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topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+arithmetic+geometry">higher arithmetic geometry</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+arithmetic+geometry">E-∞ arithmetic geometry</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/number">number</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/natural+number">natural number</a>, <a class="existingWikiWord" href="/nlab/show/integer+number">integer number</a>, <a class="existingWikiWord" href="/nlab/show/rational+number">rational number</a>, <a class="existingWikiWord" href="/nlab/show/real+number">real number</a>, <a class="existingWikiWord" href="/nlab/show/irrational+number">irrational number</a>, <a class="existingWikiWord" href="/nlab/show/complex+number">complex number</a>, <a class="existingWikiWord" href="/nlab/show/quaternion">quaternion</a>, <a class="existingWikiWord" href="/nlab/show/octonion">octonion</a>, <a class="existingWikiWord" href="/nlab/show/adic+number">adic number</a>, <a class="existingWikiWord" href="/nlab/show/cardinal+number">cardinal number</a>, <a class="existingWikiWord" href="/nlab/show/ordinal+number">ordinal number</a>, <a class="existingWikiWord" href="/nlab/show/surreal+number">surreal number</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/arithmetic">arithmetic</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Peano+arithmetic">Peano arithmetic</a>, <a class="existingWikiWord" href="/nlab/show/second-order+arithmetic">second-order arithmetic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/transfinite+arithmetic">transfinite arithmetic</a>, <a class="existingWikiWord" href="/nlab/show/cardinal+arithmetic">cardinal arithmetic</a>, <a class="existingWikiWord" href="/nlab/show/ordinal+arithmetic">ordinal arithmetic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prime+field">prime field</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+integer">p-adic integer</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+rational+number">p-adic rational number</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+complex+number">p-adic complex number</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a></strong>, <a class="existingWikiWord" href="/nlab/show/function+field+analogy">function field analogy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+scheme">arithmetic scheme</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+curve">arithmetic curve</a>, <a class="existingWikiWord" href="/nlab/show/elliptic+curve">elliptic curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+genus">arithmetic genus</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+Chern-Simons+theory">arithmetic Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+Chow+group">arithmetic Chow group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weil-%C3%A9tale+topology+for+arithmetic+schemes">Weil-étale topology for arithmetic schemes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/absolute+cohomology">absolute cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weil+conjecture+on+Tamagawa+numbers">Weil conjecture on Tamagawa numbers</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Borger%27s+absolute+geometry">Borger's absolute geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Iwasawa-Tate+theory">Iwasawa-Tate theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/arithmetic+jet+space">arithmetic jet space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adelic+integration">adelic integration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/shtuka">shtuka</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Frobenioid">Frobenioid</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Arakelov+geometry">Arakelov geometry</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+Riemann-Roch+theorem">arithmetic Riemann-Roch theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+algebraic+K-theory">differential algebraic K-theory</a></p> </li> </ul> </div></div> <h4 id="complex_geometry">Complex geometry</h4> <div class="hide"><div> <p><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a>, <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>, <a class="existingWikiWord" href="/nlab/show/complex+line">complex line</a></p> <p><strong><a class="existingWikiWord" href="/nlab/show/complex+geometry">complex geometry</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a>, <a class="existingWikiWord" href="/nlab/show/complex+structure">complex structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+analytic+space">complex analytic space</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+supermanifold">complex supermanifold</a></p> </li> </ul> <h3 id="structures">Structures</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+complex">Dolbeault complex</a>, <a class="existingWikiWord" href="/nlab/show/holomorphic+de+Rham+complex">holomorphic de Rham complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+filtration">Hodge filtration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge-filtered+differential+cohomology">Hodge-filtered differential cohomology</a></p> </li> </ul> <h3 id="examples">Examples</h3> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">dim = 1</annotation></semantics></math>: <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a>, <a class="existingWikiWord" href="/nlab/show/super+Riemann+surface">super Riemann surface</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifold">Calabi-Yau manifold</a></p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">dim = 2</annotation></semantics></math>: <a class="existingWikiWord" href="/nlab/show/K3+surface">K3 surface</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+Calabi-Yau+manifold">generalized Calabi-Yau manifold</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#automorphisms'>Automorphisms</a></li> <li><a href='#geometry_of_complex_numbers'>Geometry of complex numbers</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <p>A <strong>complex number</strong> is a <a class="existingWikiWord" href="/nlab/show/number">number</a> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>+</mo><mi mathvariant="normal">i</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">a + \mathrm{i} b</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi mathvariant="normal">i</mi> <mn>2</mn></msup><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\mathrm{i}^2 = - 1</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/imaginary+unit">imaginary unit</a>. The set of complex numbers (in fact a <a class="existingWikiWord" href="/nlab/show/field">field</a> and <a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a>) is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math>.</p> <p>This can be thought of as:</p> <ul> <li>the vector space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^2</annotation></semantics></math> made into an <a class="existingWikiWord" href="/nlab/show/associative+algebra">algebra</a> by the rule<div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mi>c</mi><mo>−</mo><mi>b</mi><mi>d</mi><mo>,</mo><mi>a</mi><mi>d</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo><mo>;</mo></mrow><annotation encoding="application/x-tex"> (a, b) \cdot (c, d) = (a c - b d, a d + b c) ;</annotation></semantics></math></div></li> <li>the subalgebra of those <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-by-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math> real <a class="existingWikiWord" href="/nlab/show/matrix">matrices</a> of the form<div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>a</mi></mtd> <mtd><mi>b</mi></mtd></mtr> <mtr><mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>b</mi></mtd> <mtd><mi>a</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>;</mo></mrow><annotation encoding="application/x-tex"> \left(\array { a & b \\ - b & a } \right);</annotation></semantics></math></div></li> <li>the <a class="existingWikiWord" href="/nlab/show/polynomial">polynomial</a> ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo stretchy="false">[</mo><mi mathvariant="normal">x</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}[\mathrm{x}]</annotation></semantics></math> modulo <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi mathvariant="normal">x</mi> <mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\mathrm{x}^2 + 1</annotation></semantics></math>;</li> <li>the result of applying the <a class="existingWikiWord" href="/nlab/show/Cayley%E2%80%93Dickson+construction">Cayley–Dickson construction</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>;</li> <li>the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/normed+division+algebra">normed division algebra</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>;</li> <li>the <a class="existingWikiWord" href="/nlab/show/Clifford+algebra">Clifford algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Cl</mi> <mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cl_{0,1}(\mathbb{R})</annotation></semantics></math>;</li> <li>the elliptic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-dimensional algebra of <a class="existingWikiWord" href="/nlab/show/hypercomplex+numbers">hypercomplex numbers</a>;</li> <li>the <a class="existingWikiWord" href="/nlab/show/complexification">complexification</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>;</li> <li>In mathematics with <a class="existingWikiWord" href="/nlab/show/weak+countable+choice">weak countable choice</a>, the <a class="existingWikiWord" href="/nlab/show/algebraic+closure">algebraic closure</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> as a field;</li> </ul> <p>We think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/subset">subset</a> (in fact <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">{\mathbb{R}}</annotation></semantics></math>-vector subspace) of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> by identifying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>+</mo><mi mathvariant="normal">i</mi><mn>0</mn></mrow><annotation encoding="application/x-tex">a + \mathrm{i} 0</annotation></semantics></math>. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> is equipped with a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>-linear <a class="existingWikiWord" href="/nlab/show/involution">involution</a> , called <strong><a class="existingWikiWord" href="/nlab/show/complex+conjugation">complex conjugation</a></strong>, that maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">i</mi></mrow><annotation encoding="application/x-tex">\mathrm{i}</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi mathvariant="normal">i</mi><mo stretchy="false">¯</mo></mover><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi mathvariant="normal">i</mi></mrow><annotation encoding="application/x-tex">\bar{\mathrm{i}} = -\mathrm{i}</annotation></semantics></math>. Concretely, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mrow><mi>a</mi><mo>+</mo><mi mathvariant="normal">i</mi><mi>b</mi></mrow><mo>¯</mo></mover><mo>=</mo><mi>a</mi><mo>−</mo><mi mathvariant="normal">i</mi><mi>b</mi></mrow><annotation encoding="application/x-tex"> \overline{a + \mathrm{i} b} = a - \mathrm{i} b </annotation></semantics></math>.</p> <p>Complex conjugation is the nontrivial <a class="existingWikiWord" href="/nlab/show/field">field</a> <a class="existingWikiWord" href="/nlab/show/automorphism">automorphism</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> which leaves <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">{\mathbb{R}}</annotation></semantics></math> invariant. In other words, the Galois group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Gal</mi><mo stretchy="false">(</mo><mi>ℂ</mi><mo stretchy="false">/</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Gal({\mathbb{C}}/\mathbb{R})</annotation></semantics></math> is cyclic of order two and generated by complex conjugation. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> also has an <a class="existingWikiWord" href="/nlab/show/absolute+value">absolute value</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mrow><mi>a</mi><mo>+</mo><mi mathvariant="normal">i</mi><mi>b</mi></mrow><mo stretchy="false">|</mo><mo>=</mo><msqrt><mrow><msup><mi>a</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>b</mi> <mn>2</mn></msup></mrow></msqrt><mo>;</mo></mrow><annotation encoding="application/x-tex"> |{a + \mathrm{i} b}| = \sqrt{a^2 + b^2} ; </annotation></semantics></math></div> <p>notice that the absolute value of a complex number is a nonnegative real number, with</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>z</mi><msup><mo stretchy="false">|</mo> <mn>2</mn></msup><mo>=</mo><mi>z</mi><mover><mi>z</mi><mo stretchy="false">¯</mo></mover><mo>.</mo></mrow><annotation encoding="application/x-tex"> |z|^2 = z \bar{z} . </annotation></semantics></math></div> <p>Most concepts in analysis can be extended from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math>, as long as they do not rely on the order in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>. Sometimes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> even works better, either because it is algebraically closed or because of Goursat's theorem. Even when the order in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> is important, often it is enough to order the absolute values of complex numbers. See <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a> for some of the concepts whose precise definition may vary with the choice of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> (or even other possibilities).</p> <h2 id="properties">Properties</h2> <h3 id="automorphisms">Automorphisms</h3> <div class="num_prop" id="AutomorphismsOfComplexNumbersIsZ2"> <h6 id="proposition">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/automorphism+group">automorphism group</a> of the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>, as an <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> over the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a>, is <a class="existingWikiWord" href="/nlab/show/Z%2F2">Z/2</a>, <a class="existingWikiWord" href="/nlab/show/action">acting</a> by <a class="existingWikiWord" href="/nlab/show/complex+conjugation">complex conjugation</a>.</p> </div> <p>See also at <em><a href="normed+division+algebra#Automorphisms">normed division algebra – automorphism</a></em>.</p> <p>Over other subfields, the automorphism group may be considerably larger. Over the <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a>, for instance, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> has transcendence degree equal to the cardinality of the continuum, i.e., there is an algebraic extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}(X) \hookrightarrow \mathbb{C}</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>X</mi><mo stretchy="false">|</mo></mrow><mo>=</mo><mi>𝔠</mi><mo>=</mo><msup><mn>2</mn> <mrow><msub><mi>ℵ</mi> <mn>0</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">{|X|} = \mathfrak{c} = 2^{\aleph_0}</annotation></semantics></math>. Any <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X \to X</annotation></semantics></math> induces a field automorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>ℚ</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{Q}(X) \to \mathbb{Q}(X)</annotation></semantics></math> which may be extended to an automorphism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math>. Therefore the number of automorphisms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> is at least <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>2</mn> <mi>𝔠</mi></msup></mrow><annotation encoding="application/x-tex">2^\mathfrak{c}</annotation></semantics></math> (and in fact at most this as well, since the number of functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C} \to \mathbb{C}</annotation></semantics></math> is also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>2</mn> <mi>𝔠</mi></msup></mrow><annotation encoding="application/x-tex">2^\mathfrak{c}</annotation></semantics></math>).</p> <p>See also at <em><a class="existingWikiWord" href="/nlab/show/automorphism+of+the+complex+numbers">automorphism of the complex numbers</a></em>.</p> <h3 id="geometry_of_complex_numbers">Geometry of complex numbers</h3> <p>The complex numbers form a plane, the <strong>complex plane</strong>. Indeed, a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><mo>→</mo><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{C} \to \mathbb{R}^2</annotation></semantics></math> given by sending <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mi mathvariant="normal">i</mi><mi mathvariant="normal">y</mi></mrow><annotation encoding="application/x-tex">\mathrm{x} + \mathrm{i}\mathrm{y}</annotation></semantics></math> to the standard real-valued coordinates <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">x</mi><mo>,</mo><mi mathvariant="normal">y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathrm{x},\mathrm{y})</annotation></semantics></math> on this plane is a bijection. Much of <a class="existingWikiWord" href="/nlab/show/complex+analysis">complex analysis</a> can be understood through <a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a> by identifying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^2</annotation></semantics></math>, using either <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">x</mi></mrow><annotation encoding="application/x-tex">\mathrm{x}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">y</mi></mrow><annotation encoding="application/x-tex">\mathrm{y}</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">z</mi></mrow><annotation encoding="application/x-tex">\mathrm{z}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi mathvariant="normal">z</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar{\mathrm{z}}</annotation></semantics></math>. (For example, Cauchy's integral theorem is Green's/Stokes's theorem.)</p> <p>It is often convenient to use the <a class="existingWikiWord" href="/nlab/show/Alexandroff+compactification">Alexandroff compactification</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/Riemann+sphere">Riemann sphere</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><msup><mi>P</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{C}P^1</annotation></semantics></math>. One may think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><msup><mi>P</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{C}P^1</annotation></semantics></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><mo>∪</mo><mo stretchy="false">{</mo><mn>∞</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\mathbb{C} \cup \{\infty\}</annotation></semantics></math>; functions valued in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> but containing ‘poles’ may be taken to be valued in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>ℂ</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\overline{\mathbb{C}}</annotation></semantics></math>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo><mo>=</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">f(\zeta) = \infty</annotation></semantics></math> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math> is a pole of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/automorphism+of+the+complex+numbers">automorphism of the complex numbers</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/absolute+value">absolute value</a>, <a class="existingWikiWord" href="/nlab/show/phase">phase</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/real+part">real part</a>, <a class="existingWikiWord" href="/nlab/show/imaginary+part">imaginary part</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+plane">complex plane</a>, <a class="existingWikiWord" href="/nlab/show/complex+projective+space">complex projective space</a>, <a class="existingWikiWord" href="/nlab/show/Riemann+sphere">Riemann sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+vector+space">complex vector space</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/complex+line">complex line</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conjugate+transpose+matrix">conjugate transpose matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+vector+bundle">complex vector bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/complex+line+bundle">complex line bundle</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/contour+integration">contour integration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+geometry">complex geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polydisc">polydisc</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/p-adic+complex+number">p-adic complex number</a></p> </li> </ul> <div> <p><strong><a href="spin+group#ExceptionalIsomorphisms">exceptional</a> <a class="existingWikiWord" href="/nlab/show/spin+representation">spinors</a> and <a class="existingWikiWord" href="/nlab/show/real+numbers">real</a> <a class="existingWikiWord" href="/nlab/show/normed+division+algebras">normed division algebras</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/Lorentzian+spacetime">Lorentzian</a> <br /> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <br /> <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>AA</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{AA}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></th><th><a class="existingWikiWord" href="/nlab/show/normed+division+algebra">normed division algebra</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a> entry</th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>3</mn><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">3 = 2+1</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>SL</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(2,1) \simeq SL(2,\mathbb{R})</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+1-brane+in+3d">super 1-brane in 3d</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>4</mn><mo>=</mo><mn>3</mn><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">4 = 3+1</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>SL</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(3,1) \simeq SL(2, \mathbb{C})</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+2-brane+in+4d">super 2-brane in 4d</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>6</mn><mo>=</mo><mn>5</mn><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">6 = 5+1</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>5</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo></mrow><annotation encoding="application/x-tex">Spin(5,1) \simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/SL%282%2CH%29">SL(2,H)</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℍ</mi></mrow><annotation encoding="application/x-tex">\mathbb{H}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/quaternions">quaternions</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/little+string">little string</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>10</mn><mo>=</mo><mn>9</mn><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">10 = 9+1</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%289%2C1%29">Spin(9,1)</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">{\simeq}</annotation></semantics></math> “<a class="existingWikiWord" href="/nlab/show/SL%282%2CO%29">SL(2,O)</a>”</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕆</mi></mrow><annotation encoding="application/x-tex">\mathbb{O}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/octonions">octonions</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/heterotic+string">heterotic</a>/<a class="existingWikiWord" href="/nlab/show/type+II+string">type II string</a></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <p>On the history of the notion of complex numbers:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jean-Pierre+Tignol">Jean-Pierre Tignol</a>, p. 19 of: <em>Galois’ Theory of Algebraic Equations</em>, World Scientific (2001) [<a href="https://doi.org/10.1142/4628">doi:10.1142/4628</a>]</p> </li> <li> <p>Orlando Merino, <em>A Short History of Complex Numbers</em> (2006) [<a href="https://www.math.uri.edu/~merino/spring06/mth562/ShortHistoryComplexNumbers2006.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Merino-ComplexNumbers.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Leo+Corry">Leo Corry</a>, <em>A Brief History of Numbers</em>, Oxford University Press (2015) [<a href="https://global.oup.com/academic/product/a-brief-history-of-numbers-9780198702597">ISBN:9780198702597</a>]</p> </li> </ul> <p>See also:</p> <ul> <li> <p>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Complex_number">Complex numbers</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tom+Leinster">Tom Leinster</a>, <em>Objects of categories as complex numbers</em>, <a href="http://arxiv.org/abs/math/0212377">arXiv:math/0212377</a></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on April 22, 2023 at 16:59:12. 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