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viewBox="0 0 28.21 25"> <path d="M27.91,21.7L16.01,1.1c-.84-1.47-2.96-1.47-3.81,0L.3,21.7c-.85,1.47,.21,3.3,1.9,3.3H26c1.69,0,2.75-1.83,1.91-3.3ZM15.75,7.44l-.46,9.42h-2.37l-.48-9.42h3.31Zm-1.65,14.25c-1.12,0-1.87-.81-1.87-1.89s.77-1.89,1.87-1.89,1.83,.79,1.85,1.89c0,1.08-.73,1.89-1.85,1.89Z" fill="#f47821"/> <polygon points="15.77 7.47 15.31 16.89 12.94 16.89 12.46 7.47 15.77 7.47" fill="#fff"/> <path d="M14.1,21.69c-1.12,0-1.87-.81-1.87-1.89s.77-1.89,1.87-1.89,1.83,.79,1.85,1.89c0,1.08-.73,1.89-1.85,1.89Z" fill="#fff"/> </symbol> </g> </svg> </div> <div id="gl-inner-wrapper">  <header> <div class="latest-edition-header"> <div class="latest-edition-text">Explore the latest version of <em>An Elementary Introduction to the Wolfram Language</em> »</div> </div> <div class="header-inner"> <div class="eiwl-header-title"><a href="/language/elementary-introduction/2nd-ed/" id="nav-header-link" class="nav-header-link"><div class="author-name">Stephen Wolfram</div>An Elementary Introduction to the Wolfram Language<div class="edition-text">Second Edition (2017)</div></a> </div> <span class="prevnext"><a href="22-machine-learning.html" id="EIWL-WEB-2nd-ed-prev-section-header-23-more-about-numbers" class="km-track-click-no-url"><span><img src="/language/elementary-introduction/2nd-ed/img/top-previous-arrow.png" width="12" height="20" alt=""></span></a><a href="24-more-forms-of-visualization.html" id="EIWL-WEB-2nd-ed-next-section-header-23-more-about-numbers" class="km-track-click-no-url"><span><img src="/language/elementary-introduction/2nd-ed/img/top-next-arrow.png" width="13" height="21" alt=""></span></a></span> </div> </header> <div class="sidebarwrapper"> <div class="sidebarHamburger" id="sidebarHamburger"> <a href="#" id="showSidebar"><img src="/language/elementary-introduction/2nd-ed/img/hamburger.png" width="45" height="28" /></a> </div> <div class="sidebar" id="sidebar"> <div class="sidebar-close-button"> <a href="#" id="hide-sidebar-button"><img src="/language/elementary-introduction/2nd-ed/img/x-icon.png" width="13" height="13" /></a> </div> <nav id="nav" class="pageNav"><ul> <li> <a class="km-track-click-no-url tocFrontmatterLink" href="/language/elementary-introduction/2nd-ed/preface.html" id="EIWL-WEB-sidebar-preface">Preface</a> </li> <li> <a class="km-track-click-no-url tocFrontmatterLink" href="/language/elementary-introduction/2nd-ed/what-is-the-wolfram-language.html" id="EIWL-WEB-sidebar-what-is-the-wolfram-language">What Is the Wolfram Language?</a> </li> <li> <a class="km-track-click-no-url tocFrontmatterLink" href="/language/elementary-introduction/2nd-ed/practicalities-of-using-the-wolfram-language.html" id="EIWL-WEB-sidebar-practicalities-of-using-the-wolfram-language">Practicalities of Using the Wolfram Language</a> </li> <li> <a class="km-track-click-no-url tocFrontmatterLink" href="/language/elementary-introduction/2nd-ed/other-resources.html" id="EIWL-WEB-sidebar-other-resources">Other Resources</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/01-starting-out-elementary-arithmetic.html" id="EIWL-WEB-sidebar-01-starting-out-elementary-arithmetic"><span class="tocSectionNumber">1</span>Starting Out: Elementary Arithmetic</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/02-introducing-functions.html" id="EIWL-WEB-sidebar-02-introducing-functions"><span class="tocSectionNumber">2</span>Introducing Functions</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/03-first-look-at-lists.html" id="EIWL-WEB-sidebar-03-first-look-at-lists"><span class="tocSectionNumber">3</span>First Look at Lists</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/04-displaying-lists.html" id="EIWL-WEB-sidebar-04-displaying-lists"><span class="tocSectionNumber">4</span>Displaying Lists</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/05-operations-on-lists.html" id="EIWL-WEB-sidebar-05-operations-on-lists"><span class="tocSectionNumber">5</span>Operations on Lists</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/06-making-tables.html" id="EIWL-WEB-sidebar-06-making-tables"><span class="tocSectionNumber">6</span>Making Tables</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/07-colors-and-styles.html" id="EIWL-WEB-sidebar-07-colors-and-styles"><span class="tocSectionNumber">7</span>Colors and Styles</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/08-basic-graphics-objects.html" id="EIWL-WEB-sidebar-08-basic-graphics-objects"><span class="tocSectionNumber">8</span>Basic Graphics Objects</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/09-interactive-manipulation.html" id="EIWL-WEB-sidebar-09-interactive-manipulation"><span class="tocSectionNumber">9</span>Interactive Manipulation</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/10-images.html" id="EIWL-WEB-sidebar-10-images"><span class="tocSectionNumber">10</span>Images</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/11-strings-and-text.html" id="EIWL-WEB-sidebar-11-strings-and-text"><span class="tocSectionNumber">11</span>Strings and Text</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/12-sound.html" id="EIWL-WEB-sidebar-12-sound"><span class="tocSectionNumber">12</span>Sound</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/13-arrays-or-lists-of-lists.html" id="EIWL-WEB-sidebar-13-arrays-or-lists-of-lists"><span class="tocSectionNumber">13</span>Arrays, or Lists of Lists</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/14-coordinates-and-graphics.html" id="EIWL-WEB-sidebar-14-coordinates-and-graphics"><span class="tocSectionNumber">14</span>Coordinates and Graphics</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/15-the-scope-of-the-wolfram-language.html" id="EIWL-WEB-sidebar-15-the-scope-of-the-wolfram-language"><span class="tocSectionNumber">15</span>Scope of the Wolfram Language</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/16-real-world-data.html" id="EIWL-WEB-sidebar-16-real-world-data"><span class="tocSectionNumber">16</span>Real‐World Data</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/17-units.html" id="EIWL-WEB-sidebar-17-units"><span class="tocSectionNumber">17</span>Units</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/18-geocomputation.html" id="EIWL-WEB-sidebar-18-geocomputation"><span class="tocSectionNumber">18</span>Geocomputation</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/19-dates-and-times.html" id="EIWL-WEB-sidebar-19-dates-and-times"><span class="tocSectionNumber">19</span>Dates and Times</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/20-options.html" id="EIWL-WEB-sidebar-20-options"><span class="tocSectionNumber">20</span>Options</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/21-graphs-and-networks.html" id="EIWL-WEB-sidebar-21-graphs-and-networks"><span class="tocSectionNumber">21</span>Graphs and Networks</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/22-machine-learning.html" id="EIWL-WEB-sidebar-22-machine-learning"><span class="tocSectionNumber">22</span>Machine Learning</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/23-more-about-numbers.html" id="EIWL-WEB-sidebar-23-more-about-numbers"><span class="tocSectionNumber">23</span>More about Numbers</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/24-more-forms-of-visualization.html" id="EIWL-WEB-sidebar-24-more-forms-of-visualization"><span class="tocSectionNumber">24</span>More Forms of Visualization</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/25-ways-to-apply-functions.html" id="EIWL-WEB-sidebar-25-ways-to-apply-functions"><span class="tocSectionNumber">25</span>Ways to Apply Functions</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/26-pure-anonymous-functions.html" id="EIWL-WEB-sidebar-26-pure-anonymous-functions"><span class="tocSectionNumber">26</span>Pure Anonymous Functions</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/27-applying-functions-repeatedly.html" id="EIWL-WEB-sidebar-27-applying-functions-repeatedly"><span class="tocSectionNumber">27</span>Applying Functions Repeatedly</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/28-tests-and-conditionals.html" id="EIWL-WEB-sidebar-28-tests-and-conditionals"><span class="tocSectionNumber">28</span>Tests and Conditionals</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/29-more-about-pure-functions.html" id="EIWL-WEB-sidebar-29-more-about-pure-functions"><span class="tocSectionNumber">29</span>More about Pure Functions</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/30-rearranging-lists.html" id="EIWL-WEB-sidebar-30-rearranging-lists"><span class="tocSectionNumber">30</span>Rearranging Lists</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/31-parts-of-lists.html" id="EIWL-WEB-sidebar-31-parts-of-lists"><span class="tocSectionNumber">31</span>Parts of Lists</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/32-patterns.html" id="EIWL-WEB-sidebar-32-patterns"><span class="tocSectionNumber">32</span>Patterns</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/33-expressions-and-their-structure.html" id="EIWL-WEB-sidebar-33-expressions-and-their-structure"><span class="tocSectionNumber">33</span>Expressions and Their Structure</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/34-associations.html" id="EIWL-WEB-sidebar-34-associations"><span class="tocSectionNumber">34</span>Associations</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/35-natural-language-understanding.html" id="EIWL-WEB-sidebar-35-natural-language-understanding"><span class="tocSectionNumber">35</span>Natural Language Understanding</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/36-creating-websites-and-apps.html" id="EIWL-WEB-sidebar-36-creating-websites-and-apps"><span class="tocSectionNumber">36</span>Creating Websites and Apps</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/37-layout-and-display.html" id="EIWL-WEB-sidebar-37-layout-and-display"><span class="tocSectionNumber">37</span>Layout and Display</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/38-assigning-names-to-things.html" id="EIWL-WEB-sidebar-38-assigning-names-to-things"><span class="tocSectionNumber">38</span>Assigning Names to Things</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/39-immediate-and-delayed-values.html" id="EIWL-WEB-sidebar-39-immediate-and-delayed-values"><span class="tocSectionNumber">39</span>Immediate and Delayed Values</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/40-defining-your-own-functions.html" id="EIWL-WEB-sidebar-40-defining-your-own-functions"><span class="tocSectionNumber">40</span>Defining Your Own Functions</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/41-more-about-patterns.html" id="EIWL-WEB-sidebar-41-more-about-patterns"><span class="tocSectionNumber">41</span>More about Patterns</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/42-string-patterns-and-templates.html" id="EIWL-WEB-sidebar-42-string-patterns-and-templates"><span class="tocSectionNumber">42</span>String Patterns and Templates </a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/43-storing-things.html" id="EIWL-WEB-sidebar-43-storing-things"><span class="tocSectionNumber">43</span>Storing Things</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/44-importing-and-exporting.html" id="EIWL-WEB-sidebar-44-importing-and-exporting"><span class="tocSectionNumber">44</span>Importing and Exporting</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/45-datasets.html" id="EIWL-WEB-sidebar-45-datasets"><span class="tocSectionNumber">45</span>Datasets</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/46-writing-good-code.html" id="EIWL-WEB-sidebar-46-writing-good-code"><span class="tocSectionNumber">46</span>Writing Good Code</a> </li> <li> <a class="km-track-click-no-url tocChapterLink" href="/language/elementary-introduction/2nd-ed/47-debugging-your-code.html" id="EIWL-WEB-sidebar-47-debugging-your-code"><span class="tocSectionNumber">47</span>Debugging Your Code</a> </li> <li> <a class="km-track-click-no-url tocFrontmatterLink" href="/language/elementary-introduction/2nd-ed/what-we-havent-discussed.html" id="EIWL-WEB-sidebar-what-we-havent-discussed">What We Haven’t Discussed</a> </li> <li> <a class="km-track-click-no-url tocFrontmatterLink" href="/language/elementary-introduction/2nd-ed/being-a-programmer.html" id="EIWL-WEB-sidebar-being-a-programmer">Afterword: Being a Programmer</a> </li> <li> <a class="km-track-click-no-url tocFrontmatterLink" href="/language/elementary-introduction/2nd-ed/answers-to-exercises.html" id="EIWL-WEB-sidebar-answers-to-exercises">Answers to Exercises</a> </li> <li id="book-index"> <a class="km-track-click-no-url tocFrontmatterLink" href="/language/elementary-introduction/2nd-ed/book-index.html" id="EIWL-WEB-sidebar-book-index">Index</a> </li> <li class="goToPageClass"> <form action="javascript:pageURL();" autocomplete="off"><span class="goToPageText">Jump to Book Page:</span><input id="goToPageInput" type="text" size="6" onkeypress="handle(event)" /></form> </li> </ul></nav> </div> </div> <section> <div class="section-inner"> <div class="wrapper mainOuter"> <div class="mainContent"> <div> <a id="p-129" /> <table class="SectionTitleTable"> <tr><td id="SectionNumber"><span class="SectionNumber">23</span></td><td id="Section"><span class="Section">More about Numbers</span></td></tr> </table> <div class="bodytext"><a id="i-1" />When you do a computation with whole numbers, the Wolfram Language gives you an exact answer. It does the same with exact fractions.</div> <div class="CodeCaption">Adding 1/2+1/3 gives an exact answer as a fraction:</div> <div class="IO"> <table class="InCell"> <tr><td class="lab">In[1]:=</td><td class="Input"><div class="copyable-input"> <img src="Files/23-more-about-numbers-io_1-i.png.en" height="20" width="60" alt="Click for copyable input" id="EIWL-WEB-2nd-ed-show-copy-input-io_1-23-more-about-numbers" class="km-track-click-no-url" /> </div><div id="io_1_out" class="IFL"></div></td></tr> </table> <table class="OCell"> <tr><td class="lab">Out[1]=</td><td class="Output"><div class="wideimage"> <img src="Files/23-more-about-numbers-io-1-o.png.en" width="12" height="35" id="23-more-about-numbers-io-1-o" alt="" /> </div></td></tr> </table> </div> <div class="bodytext"><a id="N" /><a id="i-2" />Often you<span class="special-character CloseCurlyQuote">’</span>ll just want a numerical or decimal approximation. You can get that using the function <a href="http://reference.wolfram.com/language/ref/N.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">N</a> (for <span class="special-character OpenCurlyDoubleQuote">“</span>numerical<span class="special-character CloseCurlyDoubleQuote">”</span>).</div> <div class="CodeCaption">Get an approximate numerical answer:</div> <div class="IO"> <table class="InCell"> <tr><td class="lab">In[2]:=</td><td class="Input"><div class="copyable-input"> <img src="Files/23-more-about-numbers-io_2-i.png.en" height="20" width="79" alt="Click for copyable input" id="EIWL-WEB-2nd-ed-show-copy-input-io_2-23-more-about-numbers" class="km-track-click-no-url" /> </div><div id="io_2_out" class="IFL"></div></td></tr> </table> <table class="OCell"> <tr><td class="lab">Out[2]=</td><td class="Output"><div class="wideimage"> <img src="Files/23-more-about-numbers-io-2-o.png.en" width="56" height="19" id="23-more-about-numbers-io-2-o" alt="" /> </div></td></tr> </table> </div> <div class="bodytext">If there<span class="special-character CloseCurlyQuote">’</span>s any decimal number in your input, the Wolfram Language will automatically give you an approximate answer.</div> <div class="CodeCaption">The presence of a decimal number makes the result be approximate:</div> <div class="IO"> <table class="InCell"> <tr><td class="lab">In[3]:=</td><td class="Input"><div class="copyable-input"> <img src="Files/23-more-about-numbers-io_3-i.png.en" height="20" width="72" alt="Click for copyable input" id="EIWL-WEB-2nd-ed-show-copy-input-io_3-23-more-about-numbers" class="km-track-click-no-url" /> </div><div id="io_3_out" class="IFL"></div></td></tr> </table> <table class="OCell"> <tr><td class="lab">Out[3]=</td><td class="Output"><div class="wideimage"> <img src="Files/23-more-about-numbers-io-3-o.png.en" width="49" height="19" id="23-more-about-numbers-io-3-o" alt="" /> </div></td></tr> </table> </div> <div class="CodeCaption"><a id="i-3" />It<span class="special-character CloseCurlyQuote">’</span>s enough just to have a decimal point at the end of a number:</div> <div class="IO"> <table class="InCell"> <tr><td class="lab">In[4]:=</td><td class="Input"><div class="copyable-input"> <img src="Files/23-more-about-numbers-io_4-i.png.en" height="20" width="64" alt="Click for copyable input" id="EIWL-WEB-2nd-ed-show-copy-input-io_4-23-more-about-numbers" class="km-track-click-no-url" /> </div><div id="io_4_out" class="IFL"></div></td></tr> </table> <table class="OCell"> <tr><td class="lab">Out[4]=</td><td class="Output"><div class="wideimage"> <img src="Files/23-more-about-numbers-io-4-o.png.en" width="56" height="19" id="23-more-about-numbers-io-4-o" alt="" /> </div></td></tr> </table> </div> <div class="bodytext"><a id="i-4" />The Wolfram Language can handle numbers of any size, at least so long as they fit in your computer<span class="special-character CloseCurlyQuote">’</span>s memory. </div> <div class="CodeCaption">Here<span class="special-character CloseCurlyQuote">’</span>s 2 raised to the power 1000:</div> <div class="IO"> <table class="InCell"> <tr><td class="lab">In[5]:=</td><td class="Input"><div class="copyable-input"> <img src="Files/23-more-about-numbers-io_5-i.png.en" height="20" width="49" alt="Click for copyable input" id="EIWL-WEB-2nd-ed-show-copy-input-io_5-23-more-about-numbers" class="km-track-click-no-url" /> </div><div id="io_5_out" class="IFL"></div></td></tr> </table> <table class="OCell"> <tr><td class="lab">Out[5]=</td><td class="Output"><div class="wideimage"> <img src="Files/23-more-about-numbers-io-5-o.png.en" width="583" height="116" id="23-more-about-numbers-io-5-o" alt="" /> </div></td></tr> </table> </div> <div class="CodeCaption">Get a numerical approximation:</div> <div class="IO"> <table class="InCell"> <tr><td class="lab">In[6]:=</td><td class="Input"><div class="copyable-input"> <img src="Files/23-more-about-numbers-io_6-i.png.en" height="20" width="68" alt="Click for copyable input" id="EIWL-WEB-2nd-ed-show-copy-input-io_6-23-more-about-numbers" class="km-track-click-no-url" /> </div><div id="io_6_out" class="IFL"></div></td></tr> </table> <table class="OCell"> <tr><td class="lab">Out[6]=</td><td class="Output"><div class="wideimage"> <img src="Files/23-more-about-numbers-io-6-o.png.en" width="92" height="19" id="23-more-about-numbers-io-6-o" alt="" /> </div></td></tr> </table> </div> <div class="bodytext"><a id="p-130" /><a id="i-5" />This approximate form is given in <em>scientific notation</em>. If you need to enter scientific notation, you can do it with *^.</div> <div class="CodeCaption">Enter a number in scientific notation:</div> <div class="IO"> <table class="InCell"> <tr><td class="lab">In[7]:=</td><td class="Input"><div class="copyable-input"> <img src="Files/23-more-about-numbers-io_7-i.png.en" height="20" width="43" alt="Click for copyable input" id="EIWL-WEB-2nd-ed-show-copy-input-io_7-23-more-about-numbers" class="km-track-click-no-url" /> </div><div id="io_7_out" class="IFL"></div></td></tr> </table> <table class="OCell"> <tr><td class="lab">Out[7]=</td><td class="Output"><div class="wideimage"> <img src="Files/23-more-about-numbers-io-7-o.png.en" width="52" height="19" id="23-more-about-numbers-io-7-o" alt="" /> </div></td></tr> </table> </div> <div class="bodytext"><a id="i-6" />Commonly used numbers like <span class="special-character Pi">π</span> (pi) are built into the Wolfram Language.</div> <div class="CodeCaption">Get a numerical approximation to <span class="special-character Pi">π</span>:</div> <div class="IO"> <table class="InCell"> <tr><td class="lab">In[8]:=</td><td class="Input"><div class="copyable-input"> <img src="Files/23-more-about-numbers-io_8-i.png.en" height="20" width="33" alt="Click for copyable input" id="EIWL-WEB-2nd-ed-show-copy-input-io_8-23-more-about-numbers" class="km-track-click-no-url" /> </div><div id="io_8_out" class="IFL"></div></td></tr> </table> <table class="OCell"> <tr><td class="lab">Out[8]=</td><td class="Output"><div class="wideimage"> <img src="Files/23-more-about-numbers-io-8-o.png.en" width="49" height="19" id="23-more-about-numbers-io-8-o" alt="" /> </div></td></tr> </table> </div> <div class="bodytext"><a id="i-7" />The Wolfram Language can compute to <em>arbitrary precision</em>, so for example it can find millions of digits of <span class="special-character Pi">π</span> if you want them.</div> <div class="CodeCaption">Compute 250 digits of <span class="special-character Pi">π</span>: </div> <div class="IO"> <table class="InCell"> <tr><td class="lab">In[9]:=</td><td class="Input"><div class="copyable-input"> <img src="Files/23-more-about-numbers-io_9-i.png.en" height="20" width="63" alt="Click for copyable input" id="EIWL-WEB-2nd-ed-show-copy-input-io_9-23-more-about-numbers" class="km-track-click-no-url" /> </div><div id="io_9_out" class="IFL"></div></td></tr> </table> <table class="OCell"> <tr><td class="lab">Out[9]=</td><td class="Output"><div class="wideimage"> <img src="Files/23-more-about-numbers-io-9-o.png.en" width="589" height="92" id="23-more-about-numbers-io-9-o" alt="" /> </div></td></tr> </table> </div> <div class="bodytext"><a id="i-8" />There are many functions in the Wolfram Language that handle <em>integers</em> (whole numbers). There are also many functions that handle <em>real numbers</em><span class="special-character LongDash">—</span>approximate numbers with decimals. An example is <a href="http://reference.wolfram.com/language/ref/RandomReal.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">RandomReal</a>, which gives random real numbers.</div> <div class="CodeCaption">Generate a random real number in the range 0 to 10:</div> <div class="IO"> <table class="InCell"> <tr><td class="lab">In[10]:=</td><td class="Input"><div class="copyable-input"> <img src="Files/23-more-about-numbers-io_10-i.png.en" height="20" width="108" alt="Click for copyable input" id="EIWL-WEB-2nd-ed-show-copy-input-io_10-23-more-about-numbers" class="km-track-click-no-url" /> </div><div id="io_10_out" class="IFL"></div></td></tr> </table> <table class="OCell"> <tr><td class="lab">Out[10]=</td><td class="Output"><div class="wideimage"> <img src="Files/23-more-about-numbers-io-10-o.png.en" width="49" height="19" id="23-more-about-numbers-io-10-o" alt="" /> </div></td></tr> </table> </div> <div class="CodeCaption">Generate 5 random real numbers:</div> <div class="IO"> <table class="InCell"> <tr><td class="lab">In[11]:=</td><td class="Input"><div class="copyable-input"> <img src="Files/23-more-about-numbers-io_11-i.png.en" height="20" width="168" alt="Click for copyable input" id="EIWL-WEB-2nd-ed-show-copy-input-io_11-23-more-about-numbers" class="km-track-click-no-url" /> </div><div id="io_11_out" class="IFL"></div></td></tr> </table> <table class="OCell"> <tr><td class="lab">Out[11]=</td><td class="Output"><div class="wideimage"> <img src="Files/23-more-about-numbers-io-11-o.png.en" width="280" height="19" id="23-more-about-numbers-io-11-o" alt="" /> </div></td></tr> </table> </div> <div class="CodeCaption">An alternative way to ask for 5 random real numbers:</div> <div class="IO"> <table class="InCell"> <tr><td class="lab">In[12]:=</td><td class="Input"><div class="copyable-input"> <img src="Files/23-more-about-numbers-io_12-i.png.en" height="20" width="123" alt="Click for copyable input" id="EIWL-WEB-2nd-ed-show-copy-input-io_12-23-more-about-numbers" class="km-track-click-no-url" /> </div><div id="io_12_out" class="IFL"></div></td></tr> </table> <table class="OCell"> <tr><td class="lab">Out[12]=</td><td class="Output"><div class="wideimage"> <img src="Files/23-more-about-numbers-io-12-o.png.en" width="280" height="19" id="23-more-about-numbers-io-12-o" alt="" /> </div></td></tr> </table> </div> <div class="CodeCaption"><a id="p-131" /><a id="i-9" />Random real numbers in the range 20 to 30:</div> <div class="IO"> <table class="InCell"> <tr><td class="lab">In[13]:=</td><td class="Input"><div class="copyable-input"> <img src="Files/23-more-about-numbers-io_13-i.png.en" height="20" width="157" alt="Click for copyable input" id="EIWL-WEB-2nd-ed-show-copy-input-io_13-23-more-about-numbers" class="km-track-click-no-url" /> </div><div id="io_13_out" class="IFL"></div></td></tr> </table> <table class="OCell"> <tr><td class="lab">Out[13]=</td><td class="Output"><div class="wideimage"> <img src="Files/23-more-about-numbers-io-13-o.png.en" width="272" height="19" id="23-more-about-numbers-io-13-o" alt="" /> </div></td></tr> </table> </div> <div class="bodytext">The Wolfram Language has a huge range of mathematical functions built in, from basic to very sophisticated. </div> <div class="CodeCaption"><a id="i-10" />Find the 100th prime number:</div> <div class="IO"> <table class="InCell"> <tr><td class="lab">In[14]:=</td><td class="Input"><div class="copyable-input"> <img src="Files/23-more-about-numbers-io_14-i.png.en" height="20" width="71" alt="Click for copyable input" id="EIWL-WEB-2nd-ed-show-copy-input-io_14-23-more-about-numbers" class="km-track-click-no-url" /> </div><div id="io_14_out" class="IFL"></div></td></tr> </table> <table class="OCell"> <tr><td class="lab">Out[14]=</td><td class="Output"><div class="wideimage"> <img src="Files/23-more-about-numbers-io-14-o.png.en" width="23" height="19" id="23-more-about-numbers-io-14-o" alt="" /> </div></td></tr> </table> </div> <div class="CodeCaption">Find the millionth prime number:</div> <div class="IO"> <table class="InCell"> <tr><td class="lab">In[15]:=</td><td class="Input"><div class="copyable-input"> <img src="Files/23-more-about-numbers-io_15-i.png.en" height="20" width="106" alt="Click for copyable input" id="EIWL-WEB-2nd-ed-show-copy-input-io_15-23-more-about-numbers" class="km-track-click-no-url" /> </div><div id="io_15_out" class="IFL"></div></td></tr> </table> <table class="OCell"> <tr><td class="lab">Out[15]=</td><td class="Output"><div class="wideimage"> <img src="Files/23-more-about-numbers-io-15-o.png.en" width="64" height="19" id="23-more-about-numbers-io-15-o" alt="" /> </div></td></tr> </table> </div> <div class="CodeCaption">Plot the first 50 primes:</div> <div class="IO"> <table class="InCell"> <tr><td class="lab">In[16]:=</td><td class="Input"><div class="copyable-input"> <img src="Files/23-more-about-numbers-io_16-i.png.en" height="20" width="211" alt="Click for copyable input" id="EIWL-WEB-2nd-ed-show-copy-input-io_16-23-more-about-numbers" class="km-track-click-no-url" /> </div><div id="io_16_out" class="IFL"></div></td></tr> </table> <table class="OCell"> <tr><td class="lab">Out[16]=</td><td class="Output"><div class="wideimage"> <img src="Files/23-more-about-numbers-io-16-o.png.en" width="205" height="130" id="23-more-about-numbers-io-16-o" alt="" /> </div></td></tr> </table> </div> <div class="bodytext"><a id="i-11" />Three functions common in many practical situations are <a href="http://reference.wolfram.com/language/ref/Sqrt.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Sqrt</a> (square root), <a href="http://reference.wolfram.com/language/ref/Log10.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Log10</a> (logarithm to base 10) and <a href="http://reference.wolfram.com/language/ref/Log.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Log</a> (natural logarithm).</div> <div class="CodeCaption">The square root of 16 is 4:</div> <div class="IO"> <table class="InCell"> <tr><td class="lab">In[17]:=</td><td class="Input"><div class="copyable-input"> <img src="Files/23-more-about-numbers-io_17-i.png.en" height="20" width="53" alt="Click for copyable input" id="EIWL-WEB-2nd-ed-show-copy-input-io_17-23-more-about-numbers" class="km-track-click-no-url" /> </div><div id="io_17_out" class="IFL"></div></td></tr> </table> <table class="OCell"> <tr><td class="lab">Out[17]=</td><td class="Output"><div class="wideimage"> <img src="Files/23-more-about-numbers-io-17-o.png.en" width="8" height="19" id="23-more-about-numbers-io-17-o" alt="" /> </div></td></tr> </table> </div> <div class="CodeCaption">If you don<span class="special-character CloseCurlyQuote">’</span>t ask for a numerical approximation, you<span class="special-character CloseCurlyQuote">’</span>ll get an exact formula:</div> <div class="IO"> <table class="InCell"> <tr><td class="lab">In[18]:=</td><td class="Input"><div class="copyable-input"> <img src="Files/23-more-about-numbers-io_18-i.png.en" height="20" width="60" alt="Click for copyable input" id="EIWL-WEB-2nd-ed-show-copy-input-io_18-23-more-about-numbers" class="km-track-click-no-url" /> </div><div id="io_18_out" class="IFL"></div></td></tr> </table> <table class="OCell"> <tr><td class="lab">Out[18]=</td><td class="Output"><div class="wideimage"> <img src="Files/23-more-about-numbers-io-18-o.png.en" width="41" height="20" id="23-more-about-numbers-io-18-o" alt="" /> </div></td></tr> </table> </div> <div class="CodeCaption"><a href="http://reference.wolfram.com/language/ref/N.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">N</a> gives a numerical approximation:</div> <div class="IO"> <table class="InCell"> <tr><td class="lab">In[19]:=</td><td class="Input"><div class="copyable-input"> <img src="Files/23-more-about-numbers-io_19-i.png.en" height="20" width="79" alt="Click for copyable input" id="EIWL-WEB-2nd-ed-show-copy-input-io_19-23-more-about-numbers" class="km-track-click-no-url" /> </div><div id="io_19_out" class="IFL"></div></td></tr> </table> <table class="OCell"> <tr><td class="lab">Out[19]=</td><td class="Output"><div class="wideimage"> <img src="Files/23-more-about-numbers-io-19-o.png.en" width="49" height="19" id="23-more-about-numbers-io-19-o" alt="" /> </div></td></tr> </table> </div> <div class="bodytext"><a id="p-132" /><a id="i-12" />Logarithms are often useful when you<span class="special-character CloseCurlyQuote">’</span>re dealing with numbers that have a wide range of sizes. Let<span class="special-character CloseCurlyQuote">’</span>s plot the masses of the planets. With <a href="http://reference.wolfram.com/language/ref/ListPlot.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">ListPlot</a> one can<span class="special-character CloseCurlyQuote">’</span>t tell anything about the planets before Jupiter. But <a href="http://reference.wolfram.com/language/ref/ListLogPlot.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">ListLogPlot</a> shows the relative sizes much more clearly.</div> <div class="CodeCaption">Make an ordinary <a href="http://reference.wolfram.com/language/ref/ListPlot.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">ListPlot</a> of the masses of the planets:</div> <div class="IO"> <table class="InCell"> <tr><td class="lab">In[20]:=</td><td class="Input"><div class="copyable-input"> <img src="Files/23-more-about-numbers-io_20-i.png.en" height="26" width="256" alt="Click for copyable input" id="EIWL-WEB-2nd-ed-show-copy-input-io_20-23-more-about-numbers" class="km-track-click-no-url" /> </div><div id="io_20_out" class="IFL"></div></td></tr> </table> <table class="OCell"> <tr><td class="lab">Out[20]=</td><td class="Output"><div class="wideimage"> <img src="Files/23-more-about-numbers-io-20-o.png.en" width="194" height="112" id="23-more-about-numbers-io-20-o" alt="" /> </div></td></tr> </table> </div> <div class="CodeCaption"><a id="i-13" />Make a log plot:</div> <div class="IO"> <table class="InCell"> <tr><td class="lab">In[21]:=</td><td class="Input"><div class="copyable-input"> <img src="Files/23-more-about-numbers-io_21-i.png.en" height="26" width="279" alt="Click for copyable input" id="EIWL-WEB-2nd-ed-show-copy-input-io_21-23-more-about-numbers" class="km-track-click-no-url" /> </div><div id="io_21_out" class="IFL"></div></td></tr> </table> <table class="OCell"> <tr><td class="lab">Out[21]=</td><td class="Output"><div class="wideimage"> <img src="Files/23-more-about-numbers-io-21-o.png.en" width="195" height="123" id="23-more-about-numbers-io-21-o" alt="" /> </div></td></tr> </table> </div> <div class="bodytext"><a id="i-14" />There are a few more functions that show up frequently in general programming. First, there<span class="special-character CloseCurlyQuote">’</span>s the almost trivial function <a href="http://reference.wolfram.com/language/ref/Abs.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Abs</a>, that finds the absolute value, or positive part, of a number.</div> <div class="CodeCaption"><a href="http://reference.wolfram.com/language/ref/Abs.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Abs</a> effectively just drops minus signs:</div> <div class="IO"> <table class="InCell"> <tr><td class="lab">In[22]:=</td><td class="Input"><div class="copyable-input"> <img src="Files/23-more-about-numbers-io_22-i.png.en" height="20" width="108" alt="Click for copyable input" id="EIWL-WEB-2nd-ed-show-copy-input-io_22-23-more-about-numbers" class="km-track-click-no-url" /> </div><div id="io_22_out" class="IFL"></div></td></tr> </table> <table class="OCell"> <tr><td class="lab">Out[22]=</td><td class="Output"><div class="wideimage"> <img src="Files/23-more-about-numbers-io-22-o.png.en" width="31" height="19" id="23-more-about-numbers-io-22-o" alt="" /> </div></td></tr> </table> </div> <div class="bodytext"><a id="i-15" />Next there<span class="special-character CloseCurlyQuote">’</span>s <a href="http://reference.wolfram.com/language/ref/Round.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Round</a>, which rounds to the nearest whole number.</div> <div class="CodeCaption"><a href="http://reference.wolfram.com/language/ref/Round.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Round</a> rounds to the nearest whole number:</div> <div class="IO"> <table class="InCell"> <tr><td class="lab">In[23]:=</td><td class="Input"><div class="copyable-input"> <img src="Files/23-more-about-numbers-io_23-i.png.en" height="20" width="318" alt="Click for copyable input" id="EIWL-WEB-2nd-ed-show-copy-input-io_23-23-more-about-numbers" class="km-track-click-no-url" /> </div><div id="io_23_out" class="IFL"></div></td></tr> </table> <table class="OCell"> <tr><td class="lab">Out[23]=</td><td class="Output"><div class="wideimage"> <img src="Files/23-more-about-numbers-io-23-o.png.en" width="60" height="19" id="23-more-about-numbers-io-23-o" alt="" /> </div></td></tr> </table> </div> <div class="bodytext"><a id="p-133" /><a id="i-16" />Another function that<span class="special-character CloseCurlyQuote">’</span>s very useful is <a href="http://reference.wolfram.com/language/ref/Mod.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Mod</a>. Let<span class="special-character CloseCurlyQuote">’</span>s say you<span class="special-character CloseCurlyQuote">’</span>re counting up minutes in an hour. When you reach 60, you<span class="special-character CloseCurlyQuote">’</span>ll want to start again from 0. That<span class="special-character CloseCurlyQuote">’</span>s what <a href="http://reference.wolfram.com/language/ref/Mod.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Mod</a> lets you do. </div> <div class="CodeCaption">Compute a sequence of numbers mod 60:</div> <div class="IO"> <table class="InCell"> <tr><td class="lab">In[24]:=</td><td class="Input"><div class="copyable-input"> <img src="Files/23-more-about-numbers-io_24-i.png.en" height="20" width="419" alt="Click for copyable input" id="EIWL-WEB-2nd-ed-show-copy-input-io_24-23-more-about-numbers" class="km-track-click-no-url" /> </div><div id="io_24_out" class="IFL"></div></td></tr> </table> <table class="OCell"> <tr><td class="lab">Out[24]=</td><td class="Output"><div class="wideimage"> <img src="Files/23-more-about-numbers-io-24-o.png.en" width="97" height="19" id="23-more-about-numbers-io-24-o" alt="" /> </div></td></tr> </table> </div> <div class="VocabularySection"><a id="Vocabulary" />Vocabulary</div> <div class="VocabularyTable"> <table class="nogridboxdividers"> <col /> <col /> <col /> <tr class="lineAbove" style=""><td><a href="http://reference.wolfram.com/language/ref/N.html" class="codelink km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">N</a>[<span class="CodeVariableLrg">expr</span>]</td><td> </td><td><span class="VocabularyText">numerical approximation </span></td></tr> <tr class="" style=""><td><a href="http://reference.wolfram.com/language/ref/Pi.html" class="codelink km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Pi</a></td><td> </td><td><span class="VocabularyText">the number <span class="special-character Pi">π</span> (pi)<span class="special-character TildeEqual">≃</span>3.14 </span></td></tr> <tr class="" style=""><td><a href="http://reference.wolfram.com/language/ref/Sqrt.html" class="codelink km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Sqrt</a>[<span class="CodeVariableLrg">x</span>]</td><td> </td><td><span class="VocabularyText">square root </span></td></tr> <tr class="" style=""><td><a href="http://reference.wolfram.com/language/ref/Log10.html" class="codelink km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Log10</a>[<span class="CodeVariableLrg">x</span>]</td><td> </td><td><span class="VocabularyText">logarithm to base 10 </span></td></tr> <tr class="" style=""><td><a href="http://reference.wolfram.com/language/ref/Log.html" class="codelink km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Log</a>[<span class="CodeVariableLrg">x</span>]</td><td> </td><td><span class="VocabularyText">natural logarithm (ln) </span></td></tr> <tr class="" style=""><td><a href="http://reference.wolfram.com/language/ref/Abs.html" class="codelink km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Abs</a>[<span class="CodeVariableLrg">x</span>]</td><td> </td><td><span class="VocabularyText">absolute value (drop minus signs) </span></td></tr> <tr class="" style=""><td><a href="http://reference.wolfram.com/language/ref/Round.html" class="codelink km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Round</a>[<span class="CodeVariableLrg">x</span>]</td><td> </td><td><span class="VocabularyText">round to nearest integer </span></td></tr> <tr class="" style=""><td><a href="http://reference.wolfram.com/language/ref/Prime.html" class="codelink km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Prime</a>[<span class="CodeVariableLrg">n</span>]</td><td> </td><td><span class="VocabularyText"><span class="InlineCode"><span class="CodeVariable">n</span></span>th prime number</span></td></tr> <tr class="" style=""><td><a href="http://reference.wolfram.com/language/ref/Mod.html" class="codelink km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Mod</a>[<span class="CodeVariableLrg">x</span>,<span class="CodeVariableLrg">n</span>]</td><td> </td><td><span class="VocabularyText">modulo (<span class="special-character OpenCurlyDoubleQuote">“</span>clock arithmetic<span class="special-character CloseCurlyDoubleQuote">”</span>) </span></td></tr> <tr class="" style=""><td><a href="http://reference.wolfram.com/language/ref/RandomReal.html" class="codelink km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">RandomReal</a>[<span class="CodeVariableLrg">max</span>]</td><td> </td><td><span class="VocabularyText">random real number between 0 and <span class="InlineCode"><span class="CodeVariable">max</span></span> </span></td></tr> <tr class="" style=""><td><a href="http://reference.wolfram.com/language/ref/RandomReal.html" class="codelink km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">RandomReal</a>[<span class="CodeVariableLrg">max</span>,<span class="CodeVariableLrg">n</span>]</td><td> </td><td><span class="VocabularyText">list of <span class="InlineCode"><span class="CodeVariable">n</span></span> random real numbers</span></td></tr> <tr class="" style=""><td><a href="http://reference.wolfram.com/language/ref/ListLogPlot.html" class="codelink km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">ListLogPlot</a>[<span class="CodeVariableLrg">data</span>]</td><td> </td><td><span class="VocabularyText">plot on a logarithmic scale </span></td></tr> </table> </div> <div class="ExerciseSection" id="ExerciseSection"><a id="Exercises" />Exercises<div class="NewCheckYourAnswers"> <a href="https://lab.wolframcloud.com/app/view/openEIWL?parentFilePath=elementary-introduction/23-more-about-numbers-exercises.nb#sidebar=eiwl/23-more-about-numbers" class="km-track-click-no-url ExerciseCloudLink" target="_blank" id="EIWL-WEB-2nd-ed-new-check-your-exercises-in-cloud-23-more-about-numbers">Check your answers in the Wolfram Cloud</a> </div></div> <div class="Exercise" id="23.1"><span class="ExerciseNumber">23.1</span>Find <img src="Files/23-more-about-numbers-box-1.png.en" height="20" width="22" alt="" /> to 500-digit precision. <a href="javascript:OpenCloseExercises('23.1');" class="km-track-click-no-url ExerciseOpenCloseButton" id="EIWL-WEB-2nd-ed-show-exercise-23.1-23-more-about-numbers">»</a></div> <div class="ExerciseInputBox"> <div class="closePopup"> <a href="javascript:OpenCloseExercises('23.1');" id="EIWL-WEB-2nd-ed-close-exercise-23.1-23-more-about-numbers" class="km-track-click-no-url"><img src="img/x-static-icon.png" class="ExerciseBoxCloseImage" alt=" " /></a> </div> <div class="ExerciseInputBoxHeading"> <div class="ExerciseBoxExpectedOutput">Expected output:</div> <table class="ExerciseExpectedOutputTable"> <tr><td class="ExerciseExpectedOutputCellLabel">Out[]=</td><td><div class="ExerciseBoxExpectedOutputImg"> <img src="Files/answers-to-exercises-23.1-out.png" class="ExerciseBoxOutputImage" width="599" alt=" " /> </div></td></tr> </table> <div class="ExerciseGoToWolframCloud" id="ExerciseGoToWolframCloud"> <img src="img/open-exercise-in-cloud.png" class="ExerciseGoToWolframCloudImage" alt=" " /> <span class="ExerciseGoToWolframCloudSpan"><a href="https://lab.wolframcloud.com/app/view/openEIWL?parentFilePath=elementary-introduction/23-more-about-numbers-exercises.nb#sidebar=eiwl/23-more-about-numbers" class="km-track-click-no-url ExerciseCloudLink" target="_blank" id="EIWL-WEB-2nd-ed-goto-cloud-link-in-exercise-23.1-23-more-about-numbers">Answer & check your solution</a></span> </div> </div> </div> <div class="Exercise" id="23.2"><span class="ExerciseNumber">23.2</span>Generate 10 random real numbers between 0 and 1. <a href="javascript:OpenCloseExercises('23.2');" class="km-track-click-no-url ExerciseOpenCloseButton" id="EIWL-WEB-2nd-ed-show-exercise-23.2-23-more-about-numbers">»</a></div> <div class="ExerciseInputBox"> <div class="closePopup"> <a href="javascript:OpenCloseExercises('23.2');" id="EIWL-WEB-2nd-ed-close-exercise-23.2-23-more-about-numbers" class="km-track-click-no-url"><img src="img/x-static-icon.png" class="ExerciseBoxCloseImage" alt=" " /></a> </div> <div class="ExerciseInputBoxHeading"> <div class="ExerciseBoxExpectedOutput">Sample expected output:</div> <table class="ExerciseExpectedOutputTable"> <tr><td class="ExerciseExpectedOutputCellLabel">Out[]=</td><td><div class="ExerciseBoxExpectedOutputImg"> <img src="Files/answers-to-exercises-23.2-out.png" class="ExerciseBoxOutputImage" width="373" alt=" " /> </div></td></tr> </table> <div class="ExerciseGoToWolframCloud" id="ExerciseGoToWolframCloud"> <img src="img/open-exercise-in-cloud.png" class="ExerciseGoToWolframCloudImage" alt=" " /> <span class="ExerciseGoToWolframCloudSpan"><a href="https://lab.wolframcloud.com/app/view/openEIWL?parentFilePath=elementary-introduction/23-more-about-numbers-exercises.nb#sidebar=eiwl/23-more-about-numbers" class="km-track-click-no-url ExerciseCloudLink" target="_blank" id="EIWL-WEB-2nd-ed-goto-cloud-link-in-exercise-23.2-23-more-about-numbers">Answer & check your solution</a></span> </div> </div> </div> <div class="Exercise" id="23.3"><span class="ExerciseNumber">23.3</span>Make a plot of 200 points with random real <em>x</em> and <em>y</em> coordinates between 0 and 1. <a href="javascript:OpenCloseExercises('23.3');" class="km-track-click-no-url ExerciseOpenCloseButton" id="EIWL-WEB-2nd-ed-show-exercise-23.3-23-more-about-numbers">»</a></div> <div class="ExerciseInputBox"> <div class="closePopup"> <a href="javascript:OpenCloseExercises('23.3');" id="EIWL-WEB-2nd-ed-close-exercise-23.3-23-more-about-numbers" class="km-track-click-no-url"><img src="img/x-static-icon.png" class="ExerciseBoxCloseImage" alt=" " /></a> </div> <div class="ExerciseInputBoxHeading"> <div class="ExerciseBoxExpectedOutput">Sample expected output:</div> <table class="ExerciseExpectedOutputTable"> <tr><td class="ExerciseExpectedOutputCellLabel">Out[]=</td><td><div class="ExerciseBoxExpectedOutputImg"> <img src="Files/answers-to-exercises-23.3-out.png" class="ExerciseBoxOutputImage" width="344" alt=" " /> </div></td></tr> </table> <div class="ExerciseGoToWolframCloud" id="ExerciseGoToWolframCloud"> <img src="img/open-exercise-in-cloud.png" class="ExerciseGoToWolframCloudImage" alt=" " /> <span class="ExerciseGoToWolframCloudSpan"><a href="https://lab.wolframcloud.com/app/view/openEIWL?parentFilePath=elementary-introduction/23-more-about-numbers-exercises.nb#sidebar=eiwl/23-more-about-numbers" class="km-track-click-no-url ExerciseCloudLink" target="_blank" id="EIWL-WEB-2nd-ed-goto-cloud-link-in-exercise-23.3-23-more-about-numbers">Answer & check your solution</a></span> </div> </div> </div> <div class="Exercise" id="23.4"><span class="ExerciseNumber">23.4</span>Create a random walk using <a href="http://reference.wolfram.com/language/ref/AnglePath.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">AnglePath</a> and 1000 random real numbers between 0 and 2<span class="special-character Pi">π</span>. <a href="javascript:OpenCloseExercises('23.4');" class="km-track-click-no-url ExerciseOpenCloseButton" id="EIWL-WEB-2nd-ed-show-exercise-23.4-23-more-about-numbers">»</a></div> <div class="ExerciseInputBox"> <div class="closePopup"> <a href="javascript:OpenCloseExercises('23.4');" id="EIWL-WEB-2nd-ed-close-exercise-23.4-23-more-about-numbers" class="km-track-click-no-url"><img src="img/x-static-icon.png" class="ExerciseBoxCloseImage" alt=" " /></a> </div> <div class="ExerciseInputBoxHeading"> <div class="ExerciseBoxExpectedOutput">Sample expected output:</div> <table class="ExerciseExpectedOutputTable"> <tr><td class="ExerciseExpectedOutputCellLabel">Out[]=</td><td><div class="ExerciseBoxExpectedOutputImg"> <img src="Files/answers-to-exercises-23.4-out.png" class="ExerciseBoxOutputImage" width="337" alt=" " /> </div></td></tr> </table> <div class="ExerciseGoToWolframCloud" id="ExerciseGoToWolframCloud"> <img src="img/open-exercise-in-cloud.png" class="ExerciseGoToWolframCloudImage" alt=" " /> <span class="ExerciseGoToWolframCloudSpan"><a href="https://lab.wolframcloud.com/app/view/openEIWL?parentFilePath=elementary-introduction/23-more-about-numbers-exercises.nb#sidebar=eiwl/23-more-about-numbers" class="km-track-click-no-url ExerciseCloudLink" target="_blank" id="EIWL-WEB-2nd-ed-goto-cloud-link-in-exercise-23.4-23-more-about-numbers">Answer & check your solution</a></span> </div> </div> </div> <div class="Exercise" id="23.5"><span class="ExerciseNumber">23.5</span>Make a table of <span class="InlineCode"><a href="http://reference.wolfram.com/language/ref/Mod.html" class="codelink km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Mod</a>[n^2, 10]</span> for <span class="InlineCode">n</span> from 0 to 30. <a href="javascript:OpenCloseExercises('23.5');" class="km-track-click-no-url ExerciseOpenCloseButton" id="EIWL-WEB-2nd-ed-show-exercise-23.5-23-more-about-numbers">»</a></div> <div class="ExerciseInputBox"> <div class="closePopup"> <a href="javascript:OpenCloseExercises('23.5');" id="EIWL-WEB-2nd-ed-close-exercise-23.5-23-more-about-numbers" class="km-track-click-no-url"><img src="img/x-static-icon.png" class="ExerciseBoxCloseImage" alt=" " /></a> </div> <div class="ExerciseInputBoxHeading"> <div class="ExerciseBoxExpectedOutput">Expected output:</div> <table class="ExerciseExpectedOutputTable"> <tr><td class="ExerciseExpectedOutputCellLabel">Out[]=</td><td><div class="ExerciseBoxExpectedOutputImg"> <img src="Files/answers-to-exercises-23.5-out.png" class="ExerciseBoxOutputImage" width="405" alt=" " /> </div></td></tr> </table> <div class="ExerciseGoToWolframCloud" id="ExerciseGoToWolframCloud"> <img src="img/open-exercise-in-cloud.png" class="ExerciseGoToWolframCloudImage" alt=" " /> <span class="ExerciseGoToWolframCloudSpan"><a href="https://lab.wolframcloud.com/app/view/openEIWL?parentFilePath=elementary-introduction/23-more-about-numbers-exercises.nb#sidebar=eiwl/23-more-about-numbers" class="km-track-click-no-url ExerciseCloudLink" target="_blank" id="EIWL-WEB-2nd-ed-goto-cloud-link-in-exercise-23.5-23-more-about-numbers">Answer & check your solution</a></span> </div> </div> </div> <div class="Exercise" id="23.6"><span class="ExerciseNumber">23.6</span>Make a line plot of <span class="InlineCode"><a href="http://reference.wolfram.com/language/ref/Mod.html" class="codelink km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Mod</a>[n^n, 10]</span> for <span class="InlineCode">n</span> from 1 to 100. <a href="javascript:OpenCloseExercises('23.6');" class="km-track-click-no-url ExerciseOpenCloseButton" id="EIWL-WEB-2nd-ed-show-exercise-23.6-23-more-about-numbers">»</a></div> <div class="ExerciseInputBox"> <div class="closePopup"> <a href="javascript:OpenCloseExercises('23.6');" id="EIWL-WEB-2nd-ed-close-exercise-23.6-23-more-about-numbers" class="km-track-click-no-url"><img src="img/x-static-icon.png" class="ExerciseBoxCloseImage" alt=" " /></a> </div> <div class="ExerciseInputBoxHeading"> <div class="ExerciseBoxExpectedOutput">Expected output:</div> <table class="ExerciseExpectedOutputTable"> <tr><td class="ExerciseExpectedOutputCellLabel">Out[]=</td><td><div class="ExerciseBoxExpectedOutputImg"> <img src="Files/answers-to-exercises-23.6-out.png" class="ExerciseBoxOutputImage" width="240" alt=" " /> </div></td></tr> </table> <div class="ExerciseGoToWolframCloud" id="ExerciseGoToWolframCloud"> <img src="img/open-exercise-in-cloud.png" class="ExerciseGoToWolframCloudImage" alt=" " /> <span class="ExerciseGoToWolframCloudSpan"><a href="https://lab.wolframcloud.com/app/view/openEIWL?parentFilePath=elementary-introduction/23-more-about-numbers-exercises.nb#sidebar=eiwl/23-more-about-numbers" class="km-track-click-no-url ExerciseCloudLink" target="_blank" id="EIWL-WEB-2nd-ed-goto-cloud-link-in-exercise-23.6-23-more-about-numbers">Answer & check your solution</a></span> </div> </div> </div> <div class="Exercise" id="23.7"><span class="ExerciseNumber">23.7</span>Make a table of the first 10 powers of <span class="special-character Pi">π</span>, rounded to integers. <a href="javascript:OpenCloseExercises('23.7');" class="km-track-click-no-url ExerciseOpenCloseButton" id="EIWL-WEB-2nd-ed-show-exercise-23.7-23-more-about-numbers">»</a></div> <div class="ExerciseInputBox"> <div class="closePopup"> <a href="javascript:OpenCloseExercises('23.7');" id="EIWL-WEB-2nd-ed-close-exercise-23.7-23-more-about-numbers" class="km-track-click-no-url"><img src="img/x-static-icon.png" class="ExerciseBoxCloseImage" alt=" " /></a> </div> <div class="ExerciseInputBoxHeading"> <div class="ExerciseBoxExpectedOutput">Expected output:</div> <table class="ExerciseExpectedOutputTable"> <tr><td class="ExerciseExpectedOutputCellLabel">Out[]=</td><td><div class="ExerciseBoxExpectedOutputImg"> <img src="Files/answers-to-exercises-23.7-out.png" class="ExerciseBoxOutputImage" width="281" alt=" " /> </div></td></tr> </table> <div class="ExerciseGoToWolframCloud" id="ExerciseGoToWolframCloud"> <img src="img/open-exercise-in-cloud.png" class="ExerciseGoToWolframCloudImage" alt=" " /> <span class="ExerciseGoToWolframCloudSpan"><a href="https://lab.wolframcloud.com/app/view/openEIWL?parentFilePath=elementary-introduction/23-more-about-numbers-exercises.nb#sidebar=eiwl/23-more-about-numbers" class="km-track-click-no-url ExerciseCloudLink" target="_blank" id="EIWL-WEB-2nd-ed-goto-cloud-link-in-exercise-23.7-23-more-about-numbers">Answer & check your solution</a></span> </div> </div> </div> <div class="Exercise" id="23.8"><span class="ExerciseNumber">23.8</span>Make a graph by connecting <span class="InlineCode">n</span> with <span class="InlineCode"><a href="http://reference.wolfram.com/language/ref/Mod.html" class="codelink km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Mod</a>[n^2, 100]</span> for <span class="InlineCode">n</span> from 0 to 99. <a href="javascript:OpenCloseExercises('23.8');" class="km-track-click-no-url ExerciseOpenCloseButton" id="EIWL-WEB-2nd-ed-show-exercise-23.8-23-more-about-numbers">»</a></div> <div class="ExerciseInputBox"> <div class="closePopup"> <a href="javascript:OpenCloseExercises('23.8');" id="EIWL-WEB-2nd-ed-close-exercise-23.8-23-more-about-numbers" class="km-track-click-no-url"><img src="img/x-static-icon.png" class="ExerciseBoxCloseImage" alt=" " /></a> </div> <div class="ExerciseInputBoxHeading"> <div class="ExerciseBoxExpectedOutput">Expected output:</div> <table class="ExerciseExpectedOutputTable"> <tr><td class="ExerciseExpectedOutputCellLabel">Out[]=</td><td><div class="ExerciseBoxExpectedOutputImg"> <img src="Files/answers-to-exercises-23.8-out.png" class="ExerciseBoxOutputImage" width="240" alt=" " /> </div></td></tr> </table> <div class="ExerciseGoToWolframCloud" id="ExerciseGoToWolframCloud"> <img src="img/open-exercise-in-cloud.png" class="ExerciseGoToWolframCloudImage" alt=" " /> <span class="ExerciseGoToWolframCloudSpan"><a href="https://lab.wolframcloud.com/app/view/openEIWL?parentFilePath=elementary-introduction/23-more-about-numbers-exercises.nb#sidebar=eiwl/23-more-about-numbers" class="km-track-click-no-url ExerciseCloudLink" target="_blank" id="EIWL-WEB-2nd-ed-goto-cloud-link-in-exercise-23.8-23-more-about-numbers">Answer & check your solution</a></span> </div> </div> </div> <a id="LastCellOfAPage" /> <div class="Exercise" id="23.9"><span class="ExerciseNumber">23.9</span>Generate graphics of 50 circles with random real coordinates 0 to 10, random real radii from 0 to 2, and random colors. <a href="javascript:OpenCloseExercises('23.9');" class="km-track-click-no-url ExerciseOpenCloseButton" id="EIWL-WEB-2nd-ed-show-exercise-23.9-23-more-about-numbers">»</a></div> <div class="ExerciseInputBox"> <div class="closePopup"> <a href="javascript:OpenCloseExercises('23.9');" id="EIWL-WEB-2nd-ed-close-exercise-23.9-23-more-about-numbers" class="km-track-click-no-url"><img src="img/x-static-icon.png" class="ExerciseBoxCloseImage" alt=" " /></a> </div> <div class="ExerciseInputBoxHeading"> <div class="ExerciseBoxExpectedOutput">Sample expected output:</div> <table class="ExerciseExpectedOutputTable"> <tr><td class="ExerciseExpectedOutputCellLabel">Out[]=</td><td><div class="ExerciseBoxExpectedOutputImg"> <img src="Files/answers-to-exercises-23.9-out.png" class="ExerciseBoxOutputImage" width="240" alt=" " /> </div></td></tr> </table> <div class="ExerciseGoToWolframCloud" id="ExerciseGoToWolframCloud"> <img src="img/open-exercise-in-cloud.png" class="ExerciseGoToWolframCloudImage" alt=" " /> <span class="ExerciseGoToWolframCloudSpan"><a href="https://lab.wolframcloud.com/app/view/openEIWL?parentFilePath=elementary-introduction/23-more-about-numbers-exercises.nb#sidebar=eiwl/23-more-about-numbers" class="km-track-click-no-url ExerciseCloudLink" target="_blank" id="EIWL-WEB-2nd-ed-goto-cloud-link-in-exercise-23.9-23-more-about-numbers">Answer & check your solution</a></span> </div> </div> </div> <a id="p-134" /> <div class="Exercise" id="23.10"><span class="ExerciseNumber">23.10</span> Make a plot of the <span class="InlineCode">n</span>th prime divided by <span class="InlineCode">n*log(n)</span>, for <span class="InlineCode">n</span> from 2 to 1000. <a href="javascript:OpenCloseExercises('23.10');" class="km-track-click-no-url ExerciseOpenCloseButton" id="EIWL-WEB-2nd-ed-show-exercise-23.10-23-more-about-numbers">»</a></div> <div class="ExerciseInputBox"> <div class="closePopup"> <a href="javascript:OpenCloseExercises('23.10');" id="EIWL-WEB-2nd-ed-close-exercise-23.10-23-more-about-numbers" class="km-track-click-no-url"><img src="img/x-static-icon.png" class="ExerciseBoxCloseImage" alt=" " /></a> </div> <div class="ExerciseInputBoxHeading"> <div class="ExerciseBoxExpectedOutput">Expected output:</div> <table class="ExerciseExpectedOutputTable"> <tr><td class="ExerciseExpectedOutputCellLabel">Out[]=</td><td><div class="ExerciseBoxExpectedOutputImg"> <img src="Files/answers-to-exercises-23.10-out.png" class="ExerciseBoxOutputImage" width="312" alt=" " /> </div></td></tr> </table> <div class="ExerciseGoToWolframCloud" id="ExerciseGoToWolframCloud"> <img src="img/open-exercise-in-cloud.png" class="ExerciseGoToWolframCloudImage" alt=" " /> <span class="ExerciseGoToWolframCloudSpan"><a href="https://lab.wolframcloud.com/app/view/openEIWL?parentFilePath=elementary-introduction/23-more-about-numbers-exercises.nb#sidebar=eiwl/23-more-about-numbers" class="km-track-click-no-url ExerciseCloudLink" target="_blank" id="EIWL-WEB-2nd-ed-goto-cloud-link-in-exercise-23.10-23-more-about-numbers">Answer & check your solution</a></span> </div> </div> </div> <div class="Exercise" id="23.11"><span class="ExerciseNumber">23.11</span>Make a line plot of the differences between successive primes up to 100. <a href="javascript:OpenCloseExercises('23.11');" class="km-track-click-no-url ExerciseOpenCloseButton" id="EIWL-WEB-2nd-ed-show-exercise-23.11-23-more-about-numbers">»</a></div> <div class="ExerciseInputBox"> <div class="closePopup"> <a href="javascript:OpenCloseExercises('23.11');" id="EIWL-WEB-2nd-ed-close-exercise-23.11-23-more-about-numbers" class="km-track-click-no-url"><img src="img/x-static-icon.png" class="ExerciseBoxCloseImage" alt=" " /></a> </div> <div class="ExerciseInputBoxHeading"> <div class="ExerciseBoxExpectedOutput">Expected output:</div> <table class="ExerciseExpectedOutputTable"> <tr><td class="ExerciseExpectedOutputCellLabel">Out[]=</td><td><div class="ExerciseBoxExpectedOutputImg"> <img src="Files/answers-to-exercises-23.11-out.png" class="ExerciseBoxOutputImage" width="304" alt=" " /> </div></td></tr> </table> <div class="ExerciseGoToWolframCloud" id="ExerciseGoToWolframCloud"> <img src="img/open-exercise-in-cloud.png" class="ExerciseGoToWolframCloudImage" alt=" " /> <span class="ExerciseGoToWolframCloudSpan"><a href="https://lab.wolframcloud.com/app/view/openEIWL?parentFilePath=elementary-introduction/23-more-about-numbers-exercises.nb#sidebar=eiwl/23-more-about-numbers" class="km-track-click-no-url ExerciseCloudLink" target="_blank" id="EIWL-WEB-2nd-ed-goto-cloud-link-in-exercise-23.11-23-more-about-numbers">Answer & check your solution</a></span> </div> </div> </div> <div class="Exercise" id="23.12"><span class="ExerciseNumber">23.12</span>Generate a sequence of 20 middle C notes with random durations between 0 and 0.5 seconds. <a href="javascript:OpenCloseExercises('23.12');" class="km-track-click-no-url ExerciseOpenCloseButton" id="EIWL-WEB-2nd-ed-show-exercise-23.12-23-more-about-numbers">»</a></div> <div class="ExerciseInputBox"> <div class="closePopup"> <a href="javascript:OpenCloseExercises('23.12');" id="EIWL-WEB-2nd-ed-close-exercise-23.12-23-more-about-numbers" class="km-track-click-no-url"><img src="img/x-static-icon.png" class="ExerciseBoxCloseImage" alt=" " /></a> </div> <div class="ExerciseInputBoxHeading"> <div class="ExerciseBoxExpectedOutput">Sample expected output:</div> <table class="ExerciseExpectedOutputTable"> <tr><td class="ExerciseExpectedOutputCellLabel">Out[]=</td><td><div class="ExerciseBoxExpectedOutputImg"> <img src="Files/answers-to-exercises-23.12-out.png" class="ExerciseBoxOutputImage" width="250" alt=" " /> </div></td></tr> </table> <div class="ExerciseGoToWolframCloud" id="ExerciseGoToWolframCloud"> <img src="img/open-exercise-in-cloud.png" class="ExerciseGoToWolframCloudImage" alt=" " /> <span class="ExerciseGoToWolframCloudSpan"><a href="https://lab.wolframcloud.com/app/view/openEIWL?parentFilePath=elementary-introduction/23-more-about-numbers-exercises.nb#sidebar=eiwl/23-more-about-numbers" class="km-track-click-no-url ExerciseCloudLink" target="_blank" id="EIWL-WEB-2nd-ed-goto-cloud-link-in-exercise-23.12-23-more-about-numbers">Answer & check your solution</a></span> </div> </div> </div> <div class="Exercise" id="23.13"><span class="ExerciseNumber">23.13</span>Make an array plot of <span class="InlineCode"><a href="http://reference.wolfram.com/language/ref/Mod.html" class="codelink km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Mod</a>[i, j]</span> for <span class="InlineCode">i</span> and <span class="InlineCode">j</span> up to 50. <a href="javascript:OpenCloseExercises('23.13');" class="km-track-click-no-url ExerciseOpenCloseButton" id="EIWL-WEB-2nd-ed-show-exercise-23.13-23-more-about-numbers">»</a></div> <div class="ExerciseInputBox"> <div class="closePopup"> <a href="javascript:OpenCloseExercises('23.13');" id="EIWL-WEB-2nd-ed-close-exercise-23.13-23-more-about-numbers" class="km-track-click-no-url"><img src="img/x-static-icon.png" class="ExerciseBoxCloseImage" alt=" " /></a> </div> <div class="ExerciseInputBoxHeading"> <div class="ExerciseBoxExpectedOutput">Expected output:</div> <table class="ExerciseExpectedOutputTable"> <tr><td class="ExerciseExpectedOutputCellLabel">Out[]=</td><td><div class="ExerciseBoxExpectedOutputImg"> <img src="Files/answers-to-exercises-23.13-out.png" class="ExerciseBoxOutputImage" width="240" alt=" " /> </div></td></tr> </table> <div class="ExerciseGoToWolframCloud" id="ExerciseGoToWolframCloud"> <img src="img/open-exercise-in-cloud.png" class="ExerciseGoToWolframCloudImage" alt=" " /> <span class="ExerciseGoToWolframCloudSpan"><a href="https://lab.wolframcloud.com/app/view/openEIWL?parentFilePath=elementary-introduction/23-more-about-numbers-exercises.nb#sidebar=eiwl/23-more-about-numbers" class="km-track-click-no-url ExerciseCloudLink" target="_blank" id="EIWL-WEB-2nd-ed-goto-cloud-link-in-exercise-23.13-23-more-about-numbers">Answer & check your solution</a></span> </div> </div> </div> <div class="Exercise" id="23.14"><span class="ExerciseNumber">23.14</span>Make a list for <span class="InlineCode">n</span> from 2 to 10 of array plots for <span class="InlineCode">x</span> and <span class="InlineCode">y</span> up to 50 of <span class="InlineCode">x^y mod n</span>. <a href="javascript:OpenCloseExercises('23.14');" class="km-track-click-no-url ExerciseOpenCloseButton" id="EIWL-WEB-2nd-ed-show-exercise-23.14-23-more-about-numbers">»</a></div> <div class="ExerciseInputBox"> <div class="closePopup"> <a href="javascript:OpenCloseExercises('23.14');" id="EIWL-WEB-2nd-ed-close-exercise-23.14-23-more-about-numbers" class="km-track-click-no-url"><img src="img/x-static-icon.png" class="ExerciseBoxCloseImage" alt=" " /></a> </div> <div class="ExerciseInputBoxHeading"> <div class="ExerciseBoxExpectedOutput">Expected output:</div> <table class="ExerciseExpectedOutputTable"> <tr><td class="ExerciseExpectedOutputCellLabel">Out[]=</td><td><div class="ExerciseBoxExpectedOutputImg"> <img src="Files/answers-to-exercises-23.14-out.png" class="ExerciseBoxOutputImage" width="506" alt=" " /> </div></td></tr> </table> <div class="ExerciseGoToWolframCloud" id="ExerciseGoToWolframCloud"> <img src="img/open-exercise-in-cloud.png" class="ExerciseGoToWolframCloudImage" alt=" " /> <span class="ExerciseGoToWolframCloudSpan"><a href="https://lab.wolframcloud.com/app/view/openEIWL?parentFilePath=elementary-introduction/23-more-about-numbers-exercises.nb#sidebar=eiwl/23-more-about-numbers" class="km-track-click-no-url ExerciseCloudLink" target="_blank" id="EIWL-WEB-2nd-ed-goto-cloud-link-in-exercise-23.14-23-more-about-numbers">Answer & check your solution</a></span> </div> </div> </div> <div class="Exercise" id="x23.1"><span class="ExerciseWebNumber">+23.1</span>Use <a href="http://reference.wolfram.com/language/ref/Round.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Round</a> to compute the fractional part of <span class="special-character Pi">π</span> to 50 digits. <a href="javascript:OpenCloseExercises('x23.1');" class="km-track-click-no-url ExerciseOpenCloseButton" id="EIWL-WEB-2nd-ed-show-exercise-x23.1-23-more-about-numbers">»</a></div> <div class="ExerciseInputBox"> <div class="closePopup"> <a href="javascript:OpenCloseExercises('x23.1');" id="EIWL-WEB-2nd-ed-close-exercise-x23.1-23-more-about-numbers" class="km-track-click-no-url"><img src="img/x-static-icon.png" class="ExerciseBoxCloseImage" alt=" " /></a> </div> <div class="ExerciseInputBoxHeading"> <div class="ExerciseBoxExpectedOutput">Expected output:</div> <table class="ExerciseExpectedOutputTable"> <tr><td class="ExerciseExpectedOutputCellLabel">Out[]=</td><td><div class="ExerciseBoxExpectedOutputImg"> <img src="Files/answers-to-exercises-x23.1-out.png" class="ExerciseBoxOutputImage" width="358" alt=" " /> </div></td></tr> </table> <div class="ExerciseGoToWolframCloud" id="ExerciseGoToWolframCloud"> <img src="img/open-exercise-in-cloud.png" class="ExerciseGoToWolframCloudImage" alt=" " /> <span class="ExerciseGoToWolframCloudSpan"><a href="https://lab.wolframcloud.com/app/view/openEIWL?parentFilePath=elementary-introduction/23-more-about-numbers-exercises.nb#sidebar=eiwl/23-more-about-numbers" class="km-track-click-no-url ExerciseCloudLink" target="_blank" id="EIWL-WEB-2nd-ed-goto-cloud-link-in-exercise-x23.1-23-more-about-numbers">Answer & check your solution</a></span> </div> </div> </div> <div class="Exercise" id="x23.2"><span class="ExerciseWebNumber">+23.2</span>Find the sum of the first 1000 prime numbers. <a href="javascript:OpenCloseExercises('x23.2');" class="km-track-click-no-url ExerciseOpenCloseButton" id="EIWL-WEB-2nd-ed-show-exercise-x23.2-23-more-about-numbers">»</a></div> <div class="ExerciseInputBox"> <div class="closePopup"> <a href="javascript:OpenCloseExercises('x23.2');" id="EIWL-WEB-2nd-ed-close-exercise-x23.2-23-more-about-numbers" class="km-track-click-no-url"><img src="img/x-static-icon.png" class="ExerciseBoxCloseImage" alt=" " /></a> </div> <div class="ExerciseInputBoxHeading"> <div class="ExerciseBoxExpectedOutput">Expected output:</div> <table class="ExerciseExpectedOutputTable"> <tr><td class="ExerciseExpectedOutputCellLabel">Out[]=</td><td><div class="ExerciseBoxExpectedOutputImg"> <img src="Files/answers-to-exercises-x23.2-out.png" class="ExerciseBoxOutputImage" width="52" alt=" " /> </div></td></tr> </table> <div class="ExerciseGoToWolframCloud" id="ExerciseGoToWolframCloud"> <img src="img/open-exercise-in-cloud.png" class="ExerciseGoToWolframCloudImage" alt=" " /> <span class="ExerciseGoToWolframCloudSpan"><a href="https://lab.wolframcloud.com/app/view/openEIWL?parentFilePath=elementary-introduction/23-more-about-numbers-exercises.nb#sidebar=eiwl/23-more-about-numbers" class="km-track-click-no-url ExerciseCloudLink" target="_blank" id="EIWL-WEB-2nd-ed-goto-cloud-link-in-exercise-x23.2-23-more-about-numbers">Answer & check your solution</a></span> </div> </div> </div> <div class="Exercise" id="x23.3"><span class="ExerciseWebNumber">+23.3</span>Make a list of the first 100 primes modulo 4. <a href="javascript:OpenCloseExercises('x23.3');" class="km-track-click-no-url ExerciseOpenCloseButton" id="EIWL-WEB-2nd-ed-show-exercise-x23.3-23-more-about-numbers">»</a></div> <div class="ExerciseInputBox"> <div class="closePopup"> <a href="javascript:OpenCloseExercises('x23.3');" id="EIWL-WEB-2nd-ed-close-exercise-x23.3-23-more-about-numbers" class="km-track-click-no-url"><img src="img/x-static-icon.png" class="ExerciseBoxCloseImage" alt=" " /></a> </div> <div class="ExerciseInputBoxHeading"> <div class="ExerciseBoxExpectedOutput">Expected output:</div> <table class="ExerciseExpectedOutputTable"> <tr><td class="ExerciseExpectedOutputCellLabel">Out[]=</td><td><div class="ExerciseBoxExpectedOutputImg"> <img src="Files/answers-to-exercises-x23.3-out.png" class="ExerciseBoxOutputImage" width="465" alt=" " /> </div></td></tr> </table> <div class="ExerciseGoToWolframCloud" id="ExerciseGoToWolframCloud"> <img src="img/open-exercise-in-cloud.png" class="ExerciseGoToWolframCloudImage" alt=" " /> <span class="ExerciseGoToWolframCloudSpan"><a href="https://lab.wolframcloud.com/app/view/openEIWL?parentFilePath=elementary-introduction/23-more-about-numbers-exercises.nb#sidebar=eiwl/23-more-about-numbers" class="km-track-click-no-url ExerciseCloudLink" target="_blank" id="EIWL-WEB-2nd-ed-goto-cloud-link-in-exercise-x23.3-23-more-about-numbers">Answer & check your solution</a></span> </div> </div> </div> <div class="Exercise" id="x23.4"><span class="ExerciseWebNumber">+23.4</span>Make a list of the first 10000 primes modulo 4, multiply them by 90<span class="special-character Degree">°</span> and create an angle path from them. <a href="javascript:OpenCloseExercises('x23.4');" class="km-track-click-no-url ExerciseOpenCloseButton" id="EIWL-WEB-2nd-ed-show-exercise-x23.4-23-more-about-numbers">»</a></div> <div class="ExerciseInputBox"> <div class="closePopup"> <a href="javascript:OpenCloseExercises('x23.4');" id="EIWL-WEB-2nd-ed-close-exercise-x23.4-23-more-about-numbers" class="km-track-click-no-url"><img src="img/x-static-icon.png" class="ExerciseBoxCloseImage" alt=" " /></a> </div> <div class="ExerciseInputBoxHeading"> <div class="ExerciseBoxExpectedOutput">Expected output:</div> <table class="ExerciseExpectedOutputTable"> <tr><td class="ExerciseExpectedOutputCellLabel">Out[]=</td><td><div class="ExerciseBoxExpectedOutputImg"> <img src="Files/answers-to-exercises-x23.4-out.png" class="ExerciseBoxOutputImage" width="240" alt=" " /> </div></td></tr> </table> <div class="ExerciseGoToWolframCloud" id="ExerciseGoToWolframCloud"> <img src="img/open-exercise-in-cloud.png" class="ExerciseGoToWolframCloudImage" alt=" " /> <span class="ExerciseGoToWolframCloudSpan"><a href="https://lab.wolframcloud.com/app/view/openEIWL?parentFilePath=elementary-introduction/23-more-about-numbers-exercises.nb#sidebar=eiwl/23-more-about-numbers" class="km-track-click-no-url ExerciseCloudLink" target="_blank" id="EIWL-WEB-2nd-ed-goto-cloud-link-in-exercise-x23.4-23-more-about-numbers">Answer & check your solution</a></span> </div> </div> </div> <div class="QASection"><a id="QAndA" />Q&A</div> <div class="Question"><a id="i-17" />What are examples of mathematical functions in the Wolfram Language?</div> <div class="Answer"><a id="i-18" />From standard school math, ones like <a href="http://reference.wolfram.com/language/ref/Sin.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Sin</a>, <a href="http://reference.wolfram.com/language/ref/Cos.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Cos</a>, <a href="http://reference.wolfram.com/language/ref/ArcTan.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">ArcTan</a>, <a href="http://reference.wolfram.com/language/ref/Exp.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Exp</a>, as well as <a href="http://reference.wolfram.com/language/ref/GCD.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">GCD</a>, <a href="http://reference.wolfram.com/language/ref/Factorial.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Factorial</a>, <a href="http://reference.wolfram.com/language/ref/Fibonacci.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Fibonacci</a>. From physics and engineering and higher math, ones like <a href="http://reference.wolfram.com/language/ref/Gamma.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Gamma</a> (<span class="special-character OpenCurlyDoubleQuote">“</span>gamma function<span class="special-character CloseCurlyDoubleQuote">”</span>), <a href="http://reference.wolfram.com/language/ref/BesselJ.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">BesselJ</a> (<span class="special-character OpenCurlyDoubleQuote">“</span>Bessel function<span class="special-character CloseCurlyDoubleQuote">”</span>), <a href="http://reference.wolfram.com/language/ref/EllipticK.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">EllipticK</a> (<span class="special-character OpenCurlyDoubleQuote">“</span>elliptic integral<span class="special-character CloseCurlyDoubleQuote">”</span>), <a href="http://reference.wolfram.com/language/ref/Zeta.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Zeta</a> (<span class="special-character OpenCurlyDoubleQuote">“</span>Riemann zeta function<span class="special-character CloseCurlyDoubleQuote">”</span>), <a href="http://reference.wolfram.com/language/ref/PrimePi.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">PrimePi</a>, <a href="http://reference.wolfram.com/language/ref/EulerPhi.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">EulerPhi</a>. From statistics, ones like <a href="http://reference.wolfram.com/language/ref/Erf.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Erf</a>, <a href="http://reference.wolfram.com/language/ref/NormalDistribution.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">NormalDistribution</a>, <a href="http://reference.wolfram.com/language/ref/ChiSquareDistribution.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">ChiSquareDistribution</a>. Hundreds of functions altogether.</div> <div class="Question"><a id="i-19" />What is the precision of a number?</div> <div class="Answer">It<span class="special-character CloseCurlyQuote">’</span>s the total number of decimal digits quoted in the number. <span class="InlineCode"><a href="http://reference.wolfram.com/language/ref/N.html" class="codelink km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">N</a>[100/3, 5]</span> gives <span class="InlineCode">33.333</span>, which has 5 digits of precision. The number <span class="InlineCode">100/3</span> is exact; <span class="InlineCode"><a href="http://reference.wolfram.com/language/ref/N.html" class="codelink km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">N</a>[100/3, 5]</span> approximates it to 5-digit precision.</div> <div class="Question"><a id="i-20" />What does the <span class="InlineCode"><a href="http://reference.wolfram.com/language/ref/character/Continuation.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank"><span class="special-character Continuation"></span></a></span> at the end of each line in a long number mean?</div> <div class="Answer">It<span class="special-character CloseCurlyQuote">’</span>s there to show that the number continues onto the next line<span class="special-character LongDash">—</span>like a hyphen in text.</div> <div class="Question"><a id="i-21" />Can I work with numbers in bases other than 10?</div> <div class="Answer">Yes. Enter a number in base 16 as <span class="InlineCode">16^^ffa5</span>. Find digits using <span class="InlineCode"><a href="http://reference.wolfram.com/language/ref/IntegerDigits.html" class="codelink km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">IntegerDigits</a>[655, 16]</span>.</div> <div class="Question"><a id="i-22" />Can the Wolfram Language handle complex numbers?</div> <div class="Answer">Of course. The symbol <a href="http://reference.wolfram.com/language/ref/I.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">I</a> (capital <span class="special-character OpenCurlyDoubleQuote">“</span>i<span class="special-character CloseCurlyDoubleQuote">”</span>) represents the square root of <span class="special-character Minus">−</span>1.</div> <div class="Question">Why does <span class="InlineCode"><a href="http://reference.wolfram.com/language/ref/N.html" class="codelink km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">N</a>[1.5/7, 100]</span> not give me a 100-digit result?</div> <div class="Answer">Because <span class="InlineCode">1.5</span> is an approximate number with much less than 100-digit precision. <span class="InlineCode"><a href="http://reference.wolfram.com/language/ref/N.html" class="codelink km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">N</a>[15/70, 100]</span> will for example give a 100-digit-precision number.</div> <div class="TechNoteGroup"> <div class="TechNoteSection"><a id="TechNotes" />Tech Notes</div> <ul class="TechNoteList"> <li class="TechNote">The Wolfram Language does <span class="special-character OpenCurlyDoubleQuote">“</span>arbitrary-precision computation<span class="special-character CloseCurlyDoubleQuote">”</span>, meaning that it can keep as many digits in a number as you want.</li> <li class="TechNote"><a id="i-23" />When you generate a number with a certain precision using <a href="http://reference.wolfram.com/language/ref/N.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">N</a>, the Wolfram Language will automatically keep track of how that precision is affected by computations<span class="special-character LongDash">—</span>so you don<span class="special-character CloseCurlyQuote">’</span>t have to do your own numerical analysis of roundoff errors.</li> <li class="TechNote"><a id="i-24" />If you type a number like 1.5, it<span class="special-character CloseCurlyQuote">’</span>s assumed to be at the native <span class="special-character OpenCurlyDoubleQuote">“</span>machine precision<span class="special-character CloseCurlyDoubleQuote">”</span> of numbers on your computer (usually about 16 digits, though only 6 are usually displayed). Use <span class="InlineCode">1.5<span class="special-character InvisibleSpace"> </span>`100</span> to specify 100-digit precision.</li> <li class="TechNote">With exact input (like 4 or 2/3 or <a href="http://reference.wolfram.com/language/ref/Pi.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Pi</a>), the Wolfram Language always tries to give exact output. But if the input contains an approximate number (like 2.3), or if you use <a href="http://reference.wolfram.com/language/ref/N.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">N</a>, it<span class="special-character CloseCurlyQuote">’</span>ll use numerical approximation.</li> <li class="TechNote"><a id="i-25" />Numerical approximation is often crucial in making large-scale computations feasible.</li> <li class="TechNote"><a id="LastCellOfAPage" /><a id="i-26" /><a href="http://reference.wolfram.com/language/ref/PrimeQ.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">PrimeQ</a> tests if a number is prime (see <a href="/language/elementary-introduction/2nd-ed/28-tests-and-conditionals.html#PrimeQ" class="hyperlink km-track-click-no-url" id="EIWL-WEB-2nd-ed-external-link-23-more-about-numbers" target="_blank">Section 28</a>). <a href="http://reference.wolfram.com/language/ref/FactorInteger.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">FactorInteger</a> finds the factors of an integer.</li> <li class="TechNote"><a id="p-135" /><a id="i-27" /><a href="http://reference.wolfram.com/language/ref/RandomReal.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">RandomReal</a> can give numbers that aren<span class="special-character CloseCurlyQuote">’</span>t just uniformly distributed. For example, <span class="InlineCode"><a href="http://reference.wolfram.com/language/ref/RandomReal.html" class="codelink km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">RandomReal</a>[<a href="http://reference.wolfram.com/language/ref/NormalDistribution.html" class="codelink km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">NormalDistribution</a>[ ]]</span> gives normally distributed numbers.</li> <li class="TechNote"><a id="i-28" /><a href="http://reference.wolfram.com/language/ref/Round.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Round</a> rounds to the nearest integer (up or down); <a href="http://reference.wolfram.com/language/ref/Floor.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Floor</a> always rounds down; <a href="http://reference.wolfram.com/language/ref/Ceiling.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">Ceiling</a> always rounds up.</li> <li class="TechNote"><a id="i-29" /><a href="http://reference.wolfram.com/language/ref/RealDigits.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">RealDigits</a> is the analog of <a href="http://reference.wolfram.com/language/ref/IntegerDigits.html" class="link km-track-click-no-url" id="EIWL-WEB-2nd-ed-function-link-23-more-about-numbers" target="_blank">IntegerDigits</a> for real numbers.</li> </ul> </div> <div class="MoreExploreGroup"> <div class="MoreExploreSection"><a id="MoreToExplore" />More to Explore</div> <div class="MoreExploreList"> <div class="MoreExplore"> <a href="http://reference.wolfram.com/language/guide/NumericalEvaluationAndPrecision.html" class="hyperlink km-track-click-no-url" id="EIWL-WEB-2nd-ed-external-link-23-more-about-numbers" target="_blank">Guide to Numbers in the Wolfram Language<span class="special-character NonBreakingSpace"> </span><span class="special-character RightGuillemet">»</span></a> </div> <div class="MoreExplore"> <a id="LastCellOfAPage" /> <a href="http://reference.wolfram.com/language/guide/MathematicalFunctions.html" class="hyperlink km-track-click-no-url" id="EIWL-WEB-2nd-ed-external-link-23-more-about-numbers" target="_blank">Guide to Mathematical Functions in the Wolfram Language<span class="special-character NonBreakingSpace"> </span><span class="special-character RightGuillemet">»</span></a> </div> </div> </div> </div> </div> <div class="bottomSpacerCell"> </div> </div> </div> </section> <div class="GoToCloudVersion" id="GoToCloudVersion"><a href="23-more-about-numbers-cloud.html" class="km-track-click-no-url" id="EIWL-WEB-2nd-ed-switch-to-cloud-embed-23-more-about-numbers"><img src="img/switch-to-cloud.png" alt="" ></a></div> <div class="GoToHTMLVersion" id="GoToHTMLVersion"><a href="23-more-about-numbers.html" class="km-track-click-no-url" id="EIWL-WEB-2nd-ed-switch-to-html-23-more-about-numbers"><img src="img/switch-to-html.png" alt="" ></a></div> <a href="javascript:nbdownload('EIWL2-23-more-about-numbers.nb');" class="km-track-click-no-url" id="EIWL-WEB-2nd-ed-download-notebook-23-more-about-numbers"><div class="DownloadNotebook" id="DownloadNotebook"><span class='DownloadNotebookText'>Download Notebook Version</span></div></a> <a href="22-machine-learning.html" id="EIWL-WEB-2nd-ed-prev-section-footer-23-more-about-numbers" class="km-track-click-no-url"><div class="previousFooter"></div></a> <a href="24-more-forms-of-visualization.html" id="EIWL-WEB-2nd-ed-next-section-footer-23-more-about-numbers" class="km-track-click-no-url"><div class="nextFooter"></div></a>  </div> </div> <span id="gl-footer-lang-message"><i class="gl-icon gl-r7-c2"></i></span> <span id="gl-footer-lang"></span><span id="gl-footer-lang-links"><a href="/language/elementary-introduction/2nd-ed/23-more-about-numbers.html.es?footer=lang"><img src="/common/images/gl-lang-es.png" alt="es"></a> </span> <script src="/common/javascript/gl-footer-scripts.en.js"></script> <div id ="IPstripe-wrap"></div> <script src="/common/stripe/stripe.en.js"></script>  <script src="/common/javascript/kissmetrics.js"></script> <script src="/language/elementary-introduction/2nd-ed/js/chapter-cookie.js.en"></script> <script src="https://www.wolframcdn.com/notebook-embedder/0.1/wolfram-notebook-embedder.js"></script> </body> </html>