SherbyNet - /page/wikibook/entry.mu

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<title>Imaginary number</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Imaginary number</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">"Imaginary Numbers" redirects here. For the 2013 EP by The Maine, see <a href="Imaginary_Numbers_(EP)" title="Imaginary Numbers (EP)">Imaginary Numbers (EP)</a>.</div>

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<th>The powers of <span class="texhtml mvar" style="font-style:italic;">i</span><br> are cyclic:

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<annotation encoding="application/x-tex">{\displaystyle \ \vdots }</annotation>

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</math></span><img src="./4b0139441a4bdb878ce9d2fdcbb07964d6539016.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.227ex; height:3.676ex;" alt="{\displaystyle \ \vdots }" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle \ i^{-2}=-1{\phantom {i}}}</annotation>

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</math></span><img src="./5f71c3c87b8f21759a073c6df2e774fe2e01a59b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.587ex; height:2.843ex;" alt="{\displaystyle \ i^{-2}=-1{\phantom {i}}}" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle \ i^{-1}=-i{\phantom {1}}}</annotation>

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</math></span><img src="./994eff39c54fa4b015cf7f56a5e3775e93cffaf5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.587ex; height:2.843ex;" alt="{\displaystyle \ i^{-1}=-i{\phantom {1}}}" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle \ \ i^{0}\ ={\phantom {-}}1{\phantom {i}}}</annotation>

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</math></span><img src="./1c9b3fa0e665a99ee17f6d8b524f93d6fcdc8339.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.47ex; height:2.843ex;" alt="{\displaystyle \ \ i^{0}\ ={\phantom {-}}1{\phantom {i}}}" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle \ \ i^{1}\ ={\phantom {-}}i{\phantom {1}}}</annotation>

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</math></span><img src="./9e18ea5e045cae2e1abf1796a59ffd534f710fcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.47ex; height:2.843ex;" alt="{\displaystyle \ \ i^{1}\ ={\phantom {-}}i{\phantom {1}}}" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle \ \ i^{2}\ =-1{\phantom {i}}}</annotation>

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</math></span><img src="./fa510188b3fea0ce4e91e4f32db153094e67520d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.47ex; height:2.843ex;" alt="{\displaystyle \ \ i^{2}\ =-1{\phantom {i}}}" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle \ \ i^{3}\ =-i{\phantom {1}}}</annotation>

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</math></span><img src="./cf4da6ebd144486f31e7b49fe06b0f2a9fdb192f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.47ex; height:2.843ex;" alt="{\displaystyle \ \ i^{3}\ =-i{\phantom {1}}}" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle \ \ i^{4}\ ={\phantom {-}}1{\phantom {i}}}</annotation>

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</math></span><img src="./e1767cce1945b75d4e028a7bcdf0ba0206db7f44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.47ex; height:2.843ex;" alt="{\displaystyle \ \ i^{4}\ ={\phantom {-}}1{\phantom {i}}}" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle \ \ i^{5}\ ={\phantom {-}}i{\phantom {1}}}</annotation>

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</math></span><img src="./296f79eb3818ea12202ba4affd19f38622cdd855.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.47ex; height:2.843ex;" alt="{\displaystyle \ \ i^{5}\ ={\phantom {-}}i{\phantom {1}}}" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle \ \vdots }</annotation>

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</math></span><img src="./4b0139441a4bdb878ce9d2fdcbb07964d6539016.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.227ex; height:3.676ex;" alt="{\displaystyle \ \vdots }" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

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</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> is a 4th<br> <a href="Root_of_unity" title="Root of unity">root of unity</a>

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<p>An <b>imaginary number</b> is the product of a <a href="Real_number" title="Real number">real number</a> and the <a href="Imaginary_unit" title="Imaginary unit">imaginary unit</a> <span class="texhtml mvar" style="font-style:italic;">i</span>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> which is defined by its property <span class="texhtml"><i>i</i><sup>2</sup> = −1</span>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The <a href="Square_(algebra)" title="Square (algebra)">square</a> of an imaginary number <span class="texhtml mvar" style="font-style:italic;">bi</span> is <span class="texhtml">−<i>b</i><sup>2</sup></span>. For example, <span class="texhtml">5<i>i</i></span> is an imaginary number, and its square is <span class="texhtml">−25</span>. The number <a href="0" title="0">zero</a> is considered to be both real and imaginary.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>Originally coined in the 17th century by <a href="Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> as a derogatory term and regarded as fictitious or useless, the concept gained wide acceptance following the work of <a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> (in the 18th century) and <a href="Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a> and <a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> (in the early 19th century).

</p><p>An imaginary number <span class="texhtml"><i>bi</i></span> can be added to a real number <span class="texhtml mvar" style="font-style:italic;">a</span> to form a <a href="Complex_number" title="Complex number">complex number</a> of the form <span class="texhtml"><i>a</i> + <i>bi</i></span>, where the real numbers <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> are called, respectively, the <i>real part</i> and the <i>imaginary part</i> of the complex number.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="History_of_complex_numbers" class="mw-redirect" title="History of complex numbers">History of complex numbers</a></div>

<p>Although the Greek <a href="Mathematician" title="Mathematician">mathematician</a> and <a href="Engineer" title="Engineer">engineer</a> <a href="Heron_of_Alexandria" class="mw-redirect" title="Heron of Alexandria">Heron of Alexandria</a> is noted as the first to present a calculation involving the square root of a negative number,<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> it was <a href="Rafael_Bombelli" title="Rafael Bombelli">Rafael Bombelli</a> who first set down the rules for multiplication of <a href="Complex_number" title="Complex number">complex numbers</a> in 1572. The concept had appeared in print earlier, such as in work by <a href="Gerolamo_Cardano" title="Gerolamo Cardano">Gerolamo Cardano</a>. At the time, imaginary numbers and negative numbers were poorly understood and were regarded by some as fictitious or useless, much as zero once was. Many other mathematicians were slow to adopt the use of imaginary numbers, including <a href="Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a>, who wrote about them in his <i><a href="La_G%C3%A9om%C3%A9trie" title="La Géométrie">La Géométrie</a></i> in which he coined the term <i>imaginary</i> and meant it to be derogatory.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Martinez_10-0" class="reference"><a href="#cite_note-Martinez-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The use of imaginary numbers was not widely accepted until the work of <a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> (1707–1783) and <a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> (1777–1855). The geometric significance of complex numbers as points in a plane was first described by <a href="Caspar_Wessel" title="Caspar Wessel">Caspar Wessel</a> (1745–1818).<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>

</p><p>In 1843, <a href="William_Rowan_Hamilton" title="William Rowan Hamilton">William Rowan Hamilton</a> extended the idea of an axis of imaginary numbers in the plane to a four-dimensional space of <a href="Quaternion#Definition" title="Quaternion">quaternion imaginaries</a> in which three of the dimensions are analogous to the imaginary numbers in the complex field.

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<div class="mw-heading mw-heading2"><h2 id="Geometric_interpretation">Geometric interpretation</h2></div>

<p>Geometrically, imaginary numbers are found on the vertical axis of the <a href="Complex_plane" title="Complex plane">complex number plane</a>, which allows them to be presented <a href="Perpendicular" title="Perpendicular">perpendicular</a> to the real axis. One way of viewing imaginary numbers is to consider a standard <a href="Number_line" title="Number line">number line</a> positively increasing in magnitude to the right and negatively increasing in magnitude to the left. At 0 on the <span class="texhtml mvar" style="font-style:italic;">x</span>-axis, a <span class="texhtml mvar" style="font-style:italic;">y</span>-axis can be drawn with "positive" direction going up; "positive" imaginary numbers then increase in magnitude upwards, and "negative" imaginary numbers increase in magnitude downwards. This vertical axis is often called the "imaginary axis"<sup id="cite_ref-Meier_12-0" class="reference"><a href="#cite_note-Meier-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> and is denoted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\mathbb {R} ,}">

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<annotation encoding="application/x-tex">{\displaystyle i\mathbb {R} ,}</annotation>

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</math></span><img src="./4101e29dac4163483908e013cb36da367ff57077.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.127ex; height:2.509ex;" alt="{\displaystyle i\mathbb {R} ,}" loading="lazy"></span> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {I} ,}">

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</math></span><img src="./b9a597bf0d4ea70e457a2cab15df7975d456d977.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.551ex; height:2.509ex;" alt="{\displaystyle \mathbb {I} ,}" loading="lazy"></span> or <span class="texhtml">ℑ</span>.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>

</p><p>In this representation, multiplication by <span class="texhtml mvar" style="font-style:italic;">i</span> corresponds to a counterclockwise <a href="Rotation" title="Rotation">rotation</a> of 90 degrees about the origin, which is a quarter of a circle. Multiplication by <span class="texhtml">−<i>i</i></span> corresponds to a clockwise rotation of 90 degrees about the origin. Similarly, multiplying by a purely imaginary number <span class="texhtml mvar" style="font-style:italic;">bi</span>, with <span class="texhtml mvar" style="font-style:italic;">b</span> a real number, both causes a counterclockwise rotation about the origin by 90 degrees and scales the answer by a factor of <span class="texhtml mvar" style="font-style:italic;">b</span>. When <span class="texhtml"><i>b</i> < 0</span>, this can instead be described as a clockwise rotation by 90 degrees and a scaling by <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>b</i></span>|</span>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Square_roots_of_negative_numbers">Square roots of negative numbers</h2></div>

<p>Care must be used when working with imaginary numbers that are expressed as the <a href="Principal_value" title="Principal value">principal values</a> of the <a href="Square_root" title="Square root">square roots</a> of <a href="Negative_number" title="Negative number">negative numbers</a>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> For example, if <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span> are both positive real numbers, the following chain of equalities appears reasonable at first glance:

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<mtext> (fallacy) </mtext>

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<annotation encoding="application/x-tex">{\displaystyle \textstyle {\sqrt {x\cdot y{\vphantom {t}}}}={\sqrt {(-x)\cdot (-y)}}\mathrel {\stackrel {\text{ (fallacy) }}{=}} {\sqrt {-x{\vphantom {ty}}}}\cdot {\sqrt {-y{\vphantom {ty}}}}=i{\sqrt {x{\vphantom {ty}}}}\cdot i{\sqrt {y{\vphantom {ty}}}}=-{\sqrt {x\cdot y{\vphantom {ty}}}}\,.}</annotation>

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</math></span><img src="./025a42201c002a69c9ba688f02db44a32cca020a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-right: -0.961ex; width:68.706ex; height:4.176ex;" alt="{\displaystyle \textstyle {\sqrt {x\cdot y{\vphantom {t}}}}={\sqrt {(-x)\cdot (-y)}}\mathrel {\stackrel {\text{ (fallacy) }}{=}} {\sqrt {-x{\vphantom {ty}}}}\cdot {\sqrt {-y{\vphantom {ty}}}}=i{\sqrt {x{\vphantom {ty}}}}\cdot i{\sqrt {y{\vphantom {ty}}}}=-{\sqrt {x\cdot y{\vphantom {ty}}}}\,.}" loading="lazy"></span></dd></dl>

<p>But the result is clearly nonsense. The step where the square root was broken apart was illegitimate. (See <a href="Mathematical_fallacy" title="Mathematical fallacy">Mathematical fallacy</a>.)

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<li><a href="Dual_number" title="Dual number">Dual number</a></li>

<li><a href="Split-complex_number" title="Split-complex number">Split-complex number</a></li></ul>

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<caption><a href="Number_system" class="mw-redirect" title="Number system">Number systems</a>

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<td><a href="Complex_number" title="Complex number">Complex</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :\;\mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle :\;\mathbb {C} }</annotation>

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<annotation encoding="application/x-tex">{\displaystyle :\;\mathbb {R} }</annotation>

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<td><a href="Rational_number" title="Rational number">Rational</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :\;\mathbb {Q} }">

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<annotation encoding="application/x-tex">{\displaystyle :\;\mathbb {Q} }</annotation>

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</math></span><img src="./9f77b368ade52a03084dad12fba5b25129cebe0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.745ex; height:2.509ex;" alt="{\displaystyle :\;\mathbb {Q} }" loading="lazy"></span>

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<td><a href="Integer" title="Integer">Integer</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :\;\mathbb {Z} }">

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<annotation encoding="application/x-tex">{\displaystyle :\;\mathbb {Z} }</annotation>

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<td><a href="Natural_number" title="Natural number">Natural</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :\;\mathbb {N} }">

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<annotation encoding="application/x-tex">{\displaystyle :\;\mathbb {N} }</annotation>

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</math></span><img src="./51ba123110cb54a0b89909e10845ed2ee8c52e8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.615ex; height:2.176ex;" alt="{\displaystyle :\;\mathbb {N} }" loading="lazy"></span>

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<td><a href="Zero" class="mw-redirect" title="Zero">Zero</a>: 0

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<td><a href="One" class="mw-redirect" title="One">One</a>: 1

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<td><a href="Prime_number" title="Prime number">Prime numbers</a>

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<td><a href="Composite_number" title="Composite number">Composite numbers</a>

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<td><a href="Negative_integer" class="mw-redirect" title="Negative integer">Negative integers</a>

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<td><a href="Fraction" title="Fraction">Fraction</a>

</td>

<td>

<td><a href="Finite_decimal" class="mw-redirect" title="Finite decimal">Finite decimal</a>

</td></tr>

<tr>

<td><a href="Dyadic_rational" title="Dyadic rational">Dyadic (finite binary)</a>

</td></tr>

<tr>

<td><a href="Repeating_decimal" title="Repeating decimal">Repeating decimal</a>

</td>

<td>

</td></tr></tbody></table>

</td></tr>

<tr>

<td>

<table>

<td><a href="Irrational_number" title="Irrational number">Irrational</a>

</td>

<td>

<td><a href="Algebraic_number" title="Algebraic number">Algebraic irrational</a>

</td></tr>

<tr>

<td><a href="Period_(algebraic_geometry)" title="Period (algebraic geometry)">Irrational period</a>

</td></tr>

<tr>

<td><a href="Transcendental_number" title="Transcendental number">Transcendental</a>

</td></tr></tbody></table>

</td></tr>

<tr>

<td>

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><span class="texhtml mvar" style="font-style:italic;">j</span> is usually used in engineering contexts where <span class="texhtml mvar" style="font-style:italic;">i</span> has other meanings (such as electrical current)</span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

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</style><cite id="CITEREFUno_Ingard1988" class="citation book cs1">Uno Ingard, K. (1988). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SGVfGIewvxkC&pg=PA38">"Chapter 2"</a>. <i>Fundamentals of Waves and Oscillations</i>. Cambridge University Press. p. 38. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-33957-X</bdi>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ImaginaryNumber.html">"Imaginary Number"</a>. <i>mathworld.wolfram.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-10</span></span>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFSinha2008" class="citation book cs1">Sinha, K.C. (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mqdzqbPYiAUC&pg=SA11-PA2"><i>A Text Book of Mathematics Class XI</i></a> (Second ed.). Rastogi Publications. p. 11.2. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-81-7133-912-9</bdi>.</cite></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFGiaquintaModica2004" class="citation book cs1">Giaquinta, Mariano; Modica, Giuseppe (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Z6q4EDRMC2UC"><i>Mathematical Analysis: Approximation and Discrete Processes</i></a> (illustrated ed.). Springer Science & Business Media. p. 121. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8176-4337-9</bdi>.</cite> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Z6q4EDRMC2UC&pg=PA121">Extract of page 121</a></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFAufmannBarkerNation2009" class="citation book cs1">Aufmann, Richard; Barker, Vernon C.; Nation, Richard (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=fjRa8Koq-RgC&pg=PA66"><i>College Algebra: Enhanced Edition</i></a> (6th ed.). Cengage Learning. p. 66. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4390-4379-0</bdi>.</cite></span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFHargittai1992" class="citation book cs1">Hargittai, István (1992). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-Tt37ajV5ZgC&pg=PA153"><i>Fivefold Symmetry</i></a> (2 ed.). World Scientific. p. 153. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>981-02-0600-3</bdi>.</cite></span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFRoy2007" class="citation book cs1">Roy, Stephen Campbell (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=J-2BRbFa5IkC"><i>Complex Numbers: lattice simulation and zeta function applications</i></a>. Horwood. p. 1. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-904275-25-1</bdi>.</cite></span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a href="Ren%C3%A9_Descartes" title="René Descartes">Descartes, René</a>, <i>Discours de la méthode</i> (Leiden, (Netherlands): Jan Maire, 1637), appended book: <i>La Géométrie</i>, book three, p. 380. <a rel="nofollow" class="external text" href="http://gallica.bnf.fr/ark:/12148/btv1b86069594/f464.item.zoom">From page 380:</a> <i>"Au reste tant les vrayes racines que les fausses ne sont pas tousjours reelles; mais quelquefois seulement imaginaires; c'est a dire qu'on peut bien tousjours en imaginer autant que jay dit en chasque Equation; mais qu'il n'y a quelquefois aucune quantité, qui corresponde a celles qu'on imagine, comme encore qu'on en puisse imaginer trois en celle cy, x<sup>3</sup> – 6xx + 13x – 10 = 0, il n'y en a toutefois qu'une reelle, qui est 2, & pour les deux autres, quoy qu'on les augmente, ou diminue, ou multiplie en la façon que je viens d'expliquer, on ne sçauroit les rendre autres qu'imaginaires."</i> (Moreover, the true roots as well as the false [roots] are not always real; but sometimes only imaginary [quantities]; that is to say, one can always imagine as many of them in each equation as I said; but there is sometimes no quantity that corresponds to what one imagines, just as although one can imagine three of them in this [equation], x<sup>3</sup> – 6xx + 13x – 10 = 0, only one of them however is real, which is 2, and regarding the other two, although one increase, or decrease, or multiply them in the manner that I just explained, one would not be able to make them other than imaginary [quantities].)</span>

</li>

<li id="cite_note-Martinez-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-Martinez_10-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFMartinez2006" class="citation cs2">Martinez, Albert A. (2006), <i>Negative Math: How Mathematical Rules Can Be Positively Bent</i>, Princeton: Princeton University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-691-12309-8</bdi></cite>, discusses ambiguities of meaning in imaginary expressions in historical context.</span>

</li>

<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><cite id="CITEREFRozenfeld1988" class="citation book cs1">Rozenfeld, Boris Abramovich (1988). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DRLpAFZM7uwC&pg=PA382">"Chapter 10"</a>. <i>A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space</i>. Springer. p. 382. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-96458-4</bdi>.</cite></span>

</li>

<li id="cite_note-Meier-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Meier_12-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFvon_Meier2006" class="citation book cs1">von Meier, Alexandra (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bWAi22IB3lkC"><i>Electric Power Systems – A Conceptual Introduction</i></a>. <a href="John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. pp. <span class="nowrap">61–</span>62. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-17859-4</bdi><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-01-13</span></span>.</cite></span>

</li>

<li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><cite id="CITEREFWebb2018" class="citation book cs1">Webb, Stephen (2018). "5. Meaningless marks on paper". <i>Clash of Symbols – A Ride Through the Riches of Glyphs</i>. <a href="Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer Science+Business Media</a>. pp. <span class="nowrap">204–</span>205. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-71350-2_5">10.1007/978-3-319-71350-2_5</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-319-71350-2</bdi>.</cite></span>

</li>

<li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><cite id="CITEREFKuipers1999" class="citation book cs1">Kuipers, J. B. (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_2sS4mC0p-EC&pg=PA10"><i>Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality</i></a>. <a href="Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>. pp. <span class="nowrap">10–</span>11. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-691-10298-8</bdi><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-01-13</span></span>.</cite></span>

</li>

<li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><cite id="CITEREFNahin2010" class="citation book cs1">Nahin, Paul J. (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PflwJdPhBlEC"><i>An Imaginary Tale: The Story of "i" [the square root of minus one]</i></a>. Princeton University Press. p. 12. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4008-3029-9</bdi>.</cite> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PflwJdPhBlEC&pg=PA12">Extract of page 12</a></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2></div>

<ul><li><cite id="CITEREFNahin1998" class="citation book cs1">Nahin, Paul (1998). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/imaginarytales00nahi"><i>An Imaginary Tale: the Story of the Square Root of −1</i></a></span>. Princeton: Princeton University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-691-02795-1</bdi>.</cite>, explains many applications of imaginary expressions.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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<div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/Special:Search/imaginary_number" class="extiw external" title="wiktionary:Special:Search/imaginary number">imaginary number</a></b></i> in Wiktionary, the free dictionary.</div></div>

</div>

<ul><li><a rel="nofollow" class="external text" href="https://www.math.toronto.edu/mathnet/answers/imagexist.html">How can one show that imaginary numbers really do exist?</a> – an article that discusses the existence of imaginary numbers.</li>

<li><a rel="nofollow" class="external text" href="https://www.bbc.co.uk/radio4/science/5numbers4.shtml">5Numbers programme 4</a> – BBC Radio 4 programme</li>

<li><a rel="nofollow" class="external text" href="http://www2.dsu.nodak.edu/users/mberg/Imaginary/imaginary.htm">Why Use Imaginary Numbers?</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190825172656/http://www2.dsu.nodak.edu/users/mberg/Imaginary/imaginary.htm">Archived</a> 2019-08-25 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a> – Basic Explanation and Uses of Imaginary Numbers</li></ul>

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</style><div id="Complex_numbers19" style="font-size:114%;margin:0 4em"><a href="Complex_number" title="Complex number">Complex numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Complex_conjugate" title="Complex conjugate">Complex conjugate</a></li>

<li><a href="Complex_plane" title="Complex plane">Complex plane</a></li>

<li><a href="Real_number" title="Real number">Real number</a></li>

<li><a href="Unit_complex_number" class="mw-redirect" title="Unit complex number">Unit complex number</a></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Number_systems351" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Number_systems351" style="font-size:114%;margin:0 4em"><a href="Number" title="Number">Number</a> systems</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sets of <a href="Definable_number" class="mw-redirect" title="Definable number">definable numbers</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Natural_number" title="Natural number">Natural numbers</a> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation>

</semantics>

</math></span><img src="./fdf9a96b565ea202d0f4322e9195613fb26a9bed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }" loading="lazy"></span>)</li>

<li><a href="Integer" title="Integer">Integers</a> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

</semantics>

</math></span><img src="./449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy"></span>)</li>

<li><a href="Rational_number" title="Rational number">Rational numbers</a> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation>

</semantics>

</math></span><img src="./c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy"></span>)</li>

<li><a href="Constructible_number" title="Constructible number">Constructible numbers</a></li>

<li><a href="Algebraic_number" title="Algebraic number">Algebraic numbers</a> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} }</annotation>

</semantics>

</math></span><img src="./3fb423c16a5f403edbaf66438b75e7a36e725af6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {A} }" loading="lazy"></span>)</li>

<li><a href="Closed-form_expression#Closed-form_number" title="Closed-form expression">Closed-form numbers</a></li>

<li><a href="Period_(algebraic_geometry)" title="Period (algebraic geometry)">Periods</a> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation>

</semantics>

</math></span><img src="./10d6ec962de5797ba4f161c40e66dca74ae95cc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}" loading="lazy"></span>)</li>

<li><a href="Computable_number" title="Computable number">Computable numbers</a></li>

<li><a href="Definable_real_number#Definability_in_arithmetic" title="Definable real number">Arithmetical numbers</a></li>

<li><a href="Definable_real_number#Definability_in_models_of_ZFC" title="Definable real number">Set-theoretically definable numbers</a></li>

<li><a href="Gaussian_integer" title="Gaussian integer">Gaussian integers</a>

<ul><li><a href="Gaussian_rational" title="Gaussian rational">Gaussian rationals</a></li></ul></li>

<li><a href="Eisenstein_integer" title="Eisenstein integer">Eisenstein integers</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Composition_algebra" title="Composition algebra">Composition algebras</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Division_algebra" title="Division algebra">Division algebras</a>: <a href="Real_number" title="Real number">Real numbers</a> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"></span>)</li>

<li><a href="Complex_number" title="Complex number">Complex numbers</a> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

</semantics>

</math></span><img src="./f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy"></span>)</li>

<li><a href="Quaternion" title="Quaternion">Quaternions</a> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {H} }</annotation>

</semantics>

</math></span><img src="./e050965453c42bcc6bd544546703c836bdafeac9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {H} }" loading="lazy"></span>)</li>

<li><a href="Octonion" title="Octonion">Octonions</a> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {O} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {O} }</annotation>

</semantics>

</math></span><img src="./c1ed2664a4fe515e6fbed25a7193ce663b82920c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {O} }" loading="lazy"></span>)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Split<br>types</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"></span>:</li>

<li><a href="Split-complex_number" title="Split-complex number">Split-complex numbers</a></li>

<li><a href="Split-quaternion" title="Split-quaternion">Split-quaternions</a></li>

<li><a href="Split-octonion" title="Split-octonion">Split-octonions</a><br> Over <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

</semantics>

</math></span><img src="./f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy"></span>:</li>

<li><a href="Bicomplex_number" title="Bicomplex number">Bicomplex numbers</a></li>

<li><a href="Biquaternion" title="Biquaternion">Biquaternions</a></li>

<li><a href="Bioctonion" title="Bioctonion">Bioctonions</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other <a href="Hypercomplex_number" title="Hypercomplex number">hypercomplex</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dual_number" title="Dual number">Dual numbers</a></li>

<li><a href="Dual_quaternion" title="Dual quaternion">Dual quaternions</a></li>

<li><a href="Dual-complex_number" class="mw-redirect" title="Dual-complex number">Dual-complex numbers</a></li>

<li><a href="Hyperbolic_quaternion" title="Hyperbolic quaternion">Hyperbolic quaternions</a></li>

<li><a href="Sedenion" title="Sedenion">Sedenions</a>  (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation>

</semantics>

</math></span><img src="./9f9d5874c5d7f68eba1cec9da9ccbe53903303bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }" loading="lazy"></span>)</li>

<li><a href="Trigintaduonion" title="Trigintaduonion">Trigintaduonions</a>  (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {T} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {T} }</annotation>

</semantics>

</math></span><img src="./9c039979935c00b3b216cbb065999207872677f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {T} }" loading="lazy"></span>)</li>

<li><a href="Split-biquaternion" title="Split-biquaternion">Split-biquaternions</a></li>

<li><a href="Multicomplex_number" title="Multicomplex number">Multicomplex numbers</a></li>

<li><a href="Geometric_algebra" title="Geometric algebra">Geometric algebra</a>/<a href="Clifford_algebra" title="Clifford algebra">Clifford algebra</a>

<ul><li><a href="Algebra_of_physical_space" title="Algebra of physical space">Algebra of physical space</a></li>

<li><a href="Spacetime_algebra" title="Spacetime algebra">Spacetime algebra</a></li>

<li><a href="Plane-based_geometric_algebra" title="Plane-based geometric algebra">Plane-based geometric algebra</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Infinity" title="Infinity">Infinities</a> and <a href="Infinitesimal" title="Infinitesimal">infinitesimals</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cardinal_number" title="Cardinal number">Cardinal numbers</a></li>

<li><a href="Extended_natural_numbers" title="Extended natural numbers">Extended natural numbers</a></li>

<li><a href="Extended_real_number_line" title="Extended real number line">Extended real numbers</a>

<ul><li><a href="Projectively_extended_real_line" title="Projectively extended real line">Projective</a></li></ul></li>

<li><a href="Riemann_sphere" title="Riemann sphere">Extended complex numbers</a></li>

<li><a href="Hyperreal_number" title="Hyperreal number">Hyperreal numbers</a></li>

<li><a href="Levi-Civita_field" title="Levi-Civita field">Levi-Civita field</a></li>

<li><a href="Ordinal_number" title="Ordinal number">Ordinal numbers</a></li>

<li><a href="Supernatural_number" title="Supernatural number">Supernatural numbers</a></li>

<li><a href="Surreal_number" title="Surreal number">Surreal numbers</a></li>

<li><a href="Superreal_number" title="Superreal number">Superreal numbers</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other types</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Irrational_number" title="Irrational number">Irrational numbers</a></li>

<li><a href="Fuzzy_number" title="Fuzzy number">Fuzzy numbers</a></li>

<li><a href="Transcendental_number" title="Transcendental number">Transcendental numbers</a></li>

<li><a href="P-adic_number" title="P-adic number"><span class="nowrap"><i>p</i>-adic</span> numbers</a> (<a href="Solenoid_(mathematics)#p-adic_solenoids" title="Solenoid (mathematics)"><span class="nowrap"><i>p</i>-adic</span> solenoids</a>)</li>

<li><a href="Profinite_integer" title="Profinite integer">Profinite integers</a></li>

<li><a href="Normal_number" title="Normal number">Normal numbers</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div>

<ul><li><a href="Number#Main_classification" title="Number">Classification</a></li>

</div></td></tr></tbody></table></div>

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This article is issued from <a class="external text" title="Last edited on 2025-08-03" href="https://en.wikipedia.org/wiki/?title=Imaginary_number&oldid=1303940829">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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