">Philip B. Gove, ed (1963). <i>Webster's Seventh New Collegiate Dictionary</i>. Springfield, Massachusetts: G. & C. Merriam Company. pp. 1221.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Webster%27s+Seventh+New+Collegiate+Dictionary&rft.date=1963&rft.pages=pp.%26nbsp%3B1221&rft.place=Springfield%2C+Massachusetts&rft.pub=G.+%26+C.+Merriam+Company&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-NIST_heterodyne-3"><span class="mw-cite-backlink"><a href="#cite_ref-NIST_heterodyne_3-0">↑</a></span> <span class="reference-text"><span class="citation Journal">Evenson, KM (1972). "Speed of Light from Direct Frequency and Wavelength Measurements of the Methane-Stabilized Laser". <i>Physical Review Letters</i> <b>29</b> (19): 1346–49. doi:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.29.1346">10.1103/PhysRevLett.29.1346</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Speed+of+Light+from+Direct+Frequency+and+Wavelength+Measurements+of+the+Methane-Stabilized+Laser&rft.jtitle=Physical+Review+Letters&rft.aulast=Evenson&rft.aufirst=KM&rft.au=Evenson%2C%26%2332%3BKM&rft.date=1972&rft.volume=29&rft.issue=19&rft.pages=1346%E2%80%9349&rft_id=info:doi/10.1103%2FPhysRevLett.29.1346&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">Oxford English Dictionary, 2nd ed.: <a rel="nofollow" class="external text" href="http://oxforddictionaries.com/definition/english/natural%2Blogarithm">natural logarithm</a></span> </li> <li id="cite_note-Peale-5"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Peale_5-0">4.0</a></sup> <sup><a href="#cite_ref-Peale_5-1">4.1</a></sup> <sup><a href="#cite_ref-Peale_5-2">4.2</a></sup></span> <span class="reference-text"><span class="citation Journal">Peale, S. J. (June 1974). "Possible histories of the obliquity of Mercury". <i>Astronomical Journal</i> <b>79</b> (6): 722-44. doi:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F111604">10.1086/111604</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Possible+histories+of+the+obliquity+of+Mercury&rft.jtitle=Astronomical+Journal&rft.aulast=Peale%2C+S.+J.&rft.au=Peale%2C+S.+J.&rft.date=June+1974&rft.volume=79&rft.issue=6&rft.pages=722-44&rft_id=info:doi/10.1086%2F111604&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-Seidelmann-6"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Seidelmann_6-0">5.0</a></sup> <sup><a href="#cite_ref-Seidelmann_6-1">5.1</a></sup> <sup><a href="#cite_ref-Seidelmann_6-2">5.2</a></sup> <sup><a href="#cite_ref-Seidelmann_6-3">5.3</a></sup> <sup><a href="#cite_ref-Seidelmann_6-4">5.4</a></sup> <sup><a href="#cite_ref-Seidelmann_6-5">5.5</a></sup> <sup><a href="#cite_ref-Seidelmann_6-6">5.6</a></sup> <sup><a href="#cite_ref-Seidelmann_6-7">5.7</a></sup> <sup><a href="#cite_ref-Seidelmann_6-8">5.8</a></sup></span> <span class="reference-text"><span class="citation book">P. K. Seidelmann (1976). <a rel="nofollow" class="external text" href="http://www.iau.org/public/measuring/"><i>Measuring the Universe The IAU and astronomical units</i></a>. International Astronomical Union<span class="printonly">. <a rel="nofollow" class="external free" href="http://www.iau.org/public/measuring/">http://www.iau.org/public/measuring/</a></span><span class="reference-accessdate">. Retrieved 2011-11-27</span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Measuring+the+Universe+The+IAU+and+astronomical+units&rft.aulast=P.+K.+Seidelmann&rft.au=P.+K.+Seidelmann&rft.date=1976&rft.pub=International+Astronomical+Union&rft_id=http%3A%2F%2Fwww.iau.org%2Fpublic%2Fmeasuring%2F&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-Wilkins-7"><span class="mw-cite-backlink"><a href="#cite_ref-Wilkins_7-0">↑</a></span> <span class="reference-text">International Astronomical Union "<a rel="nofollow" class="external text" href="http://www.iau.org/science/publications/proceedings_rules/units/">SI units</a>" accessed February 18, 2010. (See Table 5 and section 5.15.) Reprinted from George A. Wilkins & IAU Commission 5, <a rel="nofollow" class="external text" href="http://www.iau.org/static/publications/stylemanual1989.pdf">"The IAU Style Manual (1989)"</a> (PDF file) in <i>IAU Transactions</i> Vol. XXB</span> </li> <li id="cite_note-Williams2004-8"><span class="mw-cite-backlink"><a href="#cite_ref-Williams2004_8-0">↑</a></span> <span class="reference-text"><span class="citation book">David R. Williams (September 2004). <a rel="nofollow" class="external text" href="https://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html"><i>Sun Fact Sheet</i></a>. Greenbelt, MD: NASA Goddard Space Flight Center<span class="printonly">. <a rel="nofollow" class="external free" href="https://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html">http://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html</a></span><span class="reference-accessdate">. Retrieved 2011-12-20</span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sun+Fact+Sheet&rft.aulast=David+R.+Williams&rft.au=David+R.+Williams&rft.date=September+2004&rft.place=Greenbelt%2C+MD&rft.pub=NASA+Goddard+Space+Flight+Center&rft_id=http%3A%2F%2Fnssdc.gsfc.nasa.gov%2Fplanetary%2Ffactsheet%2Fsunfact.html&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><a href="#cite_ref-9">↑</a></span> <span class="reference-text"><i>Encyclopaedia Britannica</i>, 1968, vol. 2, p. 645</span> </li> <li id="cite_note-Caspar-10"><span class="mw-cite-backlink"><a href="#cite_ref-Caspar_10-0">↑</a></span> <span class="reference-text">M Caspar, <i>Kepler</i> (1959, Abelard-Schuman), at pp.131–140; A Koyré, <i>The Astronomical Revolution: Copernicus, Kepler, Borelli</i> (1973, Methuen), pp. 277–279</span> </li> <li id="cite_note-Linton-11"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Linton_11-0">10.0</a></sup> <sup><a href="#cite_ref-Linton_11-1">10.1</a></sup></span> <span class="reference-text"><span class="citation book">Christopher M. Linton (2004). <i>From Eudoxus to Einstein—A History of Mathematical Astronomy</i>. Cambridge: Cambridge University Press. ISBN <a href="/wiki/Special:BookSources/978-0-521-82750-8" title="Special:BookSources/978-0-521-82750-8">978-0-521-82750-8</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=From+Eudoxus+to+Einstein%26mdash%3BA+History+of+Mathematical+Astronomy&rft.aulast=Christopher+M.+Linton&rft.au=Christopher+M.+Linton&rft.date=2004&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.isbn=978-0-521-82750-8&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-Williams-12"><span class="mw-cite-backlink"><a href="#cite_ref-Williams_12-0">↑</a></span> <span class="reference-text"><span class="citation book">David R. Williams. <a rel="nofollow" class="external text" href="https://nssdc.gsfc.nasa.gov/planetary/factsheet/planetfact_notes.html"><i>Planetary Fact Sheet Notes</i></a><span class="printonly">. <a rel="nofollow" class="external free" href="https://nssdc.gsfc.nasa.gov/planetary/factsheet/planetfact_notes.html">http://nssdc.gsfc.nasa.gov/planetary/factsheet/planetfact_notes.html</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Planetary+Fact+Sheet+Notes&rft.aulast=David+R.+Williams&rft.au=David+R.+Williams&rft_id=http%3A%2F%2Fnssdc.gsfc.nasa.gov%2Fplanetary%2Ffactsheet%2Fplanetfact_notes.html&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-Hays-13"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Hays_13-0">12.0</a></sup> <sup><a href="#cite_ref-Hays_13-1">12.1</a></sup></span> <span class="reference-text"><span class="citation Journal">J. D. Hays; John Imbrie; N. J. Shackleton (December 1976). <a rel="nofollow" class="external text" href="http://www.whoi.edu/science/GG/paleoseminar/ps/hays76.ps">"Variations in the Earth's Orbit: Pacemaker of the Ice Ages"</a>. <i>Science</i> <b>194</b> (4270)<span class="printonly">. <a rel="nofollow" class="external free" href="http://www.whoi.edu/science/GG/paleoseminar/ps/hays76.ps">http://www.whoi.edu/science/GG/paleoseminar/ps/hays76.ps</a></span><span class="reference-accessdate">. Retrieved 2011-11-08</span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Variations+in+the+Earth%27s+Orbit%3A+Pacemaker+of+the+Ice+Ages&rft.jtitle=Science&rft.aulast=J.+D.+Hays&rft.au=J.+D.+Hays&rft.au=John+Imbrie&rft.au=N.+J.+Shackleton&rft.date=December+1976&rft.volume=194&rft.issue=4270&rft_id=http%3A%2F%2Fwww.whoi.edu%2Fscience%2FGG%2Fpaleoseminar%2Fps%2Fhays76.ps&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-Shu-14"><span class="mw-cite-backlink"><a href="#cite_ref-Shu_14-0">↑</a></span> <span class="reference-text"><span class="citation book">Frank H Shu (1982). <a rel="nofollow" class="external text" href="http://books.google.com/?id=v_6PbAfapSAC&pg=PA261"><i>The Physical Universe</i></a>. University Science Books. p. 261. ISBN <a href="/wiki/Special:BookSources/0935702059" title="Special:BookSources/0935702059">0935702059</a><span class="printonly">. <a rel="nofollow" class="external free" href="http://books.google.com/?id=v_6PbAfapSAC&pg=PA261">http://books.google.com/?id=v_6PbAfapSAC&pg=PA261</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Physical+Universe&rft.aulast=Frank+H+Shu&rft.au=Frank+H+Shu&rft.date=1982&rft.pages=p.%26nbsp%3B261&rft.pub=University+Science+Books&rft.isbn=0935702059&rft_id=http%3A%2F%2Fbooks.google.com%2F%3Fid%3Dv_6PbAfapSAC%26pg%3DPA261&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-Binney-15"><span class="mw-cite-backlink"><a href="#cite_ref-Binney_15-0">↑</a></span> <span class="reference-text"><span class="citation book">James Binney; Michael Merrifield (1998). <a rel="nofollow" class="external text" href="http://books.google.com/?id=arYYRoYjKacC&pg=PA536"><i>Galactic Astronomy</i></a>. Princeton University Press. p. 536. ISBN <a href="/wiki/Special:BookSources/0691025657" title="Special:BookSources/0691025657">0691025657</a><span class="printonly">. <a rel="nofollow" class="external free" href="http://books.google.com/?id=arYYRoYjKacC&pg=PA536">http://books.google.com/?id=arYYRoYjKacC&pg=PA536</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Galactic+Astronomy&rft.aulast=James+Binney&rft.au=James+Binney&rft.au=Michael+Merrifield&rft.date=1998&rft.pages=p.%26nbsp%3B536&rft.pub=Princeton+University+Press&rft.isbn=0691025657&rft_id=http%3A%2F%2Fbooks.google.com%2F%3Fid%3DarYYRoYjKacC%26pg%3DPA536&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-Reid-16"><span class="mw-cite-backlink"><a href="#cite_ref-Reid_16-0">↑</a></span> <span class="reference-text"><span class="citation book">Mark Reid (2008). <a rel="nofollow" class="external text" href="http://books.google.com/?id=bP9hZqoIfhMC&pg=PA19">"Mapping the Milky Way and the Local Group"</a>. In F. Combes, Keiichi Wada. <i>Mapping the Galaxy and Nearby Galaxies</i>. Springer. pp. 19–20. ISBN <a href="/wiki/Special:BookSources/0387727671" title="Special:BookSources/0387727671">0387727671</a><span class="printonly">. <a rel="nofollow" class="external free" href="http://books.google.com/?id=bP9hZqoIfhMC&pg=PA19">http://books.google.com/?id=bP9hZqoIfhMC&pg=PA19</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=Mapping+the+Milky+Way+and+the+Local+Group&rft.atitle=Mapping+the+Galaxy+and+Nearby+Galaxies&rft.aulast=Mark+Reid&rft.au=Mark+Reid&rft.date=2008&rft.pages=pp.%26nbsp%3B19%E2%80%9320&rft.pub=Springer&rft.isbn=0387727671&rft_id=http%3A%2F%2Fbooks.google.com%2F%3Fid%3DbP9hZqoIfhMC%26pg%3DPA19&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-Binney1-17"><span class="mw-cite-backlink"><a href="#cite_ref-Binney1_17-0">↑</a></span> <span class="reference-text"><span class="citation book">Binney J.; Merrifield M.. "§10.6". <i>op. cit.</i>. ISBN <a href="/wiki/Special:BookSources/0691025657" title="Special:BookSources/0691025657">0691025657</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=%C2%A710.6&rft.atitle=op.+cit.&rft.aulast=Binney+J.&rft.au=Binney+J.&rft.au=Merrifield+M.&rft.isbn=0691025657&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-Mamajek-18"><span class="mw-cite-backlink"><a href="#cite_ref-Mamajek_18-0">↑</a></span> <span class="reference-text"><span class="citation Journal">E.E. Mamajek (2008). "On the distance to the Ophiuchus star-forming region". <i>Astron. Nachr.</i> <b>AN 329</b>: 12. doi:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fasna.200710827">10.1002/asna.200710827</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=On+the+distance+to+the+Ophiuchus+star-forming+region&rft.jtitle=Astron.+Nachr.&rft.aulast=E.E.+Mamajek&rft.au=E.E.+Mamajek&rft.date=2008&rft.volume=AN+329&rft.pages=12&rft_id=info:doi/10.1002%2Fasna.200710827&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-Majewski-19"><span class="mw-cite-backlink"><a href="#cite_ref-Majewski_19-0">↑</a></span> <span class="reference-text"><span class="citation Journal">Steven R. Majewski (2008). "Precision Astrometry, Galactic Mergers, Halo Substructure and Local Dark Matter". <i>Proceedings of IAU Symposium 248</i> <b>3</b>. doi:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS1743921308019790">10.1017/S1743921308019790</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Precision+Astrometry%2C+Galactic+Mergers%2C+Halo+Substructure+and+Local+Dark+Matter&rft.jtitle=Proceedings+of+IAU+Symposium+248&rft.aulast=Steven+R.+Majewski&rft.au=Steven+R.+Majewski&rft.date=2008&rft.volume=3&rft_id=info:doi/10.1017%2FS1743921308019790&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-Wolfram2nd-20"><span class="mw-cite-backlink"><a href="#cite_ref-Wolfram2nd_20-0">↑</a></span> <span class="reference-text">Bryant, Jeff; Pavlyk, Oleksandr. "<a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/KeplersSecondLaw/">Kepler's Second Law</a>", <i>Wolfram Demonstrations Project</i>. Retrieved December 27, 2009.</span> </li> <li id="cite_note-Schombert-21"><span class="mw-cite-backlink"><a href="#cite_ref-Schombert_21-0">↑</a></span> <span class="reference-text"><span class="citation book">James Schombert. <a rel="nofollow" class="external text" href="http://abyss.uoregon.edu/~js/ast121/lectures/lec03.html"><i>Earth Coordinate System</i></a>. University of Oregon Department of Physics<span class="printonly">. <a rel="nofollow" class="external free" href="http://abyss.uoregon.edu/~js/ast121/lectures/lec03.html">http://abyss.uoregon.edu/~js/ast121/lectures/lec03.html</a></span><span class="reference-accessdate">. Retrieved 19 March 2011</span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Earth+Coordinate+System&rft.aulast=James+Schombert&rft.au=James+Schombert&rft.pub=University+of+Oregon+Department+of+Physics&rft_id=http%3A%2F%2Fabyss.uoregon.edu%2F%7Ejs%2Fast121%2Flectures%2Flec03.html&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-Gabuzda-22"><span class="mw-cite-backlink"><a href="#cite_ref-Gabuzda_22-0">↑</a></span> <span class="reference-text"><span class="citation Journal">D. C. Gabuzda; J. F. C. Wardle; D. H. Roberts (January 15, 1989). "Superluminal motion in the BL Lacertae object OJ 287". <i>The Astrophysical Journal</i> <b>336</b> (1): L59-62. doi:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F185361">10.1086/185361</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Superluminal+motion+in+the+BL+Lacertae+object+OJ+287&rft.jtitle=The+Astrophysical+Journal&rft.aulast=D.+C.+Gabuzda&rft.au=D.+C.+Gabuzda&rft.au=J.+F.+C.+Wardle&rft.au=D.+H.+Roberts&rft.date=January+15%2C+1989&rft.volume=336&rft.issue=1&rft.pages=L59-62&rft_id=info:doi/10.1086%2F185361&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-Sinnott-23"><span class="mw-cite-backlink"><a href="#cite_ref-Sinnott_23-0">↑</a></span> <span class="reference-text">R.W. Sinnott, "Virtues of the Haversine", Sky and Telescope, vol. 68, no. 2, 1984, p. 159</span> </li> <li id="cite_note-Vincenty-24"><span class="mw-cite-backlink"><a href="#cite_ref-Vincenty_24-0">↑</a></span> <span class="reference-text"><span class="citation Journal"><a href="/w/index.php?title=Thaddeus_Vincenty&action=edit&redlink=1" class="new" title="Thaddeus Vincenty (page does not exist)">Vincenty, Thaddeus</a> (1975-04-01). <a rel="nofollow" class="external text" href="http://www.ngs.noaa.gov/PUBS_LIB/inverse.pdf">"Direct and Inverse Solutions of Geodesics on the Ellipsoid with Application of Nested Equations"</a> (PDF). <i>Survey Review</i> (Kingston Road, Tolworth, Surrey: Directorate of Overseas Surveys) <b>23</b> (176): 88–93<span class="printonly">. <a rel="nofollow" class="external free" href="http://www.ngs.noaa.gov/PUBS_LIB/inverse.pdf">http://www.ngs.noaa.gov/PUBS_LIB/inverse.pdf</a></span><span class="reference-accessdate">. Retrieved 2008-07-21</span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Direct+and+Inverse+Solutions+of+Geodesics+on+the+Ellipsoid+with+Application+of+Nested+Equations&rft.jtitle=Survey+Review&rft.aulast=Vincenty&rft.aufirst=Thaddeus&rft.au=Vincenty%2C%26%2332%3BThaddeus&rft.date=1975-04-01&rft.volume=23&rft.issue=176&rft.pages=88%E2%80%9393&rft.place=Kingston+Road%2C+Tolworth%2C+Surrey&rft.pub=Directorate+of+Overseas+Surveys&rft_id=http%3A%2F%2Fwww.ngs.noaa.gov%2FPUBS_LIB%2Finverse.pdf&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-ZG44-25"><span class="mw-cite-backlink"><a href="#cite_ref-ZG44_25-0">↑</a></span> <span class="reference-text"><a href="#CITEREFZeilikGregory1998">Zeilik & Gregory 1998</a>, p. 44<span class="error harv-error" style="display: none; font-size:100%"> harvnb error: no target: CITEREFZeilikGregory1998 (<a href="/w/index.php?title=Category:Harv_and_Sfn_template_errors&action=edit&redlink=1" class="new" title="Category:Harv and Sfn template errors (page does not exist)">help</a>)</span>.</span> </li> <li id="cite_note-apj118-26"><span class="mw-cite-backlink"><a href="#cite_ref-apj118_26-0">↑</a></span> <span class="reference-text"><span class="citation Journal" id="CITEREFBenedictG._FritzChappellNelan1999">Benedict, G. Fritz <i>et al</i>. (1999). "Interferometric Astrometry of Proxima Centauri and Barnard's Star Using HUBBLE SPACE TELESCOPE Fine Guidance Sensor 3: Detection Limits for Substellar Companions". <i>The Astronomical Journal</i> <b>118</b> (2): 1086–1100. doi:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F300975">10.1086/300975</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Interferometric+Astrometry+of+Proxima+Centauri+and+Barnard%27s+Star+Using+HUBBLE+SPACE+TELESCOPE+Fine+Guidance+Sensor+3%3A+Detection+Limits+for+Substellar+Companions&rft.jtitle=The+Astronomical+Journal&rft.aulast=Benedict&rft.au=Benedict&rft.au=G.+Fritz&rft.au=Chappell%2C%26%2332%3BD.+W.&rft.au=Nelan%2C%26%2332%3BE.&rft.au=Jefferys%2C%26%2332%3BW.+H.&rft.au=Van+Altena%2C%26%2332%3BW.&rft.au=Lee%2C%26%2332%3BJ.&rft.au=Cornell%2C%26%2332%3BD.&rft.au=Shelus%2C%26%2332%3BP.+J.&rft.date=1999&rft.volume=118&rft.issue=2&rft.pages=1086%E2%80%931100&rft_id=info:doi/10.1086%2F300975&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-Shorter-27"><span class="mw-cite-backlink"><a href="#cite_ref-Shorter_27-0">↑</a></span> <span class="reference-text"><span class="citation book"> <i>Shorter Oxford English Dictionary</i>. 1968. "Mutual inclination of two lines meeting in an angle"</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Shorter+Oxford+English+Dictionary&rft.date=1968&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-Oxford-28"><span class="mw-cite-backlink"><a href="#cite_ref-Oxford_28-0">↑</a></span> <span class="reference-text"><span class="citation book"> <a rel="nofollow" class="external text" href="http://dictionary.oed.com/cgi/entry/50171114?single=1&query_type=word&queryword=parallax&first=1&max_to_show=10"><i>Oxford English Dictionary</i></a> (Second ed.). 1989<span class="printonly">. <a rel="nofollow" class="external free" href="http://dictionary.oed.com/cgi/entry/50171114?single=1&query_type=word&queryword=parallax&first=1&max_to_show=10">http://dictionary.oed.com/cgi/entry/50171114?single=1&query_type=word&queryword=parallax&first=1&max_to_show=10</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Oxford+English+Dictionary&rft.date=1989&rft.edition=Second&rft_id=http%3A%2F%2Fdictionary.oed.com%2Fcgi%2Fentry%2F50171114%3Fsingle%3D1%26query_type%3Dword%26queryword%3Dparallax%26first%3D1%26max_to_show%3D10&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-Seidelmann2005-29"><span class="mw-cite-backlink"><a href="#cite_ref-Seidelmann2005_29-0">↑</a></span> <span class="reference-text"><span class="citation book">P. Kenneth Seidelmann (2005). <i>Explanatory Supplement to the Astronomical Almanac</i>. University Science Books. pp. 123–125. ISBN <a href="/wiki/Special:BookSources/1891389459" title="Special:BookSources/1891389459">1891389459</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Explanatory+Supplement+to+the+Astronomical+Almanac&rft.aulast=P.+Kenneth+Seidelmann&rft.au=P.+Kenneth+Seidelmann&rft.date=2005&rft.pages=pp.%26nbsp%3B123%E2%80%93125&rft.pub=University+Science+Books&rft.isbn=1891389459&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-Barbieri-30"><span class="mw-cite-backlink"><a href="#cite_ref-Barbieri_30-0">↑</a></span> <span class="reference-text"><span class="citation book">Cesare Barbieri (2007). <i>Fundamentals of astronomy</i>. CRC Press. pp. 132–135. ISBN <a href="/wiki/Special:BookSources/0750308869" title="Special:BookSources/0750308869">0750308869</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+astronomy&rft.aulast=Cesare+Barbieri&rft.au=Cesare+Barbieri&rft.date=2007&rft.pages=pp.%26nbsp%3B132%E2%80%93135&rft.pub=CRC+Press&rft.isbn=0750308869&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-aa1981-31"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-aa1981_31-0">30.0</a></sup> <sup><a href="#cite_ref-aa1981_31-1">30.1</a></sup></span> <span class="reference-text">Astronomical Almanac e.g. for 1981, section D</span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><a href="#cite_ref-32">↑</a></span> <span class="reference-text">Astronomical Almanac, e.g. for 1981: see Glossary; for formulae see Explanatory Supplement to the Astronomical Almanac, 1992, p.400</span> </li> <li id="cite_note-Gutzwiller-33"><span class="mw-cite-backlink"><a href="#cite_ref-Gutzwiller_33-0">↑</a></span> <span class="reference-text"><span class="citation Journal" id="CITEREFGutzwiller,_Martin_C.1998">Gutzwiller, Martin C. (1998). "Moon-Earth-Sun: The oldest three-body problem". <i>Reviews of Modern Physics</i> <b>70</b> (2): 589. doi:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FRevModPhys.70.589">10.1103/RevModPhys.70.589</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Moon-Earth-Sun%3A+The+oldest+three-body+problem&rft.jtitle=Reviews+of+Modern+Physics&rft.aulast=Gutzwiller%2C+Martin+C.&rft.au=Gutzwiller%2C+Martin+C.&rft.date=1998&rft.volume=70&rft.issue=2&rft.pages=589&rft_id=info:doi/10.1103%2FRevModPhys.70.589&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-Reynolds-34"><span class="mw-cite-backlink"><a href="#cite_ref-Reynolds_34-0">↑</a></span> <span class="reference-text"><span class="citation Journal">R. J. Reynolds (May 1, 1991). <a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/full/1991ApJ...372L..17R">"Line Integrals of n<sub>e</sub> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{e}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{e}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73adc8713faa70d82cc023f1e512a010bf56cf69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.449ex; height:2.843ex;" alt="{\displaystyle n_{e}^{2}}"></span> at High Galactic Latitude"</a>. <i>The Astrophysical Journal</i> <b>372</b> (05): L17-20. doi:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F186013">10.1086/186013</a><span class="printonly">. <a rel="nofollow" class="external free" href="http://adsabs.harvard.edu/full/1991ApJ...372L..17R">http://adsabs.harvard.edu/full/1991ApJ...372L..17R</a></span><span class="reference-accessdate">. Retrieved 2013-12-17</span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Line+Integrals+of+n%3Csub%3Ee%3C%2Fsub%3E+and++at+High+Galactic+Latitude&rft.jtitle=The+Astrophysical+Journal&rft.aulast=R.+J.+Reynolds&rft.au=R.+J.+Reynolds&rft.date=May+1%2C+1991&rft.volume=372&rft.issue=05&rft.pages=L17-20&rft_id=info:doi/10.1086%2F186013&rft_id=http%3A%2F%2Fadsabs.harvard.edu%2Ffull%2F1991ApJ...372L..17R&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-Foster-35"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Foster_35-0">34.0</a></sup> <sup><a href="#cite_ref-Foster_35-1">34.1</a></sup> <sup><a href="#cite_ref-Foster_35-2">34.2</a></sup></span> <span class="reference-text"><span class="citation Journal">Grant Foster (January 1996). <a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/full/1996AJ....111..555F">"Time Series Analysis by Projection. II. Tensor Methods for Time Series Analysis"</a>. <i>The Astronomical Journal</i> <b>111</b> (1): 555-65<span class="printonly">. <a rel="nofollow" class="external free" href="http://adsabs.harvard.edu/full/1996AJ....111..555F">http://adsabs.harvard.edu/full/1996AJ....111..555F</a></span><span class="reference-accessdate">. Retrieved 2013-12-16</span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Time+Series+Analysis+by+Projection.+II.+Tensor+Methods+for+Time+Series+Analysis&rft.jtitle=The+Astronomical+Journal&rft.aulast=Grant+Foster&rft.au=Grant+Foster&rft.date=January+1996&rft.volume=111&rft.issue=1&rft.pages=555-65&rft_id=http%3A%2F%2Fadsabs.harvard.edu%2Ffull%2F1996AJ....111..555F&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-Bicak-36"><span class="mw-cite-backlink"><a href="#cite_ref-Bicak_36-0">↑</a></span> <span class="reference-text"><span class="citation Journal">Jiří Bičák (2000). <a rel="nofollow" class="external text" href="http://arxiv.org/pdf/gr-qc/0004016">"Selected Solutions of Einstein's Field Equations: Their Role in General Relativity and Astrophysics, In: <i>Einstein’s Field Equations and Their Physical Implications</i>"</a>. <i>Lecture Notes in Physics</i> (Berlin: Springer Berlin Heidelberg) <b>540</b>: 1-126. doi:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-46580-4_1">10.1007/3-540-46580-4_1</a>. ISBN <a href="/wiki/Special:BookSources/978-3-540-67073-5" title="Special:BookSources/978-3-540-67073-5">978-3-540-67073-5</a><span class="printonly">. <a rel="nofollow" class="external free" href="http://arxiv.org/pdf/gr-qc/0004016">http://arxiv.org/pdf/gr-qc/0004016</a></span><span class="reference-accessdate">. Retrieved 2013-07-04</span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Selected+Solutions+of+Einstein%27s+Field+Equations%3A+Their+Role+in+General+Relativity+and+Astrophysics%2C+In%3A+%27%27Einstein%E2%80%99s+Field+Equations+and+Their+Physical+Implications%27%27&rft.jtitle=Lecture+Notes+in+Physics&rft.aulast=Ji%C5%99%C3%AD+Bi%C4%8D%C3%A1k&rft.au=Ji%C5%99%C3%AD+Bi%C4%8D%C3%A1k&rft.date=2000&rft.volume=540&rft.pages=1-126&rft.place=Berlin&rft.pub=Springer+Berlin+Heidelberg&rft_id=info:doi/10.1007%2F3-540-46580-4_1&rft.isbn=978-3-540-67073-5&rft_id=http%3A%2F%2Farxiv.org%2Fpdf%2Fgr-qc%2F0004016&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-pis-ch4-p132-37"><span class="mw-cite-backlink"><a href="#cite_ref-pis-ch4-p132_37-0">↑</a></span> <span class="reference-text"><span class="citation book">Stair, Ralph M. (2003). <i>Principles of Information Systems, Sixth Edition</i>. Thomson Learning, Inc.. pp. 132. ISBN <a href="/wiki/Special:BookSources/0-619-06489-7" title="Special:BookSources/0-619-06489-7">0-619-06489-7</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Information+Systems%2C+Sixth+Edition&rft.aulast=Stair&rft.aufirst=Ralph+M.&rft.au=Stair%2C%26%2332%3BRalph+M.&rft.date=2003&rft.pages=pp.%26nbsp%3B132&rft.pub=Thomson+Learning%2C+Inc.&rft.isbn=0-619-06489-7&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-osc-ch3-p58-38"><span class="mw-cite-backlink"><a href="#cite_ref-osc-ch3-p58_38-0">↑</a></span> <span class="reference-text"><span class="citation book">Silberschatz, Abraham (1994). <i>Operating System Concepts, Fourth Edition</i>. Addison-Wesley. pp. 58. ISBN <a href="/wiki/Special:BookSources/0-201-50480-4" title="Special:BookSources/0-201-50480-4">0-201-50480-4</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Operating+System+Concepts%2C+Fourth+Edition&rft.aulast=Silberschatz&rft.aufirst=Abraham&rft.au=Silberschatz%2C%26%2332%3BAbraham&rft.date=1994&rft.pages=pp.%26nbsp%3B58&rft.pub=Addison-Wesley&rft.isbn=0-201-50480-4&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-Feller-39"><span class="mw-cite-backlink"><a href="#cite_ref-Feller_39-0">↑</a></span> <span class="reference-text"><span class="citation book">William Feller (1968). <i>An Introduction to Probability Theory and its Applications</i>. <b>1</b>. ISBN <a href="/wiki/Special:BookSources/0-471-25708-7" title="Special:BookSources/0-471-25708-7">0-471-25708-7</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Probability+Theory+and+its+Applications&rft.aulast=William+Feller&rft.au=William+Feller&rft.date=1968&rft.volume=1&rft.isbn=0-471-25708-7&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-Dodge-40"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Dodge_40-0">39.0</a></sup> <sup><a href="#cite_ref-Dodge_40-1">39.1</a></sup></span> <span class="reference-text">Dodge, Y. (2003) <i>The Oxford Dictionary of Statistical Terms</i>, OUP. <style data-mw-deduplicate="TemplateStyles:r2527938">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:#d33}.mw-parser-output .cs1-visible-error{color:#d33}.mw-parser-output .cs1-maint{display:none;color:#3a3;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-920613-9" title="Special:BookSources/0-19-920613-9">0-19-920613-9</a></span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><a href="#cite_ref-41">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.thefreedictionary.com/dict.asp?Word=statistics">The Free Online Dictionary</a></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematics/Astronomy&veaction=edit&section=76" title="Edit section: Further reading" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mathematics/Astronomy&action=edit&section=76" title="Edit section's source code: Further reading"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation book">William Marshall Smart; Robin Michael Green (July 7, 1977). <a rel="nofollow" class="external text" href="http://books.google.com/books?id=W0f2vc2EePUC&lr=&source=gbs_navlinks_s"><i>Textbook on Spherical Astronomy, Sixth Edition</i></a>. Cambridge: University of Cambridge. pp. 431. ISBN <a href="/wiki/Special:BookSources/0_521_21516_1" title="Special:BookSources/0 521 21516 1">0 521 21516 1</a><span class="printonly">. <a rel="nofollow" class="external free" href="http://books.google.com/books?id=W0f2vc2EePUC&lr=&source=gbs_navlinks_s">http://books.google.com/books?id=W0f2vc2EePUC&lr=&source=gbs_navlinks_s</a></span><span class="reference-accessdate">. Retrieved 2012-05-18</span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Textbook+on+Spherical+Astronomy%2C+Sixth+Edition&rft.aulast=William+Marshall+Smart&rft.au=William+Marshall+Smart&rft.au=Robin+Michael+Green&rft.date=July+7%2C+1977&rft.pages=pp.%26nbsp%3B431&rft.place=Cambridge&rft.pub=University+of+Cambridge&rft.isbn=0+521+21516+1&rft_id=http%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DW0f2vc2EePUC%26lr%3D%26source%3Dgbs_navlinks_s&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></li> <li><span class="citation Journal">Tenorio-Tagle G; Bodenheimer P (1988). <a rel="nofollow" class="external text" href="http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1988ARA%26A..26..145T">"Large-scale expanding superstructures in galaxies"</a>. <i>Annual Review of Astronomy and Astrophysics</i> <b>26</b>: 145–97<span class="printonly">. <a rel="nofollow" class="external free" href="http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1988ARA%26A..26..145T">http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1988ARA%26A..26..145T</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Large-scale+expanding+superstructures+in+galaxies&rft.jtitle=Annual+Review+of+Astronomy+and+Astrophysics&rft.aulast=Tenorio-Tagle+G&rft.au=Tenorio-Tagle+G&rft.au=Bodenheimer+P&rft.date=1988&rft.volume=26&rft.pages=145%E2%80%9397&rft_id=http%3A%2F%2Farticles.adsabs.harvard.edu%2Fcgi-bin%2Fnph-iarticle_query%3F1988ARA%2526A..26..145T&rfr_id=info:sid/en.wikipedia.org:Mathematics/Astronomy"><span style="display: none;"> </span></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a 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classification/Laboratory">X-ray classification of a star</a></li> <li><a href="/wiki/X-ray_trigonometric_parallax/Laboratory" title="X-ray trigonometric parallax/Laboratory">X-ray trigonometric parallax</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="background:#FAF0BE;;width:1%;background:#FAF0BE; color:#000000;">Articles</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0;background:#FFDEAD;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Radio_Interferometer_Telescope" title="Radio Interferometer Telescope">Radio Interferometer Telescope</a></li> <li><a href="/wiki/Changes_in_the_properties_of_matter_(mass_spectrometer_and_spectral_analysis_of_stars)" class="mw-redirect" title="Changes in the properties of matter (mass spectrometer and spectral analysis of stars)">Spectral analysis of stars</a></li> <li><a href="/w/index.php?title=Vedic_mathematics&action=edit&redlink=1" class="new" title="Vedic mathematics 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href="/wiki/Category:Secondary_Math_Courses" title="Category:Secondary Math Courses">Secondary Math Courses</a></li> <li><a href="/wiki/Category:Secondary_Math_Lessons" title="Category:Secondary Math Lessons">Secondary Math Lessons</a></li> <li><a href="/w/index.php?title=Category:Secondary_Math_Quizzes&action=edit&redlink=1" class="new" title="Category:Secondary Math Quizzes (page does not exist)">Secondary Math Quizzes</a></li> <li><a href="/wiki/Category:Systems_theory" title="Category:Systems theory">Systems theory</a></li> <li><a href="/wiki/Category:Trigonometry" title="Category:Trigonometry">Trigonometry</a></li> <li><a href="/wiki/Category:Units_of_measurement" title="Category:Units of measurement">Units of measurement</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="background:#FAF0BE;;width:1%;background:#FAF0BE; color:#000000;">Courses</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0;background:#FFDEAD;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Calculus_I" class="mw-redirect" title="Calculus I">Calculus I</a></li> <li><a href="/wiki/Calculus_II" title="Calculus II">Calculus II</a></li> <li><a href="/wiki/Design_and_Analysis_of_Algorithms" title="Design and Analysis of Algorithms">Design and Analysis of Algorithms</a></li> <li><a href="/wiki/Foundations_of_mathematical_concepts" title="Foundations of mathematical concepts">Foundations of mathematical concepts</a></li> <li><a href="/wiki/Geometry" title="Geometry">Geometry</a></li> <li><a href="/wiki/Information_geometry" title="Information geometry">Information geometry</a></li> <li><a href="/wiki/Introduction_to_calculus" class="mw-redirect" title="Introduction to calculus">Introduction to calculus</a></li> <li><a href="/wiki/Introduction_to_finite_elements" class="mw-redirect" title="Introduction to finite elements">Introduction to finite elements</a></li> <li><a href="/wiki/Introduction_to_Real_Analysis" title="Introduction to Real Analysis">Introduction to Real Analysis</a></li> <li><a href="/wiki/Introduction_to_Statistical_Analysis" class="mw-redirect" title="Introduction to Statistical Analysis">Introduction to Statistical Analysis</a></li> <li><a href="/wiki/Introduction_to_Strategic_Studies" title="Introduction to Strategic Studies">Introduction to Strategic Studies</a></li> <li><a href="/wiki/Introductory_Algebra" title="Introductory Algebra">Introductory Algebra</a></li> <li><a href="/wiki/Mathematical_Methods_in_Physics" title="Mathematical Methods in Physics">Mathematical Methods in Physics</a></li> <li><a href="/wiki/Our_Playground:_The_Real_Numbers_and_Their_Development" class="mw-redirect" title="Our Playground: The Real Numbers and Their Development">Our Playground: The Real Numbers and Their Development</a></li> <li><a href="/wiki/Ordinary_differential_equations" class="mw-redirect" title="Ordinary differential equations">Ordinary differential equations</a></li> <li><a href="/wiki/The_Real_and_Complex_Number_System" title="The Real and Complex Number System">The Real and Complex Number System</a></li> <li><a href="/wiki/Vector_calculus" title="Vector calculus">Vector calculus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="background:#FAF0BE;;width:1%;background:#FAF0BE; color:#000000;">Glossaries</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0;background:#D1BEA8;"><div style="padding:0 0.25em"> <ul><li><a href="/w/index.php?title=KinderCalculus/Glossary&action=edit&redlink=1" class="new" title="KinderCalculus/Glossary (page does not exist)">KinderCalculus/Glossary</a></li> <li><a href="/wiki/Laboratory_on_Mathematics_and_Mathematics_Education/Glossary" title="Laboratory on Mathematics and Mathematics Education/Glossary">Laboratory on Mathematics and Mathematics Education/Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="background:#FAF0BE;;width:1%;background:#FAF0BE; color:#000000;">Lectures</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0;background:#FFDEAD;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_concept_generator" class="mw-redirect" title="Abstract concept generator">Abstract concept generator</a></li> <li><a href="/wiki/Actuarial_Mathematics" class="mw-redirect" title="Actuarial Mathematics">Actuarial Mathematics</a></li> <li><a href="/wiki/Actuarial_science" title="Actuarial science">Actuarial science</a></li> <li><a href="/wiki/Applied_analysis" title="Applied analysis">Applied analysis</a></li> <li><a href="/wiki/Astrophysics" title="Astrophysics">Astrophysics</a></li> <li><a href="/wiki/Calculus" title="Calculus">Calculus</a></li> <li><a href="/wiki/Christoffel_symbols" title="Christoffel symbols">Christoffel symbols</a></li> <li><a href="/wiki/Discrete_mathematics" title="Discrete mathematics">Discrete mathematics</a></li> <li><a href="/w/index.php?title=Radiation_astronomy/Empiricisms&action=edit&redlink=1" class="new" title="Radiation astronomy/Empiricisms (page does not exist)">Empirical radiation astronomy</a></li> <li><a href="/wiki/LaTeX" title="LaTeX">LaTeX</a></li> <li><a href="/wiki/Astronomy/Mathematics" class="mw-redirect" title="Astronomy/Mathematics">Mathematical astronomy</a></li> <li><a href="/wiki/Mathematical_induction" title="Mathematical induction">Mathematical induction</a></li> <li><a href="/wiki/Modelling" title="Modelling">Modelling</a></li> <li><a href="/wiki/Optimisation" title="Optimisation">Optimisation</a></li> <li><a href="/wiki/Probability" title="Probability">Probability</a></li> <li><a href="/w/index.php?title=Radiation_astronomy/Mathematics&action=edit&redlink=1" class="new" title="Radiation astronomy/Mathematics (page does not exist)">Radiation mathematics</a></li> <li><a href="/w/index.php?title=Radiation_astronomy/Scattered_disks&action=edit&redlink=1" class="new" title="Radiation astronomy/Scattered disks (page does not exist)">Scattered disc</a></li> <li><a href="/wiki/Skewness" title="Skewness">Skewness</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Taylor%27s_series" title="Taylor's series">Taylor's series</a></li> <li><a href="/wiki/Topology" title="Topology">Topology</a></li> <li><a href="/wiki/T-test" title="T-test">T-test</a></li> <li><a href="/wiki/Variable" title="Variable">Variable</a></li> <li><a href="/wiki/X-ray_trigonometric_parallax" class="mw-redirect" title="X-ray trigonometric parallax">X-ray trigonometric parallax</a></li> <li><a href="/wiki/Z-test" title="Z-test">Z-test</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="background:#FAF0BE;;width:1%;background:#FAF0BE; color:#000000;">Lessons</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0;background:#D1BEA8;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Haskell/Lesson_one" class="mw-redirect" title="Haskell/Lesson one">Haskell/Lesson one</a></li> <li><a href="/wiki/Introduction_to_calculus_-_lesson_1" title="Introduction to calculus - lesson 1">Introduction to Algebra; Introduction to Calculus./lesson01</a></li> <li><a href="/w/index.php?title=Lesson_1:_Topics_Map&action=edit&redlink=1" class="new" title="Lesson 1: Topics Map (page does not exist)">Lesson 1: Topics Map</a></li> <li><a href="/wiki/Ideas_in_Geometry/Analytic_Geometry" title="Ideas in Geometry/Analytic Geometry">Lesson Five: Analytic Geometry</a></li> <li><a href="/wiki/Factorising_quadratics" title="Factorising quadratics">Factorising quadratics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="background:#FAF0BE;;width:1%;background:#FAF0BE; color:#000000;">Lists</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0;background:#FFDEAD;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Materials_Science_and_Engineering/List_of_Topics" title="Materials Science and Engineering/List of Topics">Materials Science and Engineering/List of Topics</a></li> <li><a href="/wiki/Physics_Formulae" title="Physics Formulae">Tables of Physics Formulae</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="background:#FAF0BE;;width:1%;background:#FAF0BE; color:#000000;">Portals</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0;background:#D1BEA8;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Portal:Discrete_Mathematics_for_Computer_Science" title="Portal:Discrete Mathematics for Computer Science">Discrete Mathematics for Computer Science</a></li> <li><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="background:#FAF0BE;;width:1%;background:#FAF0BE; color:#000000;">Problem sets</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0;background:#FFDEAD;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Angular_momentum_and_energy" title="Angular momentum and energy">Angular momentum and energy</a></li> <li><a href="/wiki/Column_densities" title="Column densities">Column densities</a></li> <li><a href="/w/index.php?title=Cosmic_circuits&action=edit&redlink=1" class="new" title="Cosmic circuits (page does not exist)">Cosmic circuits</a></li> <li><a href="/wiki/Differential_equations/Assignment_1" title="Differential equations/Assignment 1">Differential equations/Assignment 1</a></li> <li><a href="/wiki/Energy_phantoms" title="Energy phantoms">Energy phantoms</a></li> <li><a href="/wiki/Furlongs_per_fortnight" title="Furlongs per fortnight">Furlongs per fortnight</a></li> <li><a href="/wiki/Lenses_and_focal_length" title="Lenses and focal length">Lenses and focal length</a></li> <li><a href="/w/index.php?title=Neutrino_emissions&action=edit&redlink=1" class="new" title="Neutrino emissions (page does not exist)">Neutrino emissions</a></li> <li><a href="/wiki/Nonlinear_finite_elements/Homework_1" title="Nonlinear finite elements/Homework 1">Nonlinear finite elements/Homework 1</a></li> <li><a href="/wiki/Planck%27s_equation" title="Planck's equation">Planck's equation</a></li> <li><a href="/w/index.php?title=Radiation_astronomy/Problem_set&action=edit&redlink=1" class="new" title="Radiation astronomy/Problem set (page does not exist)">Radiation astronomy/Problem set</a></li> <li><a href="/wiki/Radiation_dosage" title="Radiation dosage">Radiation dosage</a></li> <li><a href="/w/index.php?title=Radiation_astronomy/Mathematics/Problem_set&action=edit&redlink=1" class="new" title="Radiation astronomy/Mathematics/Problem set (page does not exist)">Radiation mathematics/Problem set</a></li> <li><a href="/wiki/Ideas_in_Geometry/Instructive_examples/Section_1.2_problem" title="Ideas in Geometry/Instructive examples/Section 1.2 problem">Section 1.2 problem</a></li> <li><a href="/wiki/Spectrographs" title="Spectrographs">Spectrographs</a></li> <li><a href="/wiki/Star_jumping" title="Star jumping">Star jumping</a></li> <li><a href="/wiki/Synchrotron_radiation/Problem_set" title="Synchrotron radiation/Problem set">Synchrotron radiation</a></li> <li><a href="/wiki/Telescopes_and_cameras" title="Telescopes and cameras">Telescopes and cameras</a></li> <li><a href="/wiki/Unknown_coordinate_systems" title="Unknown coordinate systems">Unknown coordinate systems</a></li> <li><a href="/wiki/Unusual_units" title="Unusual units">Unusual units</a></li> <li><a href="/wiki/Vectors_and_coordinates" title="Vectors and coordinates">Vectors and coordinates</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="background:#FAF0BE;;width:1%;background:#FAF0BE; color:#000000;">Projects</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0;background:#D1BEA8;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Astronomy_Project" title="Astronomy Project">Astronomy Project</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="background:#FAF0BE;;width:1%;background:#FAF0BE; color:#000000;">Quizzes</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0;background:#FFDEAD;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_concept_generator/Quiz" title="Abstract concept generator/Quiz">Abstract concept generator/Quiz</a></li> <li><a href="/wiki/Astrophysics/Quiz" title="Astrophysics/Quiz">Astrophysics/Quiz</a></li> <li><a href="/wiki/Calculus/Quiz" class="mw-redirect" title="Calculus/Quiz">Calculus/Quiz</a></li> <li><a href="/wiki/Complex_Analysis/Sample_Midterm_Exam_1" title="Complex Analysis/Sample Midterm Exam 1">Complex Analysis/Sample Midterm Exam 1</a></li> <li><a href="/w/index.php?title=Conditions/Quiz&action=edit&redlink=1" class="new" title="Conditions/Quiz (page does not exist)">Conditions/Quiz</a></li> <li><a href="/wiki/Control_groups/Quiz" title="Control groups/Quiz">Control group/Quiz</a></li> <li><a href="/wiki/Empirical_astronomy/Quiz" title="Empirical astronomy/Quiz">Empirical astronomy/Quiz</a></li> <li><a href="/w/index.php?title=Radiation_astronomy/Empiricisms/Quiz&action=edit&redlink=1" class="new" title="Radiation astronomy/Empiricisms/Quiz (page does not exist)">Empirical radiation astronomy/Quiz</a></li> <li><a href="/w/index.php?title=Logic/Quiz&action=edit&redlink=1" class="new" title="Logic/Quiz (page does not exist)">Logic/Quiz</a></li> <li><a href="/wiki/Plasmas/Magnetohydrodynamics/Quiz" title="Plasmas/Magnetohydrodynamics/Quiz">Magnetohydrodynamics/Quiz</a></li> <li><a href="/w/index.php?title=MacLaurin_series/Quiz&action=edit&redlink=1" class="new" title="MacLaurin series/Quiz (page does not exist)">MacLaurin series/Quiz</a></li> <li><a href="/wiki/Astronomy/Mathematics/Quiz" class="mw-redirect" title="Astronomy/Mathematics/Quiz">Mathematical astronomy/Quiz</a></li> <li><a href="/w/index.php?title=Mathematical_induction/Quiz&action=edit&redlink=1" class="new" title="Mathematical induction/Quiz (page does not exist)">Mathematical induction/Quiz</a></li> <li><a href="/w/index.php?title=Mathematics/Quiz&action=edit&redlink=1" class="new" title="Mathematics/Quiz (page does not exist)">Mathematics/Quiz</a></li> <li><a href="/w/index.php?title=Radiation_astronomy/Mathematics/Quiz&action=edit&redlink=1" class="new" title="Radiation astronomy/Mathematics/Quiz (page does not exist)">Radiation mathematics/Quiz</a></li> <li><a href="/w/index.php?title=Radiation_physics/Quiz&action=edit&redlink=1" class="new" title="Radiation physics/Quiz (page does not exist)">Radiation physics/Quiz</a></li> <li><a href="/w/index.php?title=Scattered_disc/Quiz&action=edit&redlink=1" class="new" title="Scattered disc/Quiz (page does not exist)">Scattered disc/Quiz</a></li> <li><a href="/wiki/X-ray_trigonometric_parallax/Quiz" title="X-ray trigonometric parallax/Quiz">X-ray trigonometric parallax/Quiz</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="background:#FAF0BE;;width:1%;background:#FAF0BE; color:#000000;">Schools</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0;background:#D1BEA8;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/School:Biomathematics" class="mw-redirect" title="School:Biomathematics">Biomathematics</a></li> <li><a href="/wiki/School:Mathematics" title="School:Mathematics">Mathematics</a></li></ul> </div></td></tr></tbody></table></div><p>{{<a href="/wiki/Template:Principles_of_radiation_astronomy" title="Template:Principles of radiation astronomy">Principles of radiation astronomy</a>}}{{<a 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