View source for Complex number

{{Short description|Number with a real and an imaginary part}}
{{pp-move}}

[[File:A plus bi.svg|thumb|upright=1.15|right|A complex number can be visually represented as a pair of numbers {{math|(''a'', ''b'')}} forming a vector on a diagram called an Argand diagram, representing the [[Complex plane|complex plane]]. ''Re'' is the real axis, ''Im'' is the imaginary axis, and {{mvar|i}} is the "[[Imaginary unit|imaginary unit]]", that satisfies {{math|1=''i''<sup>2</sup> = −1}}.]]

In [[Mathematics|mathematics]], a '''complex number''' is an element of a number system that extends the [[Real number|real number]]s with a specific element denoted {{mvar|i}}, called the [[Imaginary unit|imaginary unit]] and satisfying the [[Equation|equation]] <math>i^{2}= -1</math>; every complex number can be expressed in the form <math>a + bi</math>, where {{mvar|a}} and {{mvar|b}} are real numbers. Because no real number satisfies the above equation, {{mvar|i}} was called an [[Imaginary number|imaginary number]] by [[Biography:René Descartes|René Descartes]]. For the complex number {{nowrap|<math>a+bi</math>,}} {{mvar|a}} is called the '''{{visible anchor|real part}}''', and {{mvar|b}} is called the '''{{visible anchor|imaginary part}}'''. The set of complex numbers is denoted by either of the symbols <math>\mathbb C</math> or {{math|'''C'''}}. Despite the historical nomenclature "imaginary", complex numbers are regarded in the [[Mathematical sciences|mathematical sciences]] as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world.<ref>For an extensive account of the history of "imaginary" numbers, from initial skepticism to ultimate acceptance, see {{cite book |last=Bourbaki |first=Nicolas |year=1998 |title=Elements of the History of Mathematics |chapter=Foundations of Mathematics § Logic: Set theory |pages=18–24 |publisher=Springer}}
</ref>{{efn| "Complex numbers, as much as reals, and perhaps even more, find a unity with nature that is truly remarkable. It is as though Nature herself is as impressed by the scope and consistency of the complex-number system as we are ourselves, and has entrusted to these numbers the precise operations of her world at its minutest scales." — R. Penrose (2016, <!-- [https://books.google.com/books?id=VWTNCwAAQBAJ&pg=PA73 ] --> p.&nbsp;73)<ref>{{cite book |first=Roger |last=Penrose |year=2016 |title=The Road to Reality: A complete guide to the laws of the universe |edition=reprint |publisher=Random House |isbn=978-1-4464-1820-8 |pages=72–73 |url=https://books.google.com/books?id=VWTNCwAAQBAJ&pg=PA73}}</ref> }}

Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the [[Fundamental theorem of algebra|fundamental theorem of algebra]] asserts that every non-constant polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation
<math>(x+1)^2 = -9</math>
has no real solution, since the square of a real number cannot be negative, but has the two nonreal complex solutions <math>-1+3i</math> and <math>-1-3i</math>.

Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule <math>i^{2}=-1</math> combined with the associative, commutative, and distributive laws. Every nonzero complex number has a [[Multiplicative inverse|multiplicative inverse]]. This makes the complex numbers a [[Field (mathematics)|field]] that has the real numbers as a subfield. The complex numbers also form a real vector space of [[Two-dimensional space|dimension two]], with {{math|{{mset|1, ''i''}}}} as a [[Standard basis|standard basis]].

This standard basis makes the complex numbers a Cartesian plane, called the [[Complex plane|complex plane]]. This allows a geometric interpretation of the complex numbers and their operations, and conversely expressing in terms of complex numbers some geometric properties and constructions. For example, the real numbers form the [[Real line|real line]] which is identified to the horizontal axis of the complex plane. The complex numbers of [[Absolute value|absolute value]] one form the [[Unit circle|unit circle]]. The addition of a complex number is a [[Translation (geometry)|translation]] in the complex plane, and the multiplication by a complex number is a [[Similarity (geometry)|similarity]] centered at the origin. The complex conjugation is the [[Reflection symmetry|reflection symmetry]] with respect to the real axis. The complex absolute value is a Euclidean norm.

In summary, the complex numbers form a rich structure that is simultaneously an [[Algebraically closed field|algebraically closed field]], a commutative algebra over the reals, and a Euclidean vector space of dimension two.

==Definition==

[[File:Illustration of a complex number.svg|right|thumb|upright=1.05|An illustration of the complex number {{math|1=''z'' = ''x'' + ''iy''}} on the [[Complex plane|complex plane]]. The real part is {{mvar|x}}, and its imaginary part is {{mvar|y}}.]]

A complex number is a number [[Of the form|of the form]] {{math|1=''a'' + ''bi''}}, where {{mvar|a}} and {{mvar|b}} are real numbers, and {{math|''i''}} is an indeterminate satisfying {{math|1=''i''<sup>2</sup> = −1}}. For example, {{math|2 + 3''i''}} is a complex number.<ref>{{cite book|title=College algebra |url=https://archive.org/details/collegealgebrawi00axle |url-access=limited |last=Axler |first=Sheldon |page=[https://archive.org/details/collegealgebrawi00axle/page/n285 262]|publisher=Wiley|year=2010|isbn=9780470470770 }}</ref>

This way, a complex number is defined as a [[Polynomial|polynomial]] with real coefficients in the single indeterminate {{math|''i''}}, for which the relation {{math|''i''<sup>2</sup> + 1 {{=}} 0}} is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials. The relation {{math|''i''<sup>2</sup> + 1 {{=}} 0}} induces the equalities {{math|''i''<sup>4''k''</sup> {{=}} 1, ''i''<sup>4''k''+1</sup> {{=}} ''i'', ''i''<sup>4''k''+2</sup> {{=}} −1,}} and {{math|''i''<sup>4''k''+3</sup> {{=}} −''i'',}} which hold for all integers {{mvar|k}}; these allow the reduction of any polynomial that results from the addition and multiplication of complex numbers to a linear polynomial in {{mvar|i}}, again of the form {{math|1=''a'' + ''bi''}} with real coefficients {{mvar|a, b.}}

The real number {{mvar|a}} is called the ''real part'' of the complex number {{math|''a'' + ''bi''}}; the real number {{mvar|b}} is called its ''imaginary part''. To emphasize, the imaginary part does not include a factor {{mvar|i}}; that is, the imaginary part is {{mvar|b}}, not {{math|''bi''}}.<ref>{{cite book |last1=Spiegel |first1= M.R. |last2=Lipschutz |first2= S. |last3= Schiller |first3= J.J. |last4=Spellman |first4=D. |title= Complex Variables |edition=2nd |series=Schaum's Outline Series |publisher= McGraw Hill |isbn= 978-0-07-161569-3|date= 14 April 2009 }}</ref><ref>{{cite book |title=College Algebra and Trigonometry |edition=6 |first1=Richard N. |last1=Aufmann |first2=Vernon C. |last2=Barker |first3=Richard D. |last3=Nation |publisher=Cengage Learning |year=2007 |isbn=978-0-618-82515-8 |page=66 |chapter-url=https://books.google.com/books?id=g5j-cT-vg_wC&pg=PA66 |chapter=Chapter P}}</ref>

Formally, the complex numbers are defined as the [[Quotient ring|quotient ring]] of the [[Polynomial ring|polynomial ring]] in the indeterminate {{math|''i''}}, by the [[Ideal (ring theory)|ideal]] generated by the polynomial {{math|''i''<sup>2</sup> + 1}} (see below).<ref name="Bourbaki 1998 loc=§VIII.1">{{harvnb|Bourbaki|1998|loc=§VIII.1}}</ref>

==Notation==

A real number {{mvar|a}} can be regarded as a complex number {{math|''a'' + 0''i''}}, whose imaginary part is 0. A purely [[Imaginary number|imaginary number]] {{math|''bi''}} is a complex number {{math|0 + ''bi''}}, whose real part is zero. As with polynomials, it is common to write {{mvar|a}} for {{math|''a'' + 0''i''}} and {{math|''bi''}} for {{math|0 + ''bi''}}. Moreover, when the imaginary part is negative, that is, {{math|1=''b'' = −''{{!}}b{{!}}'' < 0}}, it is common to write {{math|''a'' − ''{{!}}b{{!}}i''}} instead of {{math|''a'' + (−''{{!}}b{{!}}'')''i''}}; for example, for {{math|1=''b'' = −4}}, {{math|3 − 4''i''}} can be written instead of {{math|3 + (−4)''i''}}.

Since the multiplication of the indeterminate {{math|''i''}} and a real is commutative in polynomials with real coefficients, the polynomial {{math|''a'' + ''bi''}} may be written as {{math|''a'' + ''ib''.}} This is often expedient for imaginary parts denoted by expressions, for example, when {{mvar|b}} is a radical.{{sfn|Ahlfors|1979}}

The real part of a complex number {{mvar|z}} is denoted by {{math|Re(''z'')}}, <math>\mathcal{Re}(z)</math>, or <math>\mathfrak{R}(z)</math>; the imaginary part of a complex number {{mvar|z}} is denoted by {{math|Im(''z'')}}, <math>\mathcal{Im}(z)</math>, or <math>\mathfrak{I}(z).</math> For example,
<math display=block>  \operatorname{Re}(2 + 3i) = 2 \quad \text{ and } \quad  \operatorname{Im}(2 + 3i) = 3~.</math>

The [[Set (mathematics)|set]] of all complex numbers is denoted by <math>\Complex</math> ([[Blackboard bold|blackboard bold]]) or {{math|'''C'''}} (upright bold).

In some disciplines, particularly in [[Physics:Electromagnetism|electromagnetism]] and [[Electrical engineering|electrical engineering]], {{mvar|j}} is used instead of {{mvar|i}} as {{mvar|i}} is frequently used to represent [[Physics:Electric current|electric current]].<ref name="Campbell_1911"/><ref name="Brown-Churchill_1996"/> In these cases, complex numbers are written as {{math|''a'' + ''bj''}}, or {{math|''a'' + ''jb''}}.

==Visualization==
{{Main|Complex plane}}
[[File:Complex number illustration.svg|thumb|right|A complex number {{mvar|z}}, as a point (black) and its position vector (blue)]]

A complex number {{mvar|z}} can thus be identified with an [[Ordered pair|ordered pair]] <math>(\Re (z),\Im (z))</math> of real numbers, which in turn may be interpreted as coordinates of a point in a two-dimensional space. The most immediate space is the Euclidean plane with suitable coordinates, which is then called ''complex plane'' or ''Argand diagram,''<ref>{{cite book |last=Pedoe |first=Dan |title=Geometry: A comprehensive course |publisher=Dover |year=1988 |isbn=978-0-486-65812-4}}</ref>{{efn| {{harvnb|Solomentsev|2001}}: "The plane <math>\R^2</math> whose points are identified with the elements of <math>\Complex</math> is called the complex plane&nbsp;... The complete geometric interpretation of complex numbers and operations on them appeared first in the work of C. Wessel (1799). The geometric representation of complex numbers, sometimes called the 'Argand diagram', came into use after the publication in 1806 and 1814 of papers by J.R. Argand, who rediscovered, largely independently, the findings of Wessel".}}<ref name=":2">{{Cite web |last=Weisstein |first=Eric W. |title=Complex Number |url=https://mathworld.wolfram.com/ComplexNumber.html |access-date=2020-08-12 |website=mathworld.wolfram.com}}</ref> named after [[Biography:Jean-Robert Argand|Jean-Robert Argand]]. Another prominent space on which the coordinates may be projected is the two-dimensional surface of a sphere, which is then called [[Riemann sphere]].

===Cartesian complex plane===
The definition of the complex numbers involving two arbitrary real values immediately suggests the use of Cartesian coordinates in the complex plane. The horizontal (''real'') axis is generally used to display the real part, with increasing values to the right, and the imaginary part marks the vertical (''imaginary'') axis, with increasing values upwards.

A charted number may be viewed either as the coordinatized point or as a position vector from the origin to this point. The coordinate values of a complex number {{mvar|z}} can hence be expressed in its ''Cartesian'', ''rectangular'', or ''algebraic'' form.

Notably, the operations of addition and multiplication take on a very natural geometric character, when complex numbers are viewed as position vectors: addition corresponds to [[Euclidean vector#Addition and subtraction|vector addition]], while multiplication (see below) corresponds to multiplying their magnitudes and adding the angles they make with the real axis. Viewed in this way, the multiplication of a complex number by {{math|''i''}} corresponds to rotating the position vector [[Orientation (geometry)|counterclockwise]] by a quarter [[Turn (geometry)|turn]] ([[Right angle|90°]]) about the origin—a fact which can be expressed algebraically as
<math display=block>(x + yi)\, i = -y + xi .</math>

===Polar complex plane {{anchor|Polar form}}=== <!-- [[Nth root]] links to this section -->
{{Main|Polar coordinate system}}

====Modulus and argument====
[[File:Complex number illustration modarg.svg|right|thumb|Argument {{mvar|φ}} and modulus {{mvar|r}} locate a point in the complex plane.]]

An alternative option for coordinates in the complex plane is the [[Polar coordinate system|polar coordinate system]] that uses the distance of the point {{mvar|z}} from the [[Origin (mathematics)|origin]] ({{mvar|O}}), and the angle subtended between the positive real axis and the line segment {{mvar|Oz}} in a counterclockwise sense. This leads to the '''polar form''' 
:<math>z=re^{i\varphi}=r(\cos\varphi +i\sin\varphi) </math>
of a complex number, where {{mvar|r}} is the [[Absolute value|absolute value]] of {{mvar|z}}, and <math>\varphi</math> is the [[Argument (complex analysis)|argument]] of {{mvar|z}}.

The ''absolute value'' (or ''modulus'' or ''magnitude'') of a complex number {{math|1=''z'' = ''x'' + ''yi''}} is{{sfn|Apostol|1981|p=18}}
<math display=block>r=|z|=\sqrt{x^2+y^2}.</math>
If {{mvar|z}} is a real number (that is, if {{math|1=''y'' = 0}}), then {{math|1=''r'' = {{!}}''x''{{!}}}}. That is, the absolute value of a real number equals its absolute value as a complex number.

By Pythagoras' theorem, the absolute value of a complex number is the distance to the origin of the point representing the complex number in the [[Complex plane|complex plane]].

The ''argument'' of {{mvar|z}} (in many applications referred to as the "phase" {{mvar|φ}})<ref name=":2" /> is the angle of the [[Radius|radius]] {{mvar|Oz}} with the positive real axis, and is written as {{math|arg ''z''}}. As with the modulus, the argument can be found from the rectangular form {{mvar|x + yi}}<ref>{{cite book
|title=Complex Variables: Theory And Applications
|edition=2nd
|chapter=Chapter 1
|first1=H.S.
|last1=Kasana
|publisher=PHI Learning Pvt. Ltd
|year=2005
|isbn=978-81-203-2641-5
|page=14
|chapter-url=https://books.google.com/books?id=rFhiJqkrALIC&pg=PA14}}</ref> by applying the inverse tangent to the quotient of imaginary-by-real parts. By using a half-angle identity, a single branch of the arctan suffices to cover the range {{open-closed|−''π'', ''π''}} of the {{math|arg}}-function, and avoids a more subtle case-by-case analysis

<math display=block>\varphi = \arg (x+yi) = \begin{cases}
   2 \arctan\left(\dfrac{y}{x + \sqrt{x^2 + y^2}}\right) &\text{if } y \neq 0 \text{ or } x > 0, \\
   \pi &\text{if } x < 0 \text{ and } y = 0, \\
   \text{undefined} &\text{if } x = 0 \text{ and } y = 0.
 \end{cases}</math>

Normally, as given above, the [[Principal value|principal value]] in the interval {{open-closed|−{{mvar|π}}, {{mvar|π}}}} is chosen. If the arg value is negative, values in the range {{open-closed|−{{mvar|π}}, {{mvar|π}}}} or {{closed-open|0, 2{{mvar|π}}}} can be obtained by adding {{math|2''π''}}.<!--don't change this into π. Doing so produces *another* complex number.--> The value of {{mvar|φ}} is expressed in [[Radian|radian]]s in this article. It can increase by any integer multiple of {{math|2''π''}} and still give the same angle, viewed as subtended by the rays of the positive real axis and from the origin through {{mvar|z}}. Hence, the arg function is sometimes considered as [[Multivalued function|multivalued]]. The polar angle for the complex number 0 is indeterminate, but arbitrary choice of the polar angle&nbsp;0 is common.

The value of {{mvar|φ}} equals the result of atan2:
<math display=block>\varphi = \operatorname{atan2}\left(\operatorname{Im}(z),\operatorname{Re}(z) \right).</math>

Together, {{mvar|r}} and {{mvar|φ}} give another way of representing complex numbers, the ''polar form'', as the combination of modulus and argument fully specify the position of a point on the plane. Recovering the original rectangular co-ordinates from the polar form is done by the formula called ''trigonometric form''
<math display=block> z = r(\cos \varphi + i\sin \varphi ).</math>

Using [[Euler's formula]] this can be written as
<math display=block>z = r e^{i \varphi} \text{ or } z = r \exp i \varphi.</math>

Using the {{math|[[Cis (mathematics)|cis]]}} function, this is sometimes abbreviated to
<math display=block> z = r \operatorname\mathrm{cis} \varphi. </math>

In [[Engineering:Angle notation|angle notation]], often used in [[Engineering:Electronics|electronics]] to represent a phasor with amplitude {{mvar|r}} and phase {{mvar|φ}}, it is written as<ref>
{{cite book
 |first1=James William |last1=Nilsson
 |first2=Susan A.      |last2=Riedel
 |year=2008
 |title=Electric circuits  |edition=8th  |page=338
 |chapter=Chapter 9
 |publisher=Prentice Hall
 |isbn=978-0-13-198925-2
 |chapter-url=https://books.google.com/books?id=sxmM8RFL99wC&pg=PA338
}}
</ref>
<math display=block>z = r \angle \varphi . </math>

===Complex graphs===
{{main|Domain coloring}}
[[File:Complex-plot.png|right|thumb|A color wheel graph of the expression
{{math|{{sfrac|(''z''<sup>2</sup> − 1)(''z'' − 2 − ''i'')<sup>2</sup>|''z''<sup>2</sup> + 2 + 2''i''}}}}]]
When visualizing [[Complex analysis|complex functions]], both a complex input and output are needed. Because each complex number is represented in two dimensions, visually graphing a complex function would require the perception of a four dimensional space, which is possible only in projections. Because of this, other ways of visualizing complex functions have been designed.

In [[Domain coloring|domain coloring]] the output dimensions are represented by color and brightness, respectively. Each point in the complex plane as domain is ''ornated'', typically with ''color'' representing the argument of the complex number, and ''brightness'' representing the magnitude. Dark spots mark moduli near zero, brighter spots are farther away from the origin, the gradation may be discontinuous, but is assumed as monotonous. The colors often vary in steps of {{sfrac|{{pi}}|3}} for {{math|0}} to {{math|2{{pi}}}} from red, yellow, green, cyan, blue, to magenta. These plots are called [[Domain coloring|color wheel graphs]]. This provides a simple way to visualize the functions without losing information. The picture shows zeros for {{math|±1, (2 + ''i'')}} and poles at <math>\pm \sqrt{{-2-2i}}.</math>

==History==
The solution in [[Nth root|radicals]] (without [[Trigonometric functions|trigonometric functions]]) of a general [[Cubic equation|cubic equation]], when all three of its roots are real numbers, contains the square roots of negative numbers, a situation that cannot be rectified by factoring aided by the rational root test, if the cubic is [[Irreducible polynomial|irreducible]]; this is the so-called  ''[[Casus irreducibilis|casus irreducibilis]]'' ("irreducible case"). This conundrum led Italian mathematician Gerolamo Cardano to conceive of complex numbers in around 1545 in his ''Ars Magna'',<ref>{{cite book|first=Morris |last= Kline|title=A history of mathematical thought, volume 1|page=253}}</ref> though his understanding was rudimentary; moreover he later described complex numbers as "as subtle as they are useless".<ref>{{Cite book|last=Jurij.|first=Kovič|url=http://worldcat.org/oclc/1080410598|title=Tristan Needham, Visual Complex Analysis, Oxford University Press Inc., New York, 1998, 592 strani|oclc=1080410598}}</ref> Cardano did use imaginary numbers, but described using them as "mental torture."<ref>O'Connor and Robertson (2016), "Girolamo Cardano."</ref> This was prior to the use of the graphical complex plane. Cardano and other Italian mathematicians, notably Scipione del Ferro, in the 1500s created an algorithm for solving cubic equations which generally had one real solution and two solutions containing an imaginary number. Since they ignored the answers with the imaginary numbers, Cardano found them useless.<ref>Nahin, Paul J. An Imaginary Tale: The Story of √−1. Princeton: Princeton University Press, 1998.</ref>

Work on the problem of general polynomials ultimately led to the [[Fundamental theorem of algebra|fundamental theorem of algebra]], which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher. Complex numbers thus form an [[Algebraically closed field|algebraically closed field]], where any polynomial equation has a [[Root of a function|root]].

Many mathematicians contributed to the development of complex numbers. The rules for addition, subtraction, multiplication, and root extraction of complex numbers were developed by the Italian mathematician [[Biography:Rafael Bombelli|Rafael Bombelli]].<ref>{{cite book |last1=Katz |first1=Victor J. |title=A History of Mathematics, Brief Version |section= 9.1.4 |publisher=Addison-Wesley |isbn=978-0-321-16193-2 |year=2004}}</ref> A more abstract formalism for the complex numbers was further developed by the Irish mathematician [[Biography:William Rowan Hamilton|William Rowan Hamilton]], who extended this abstraction to the theory of quaternions.<ref>{{cite journal |last1=Hamilton |first1=Wm. |title=On a new species of imaginary quantities connected with a theory of quaternions |journal=Proceedings of the Royal Irish Academy |date=1844 |volume=2 |pages=424–434 |url=https://babel.hathitrust.org/cgi/pt?id=njp.32101040410779&view=1up&seq=454}}</ref>

The earliest fleeting reference to [[Square root|square root]]s of [[Negative number|negative number]]s can perhaps be said to occur in the work of the Greek mathematician [[Biography:Hero of Alexandria|Hero of Alexandria]] in the 1st century AD, where in his ''[[Biography:Hero of Alexandria#Bibliography|Stereometrica]]'' he considered, apparently in error, the volume of an impossible [[Frustum|frustum]] of a pyramid to arrive at the term <math>\sqrt{81 - 144}</math> in his calculations, which today would simplify to  <math>\sqrt{-63} = 3i\sqrt{7}</math>.{{efn|In the literature the imaginary unit often precedes the radical sign, even when preceded itself by an integer.<ref>{{cite book |title=Trigonometry |author1=Cynthia Y. Young |edition=4th |publisher=John Wiley & Sons |year=2017 |isbn=978-1-119-44520-3 |page=406 |url=https://books.google.com/books?id=476ZDwAAQBAJ}} [https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA406 Extract of page 406]</ref>}} Negative quantities were not conceived of in Hellenistic mathematics and Hero merely replaced it by its positive <math>\sqrt{144 - 81} = 3\sqrt{7}.</math><ref>{{cite book |title=An Imaginary Tale: The Story of √−1 |last=Nahin |first=Paul J. |year=2007 |publisher=Princeton University Press |isbn=978-0-691-12798-9 |url=http://mathforum.org/kb/thread.jspa?forumID=149&threadID=383188&messageID=1181284 |access-date=20 April 2011 |archive-url=https://web.archive.org/web/20121012090553/http://mathforum.org/kb/thread.jspa?forumID=149&threadID=383188&messageID=1181284 |archive-date=12 October 2012 |url-status=live }}</ref>

The impetus to study complex numbers as a topic in itself first arose in the 16th century when [[Algebraic solution|algebraic solution]]s for the roots of [[Cubic equation|cubic]] and [[Quartic equation|quartic]] [[Polynomial|polynomial]]s were discovered by Italian mathematicians (Niccolò Fontana Tartaglia and Gerolamo Cardano). It was soon realized (but proved much later)<ref name=Casus/> that these formulas, even if one were interested only in real solutions, sometimes required the manipulation of square roots of negative numbers. In fact, it was proved later that the use of complex numbers [[Casus irreducibilis|is unavoidable]] when all three roots are real and distinct.{{efn|It has been proved that imaginary numbers necessarily appear in the cubic formula when the equation has three real, different roots by Pierre Laurent Wantzel in 1843, Vincenzo Mollame in 1890, Otto Hölder in 1891, and Adolf Kneser in 1892. Paolo Ruffini also provided an incomplete proof in 1799.——S. Confalonieri (2015)<ref name=Casus>{{cite book |title=The Unattainable Attempt to Avoid the Casus Irreducibilis for Cubic Equations: Gerolamo Cardano's De Regula Aliza |first=Sara |last=Confalonieri |publisher=Springer |year=2015 |pages=15–16 (note 26) |isbn=978-3658092757 }}</ref>}} However, the general formula can still be used in this case, with some care to deal with the ambiguity resulting from the existence of three cubic roots for nonzero complex numbers. Rafael Bombelli was the first to address explicitly these seemingly paradoxical solutions of cubic equations and developed the rules for complex arithmetic, trying to resolve these issues.

The term "imaginary" for these quantities was coined by [[Biography:René Descartes|René Descartes]] in 1637, who was at pains to stress their unreal nature:<ref>{{cite book |title=La Géométrie {{pipe}} The Geometry of René Descartes with a facsimile of the first edition |last=Descartes |first=René |year=1954 |orig-year=1637 |publisher=[[Social:Dover Publications|Dover Publications]] |isbn=978-0-486-60068-0 |url=https://archive.org/details/geometryofrenede00rend |access-date=20 April 2011 }}</ref>
{{blockquote|...&nbsp;sometimes only imaginary, that is one can imagine as many as I said in each equation, but sometimes there exists no quantity that matches that which we imagine.<br/>
[''...&nbsp;quelquefois seulement imaginaires c'est-à-dire que l'on peut toujours en imaginer autant que j'ai dit en chaque équation, mais qu'il n'y a quelquefois aucune quantité qui corresponde à celle qu'on imagine.'']}}
A further source of confusion was that the equation <math>\sqrt{-1}^2 = \sqrt{-1}\sqrt{-1} = -1</math> seemed to be capriciously inconsistent with the algebraic identity <math>\sqrt{a}\sqrt{b} = \sqrt{ab}</math>, which is valid for non-negative real numbers {{mvar|a}} and {{mvar|b}}, and which was also used in complex number calculations with one of {{mvar|a}}, {{mvar|b}} positive and the other negative. The incorrect use of this identity in the case when both {{mvar|a}} and {{mvar|b}} are negative, and the related identity <math display="inline">\frac{1}{\sqrt{a}} = \sqrt{\frac{1}{a}}</math>, even bedeviled [[Biography:Leonhard Euler|Leonhard Euler]]. This difficulty eventually led to the convention of using the special symbol {{math|''i''}} in place of <math>\sqrt{-1}</math> to guard against this mistake.{{Citation needed|date=April 2011}} Even so, Euler considered it natural to introduce students to complex numbers much earlier than we do today. In his elementary algebra text book, ''[[Elements of Algebra]]'', he introduces these numbers almost at once and then uses them in a natural way throughout.

In the 18th century complex numbers gained wider use, as it was noticed that formal manipulation of complex expressions could be used to simplify calculations involving trigonometric functions. For instance, in 1730 [[Biography:Abraham de Moivre|Abraham de Moivre]] noted that the identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be re-expressed by the following [[De Moivre's formula|de Moivre's formula]]:

<math display=block>(\cos \theta + i\sin \theta)^{n} = \cos n \theta + i\sin n \theta. </math>

In 1748, Euler went further and obtained [[Euler's formula]] of [[Complex analysis|complex analysis]]:<ref>{{cite book |last1=Euler |first1=Leonard |title=Introductio in Analysin Infinitorum |trans-title=Introduction to the Analysis of the Infinite |date=1748 |publisher=Marc Michel Bosquet & Co. |location=Lucerne, Switzerland |volume=1 |page=104 |url=https://books.google.com/books?id=jQ1bAAAAQAAJ&pg=PA104 |language=la}}</ref>

<math display=block>\cos \theta + i\sin \theta = e ^{i\theta } </math>

by formally manipulating complex [[Power series|power series]] and observed that this formula could be used to reduce any trigonometric identity to much simpler exponential identities.

The idea of a complex number as a point in the complex plane (above) was first described by {{wipe|Denmark}}–{{wipe|Norway}} [[Biography:Mathematician|mathematician]] [[Biography:Caspar Wessel|Caspar Wessel]] in 1799,<ref>{{cite journal |last1=Wessel |first1=Caspar |title=Om Directionens analytiske Betegning, et Forsog, anvendt fornemmelig til plane og sphæriske Polygoners Oplosning |journal=Nye Samling af det Kongelige Danske Videnskabernes Selskabs Skrifter [New Collection of the Writings of the Royal Danish Science Society] |date=1799 |volume=5 |pages=469–518 |url=https://babel.hathitrust.org/cgi/pt?id=ien.35556000979690&view=1up&seq=527 |trans-title=On the analytic representation of direction, an effort applied in particular to the determination of plane and spherical polygons |language=da}}</ref> although it had been anticipated as early as 1685 in [[Biography:John Wallis|Wallis's]] ''A Treatise of Algebra''.<ref>{{cite book |last=Wallis |first=John |date=1685 |title=A Treatise of Algebra, Both Historical and Practical ... |url=https://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/H3GRV5AU/pageimg&start=291&mode=imagepath&pn=291|location=London, England |publisher=printed by John Playford, for Richard Davis |pages=264–273 }}</ref>

Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed. In 1806 [[Biography:Jean-Robert Argand|Jean-Robert Argand]] independently issued a pamphlet on complex numbers and provided a rigorous proof of the [[Fundamental theorem of algebra#History|fundamental theorem of algebra]].<ref>{{cite book |last1=Argand |title=Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques |trans-title=Essay on a way to represent complex quantities by geometric constructions |date=1806 |publisher=Madame Veuve Blanc |location=Paris, France |url=http://www.bibnum.education.fr/mathematiques/geometrie/essai-sur-une-maniere-de-representer-des-quantites-imaginaires-dans-les-cons |language=fr}}</ref> [[Biography:Carl Friedrich Gauss|Carl Friedrich Gauss]] had earlier published an essentially [[Topology|topological]] proof of the theorem in 1797 but expressed his doubts at the time about "the true metaphysics of the square root of &minus;1".<ref>Gauss, Carl Friedrich (1799) [https://books.google.com/books?id=g3VaAAAAcAAJ&pg=PP1 ''"Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse."''] [New proof of the theorem that any rational integral algebraic function of a single variable can be resolved into real factors of the first or second degree.] Ph.D. thesis, University of Helmstedt, (Germany). (in Latin)</ref> It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane,<ref name=Ewald>{{cite book |last=Ewald |first=William B. |date=1996 |title=From Kant to Hilbert: A Source Book in the Foundations of Mathematics |volume=1 |page=313 |publisher=Oxford University Press |isbn=9780198505358|url=https://books.google.com/books?id=rykSDAAAQBAJ&pg=PA313 |access-date=18 March 2020}}</ref> largely establishing modern notation and terminology:{{sfn|Gauss|1831|p=638}}

<blockquote>If one formerly contemplated this subject from a false point of view and therefore found a mysterious darkness, this is in large part attributable to clumsy terminology. Had one not called +1, −1, <math>\sqrt{-1}</math> positive, negative, or imaginary (or even impossible) units, but instead, say, direct, inverse, or lateral units, then there could scarcely have been talk of such darkness.</blockquote>

In the beginning of the 19th century, other mathematicians discovered independently the geometrical representation of the complex numbers: Buée,<ref>{{cite web| url = https://mathshistory.st-andrews.ac.uk/Biographies/Buee/| title = Adrien Quentin Buée (1745–1845): MacTutor}}</ref><ref>{{cite journal |last1=Buée |title=Mémoire sur les quantités imaginaires |journal=Philosophical Transactions of the Royal Society of London |date=1806 |volume=96 |pages=23–88 |doi=10.1098/rstl.1806.0003 |s2cid=110394048 |url=https://royalsocietypublishing.org/doi/pdf/10.1098/rstl.1806.0003 |trans-title=Memoir on imaginary quantities |language=fr}}</ref> [[Biography:C. V. Mourey|Mourey]],<ref>{{cite book |last1=Mourey |first1=C.V. |title=La vraies théore des quantités négatives et des quantités prétendues imaginaires |trans-title=The true theory of negative quantities and of alleged imaginary quantities |date=1861 |publisher=Mallet-Bachelier |location=Paris, France |url=https://archive.org/details/bub_gb_8YxKAAAAYAAJ |language=fr}}  1861 reprint of 1828 original.</ref> Warren,<ref>{{cite book |last1=Warren |first1=John |title=A Treatise on the Geometrical Representation of the Square Roots of Negative Quantities |date=1828 |publisher=Cambridge University Press |location=Cambridge, England |url=https://archive.org/details/treatiseongeomet00warrrich}}</ref><ref>{{cite journal |last1=Warren |first1=John |title=Consideration of the objections raised against the geometrical representation of the square roots of negative quantities |journal=Philosophical Transactions of the Royal Society of London |date=1829 |volume=119 |pages=241–254 |s2cid=186211638 |doi=10.1098/rstl.1829.0022 |doi-access=free }}</ref><ref>{{cite journal |last1=Warren |first1=John |title=On the geometrical representation of the powers of quantities, whose indices involve the square roots of negative numbers |journal=Philosophical Transactions of the Royal Society of London |date=1829 |volume=119 |pages=339–359 |s2cid=125699726 |doi=10.1098/rstl.1829.0031 |doi-access=free }}</ref> Français and his brother, [[Biography:Giusto Bellavitis|Bellavitis]].<ref>{{cite journal |last1=Français |first1=J.F. |title=Nouveaux principes de géométrie de position, et interprétation géométrique des symboles imaginaires |journal=Annales des mathématiques pures et appliquées |date=1813 |volume=4 |pages=61–71 |url=https://babel.hathitrust.org/cgi/pt?id=uc1.$c126478&view=1up&seq=69 |trans-title=New principles of the geometry of position, and geometric interpretation of complex [number] symbols |language=fr}}</ref><ref>{{cite book |title=Two Cultures |editor= Kim Williams |last1=Caparrini |first1=Sandro |chapter=On the Common Origin of Some of the Works on the Geometrical Interpretation of Complex Numbers |year=2000 |publisher=Birkhäuser |isbn=978-3-7643-7186-9 |page=139 |url=https://books.google.com/books?id=voFsJ1EhCnYC |chapter-url=https://books.google.com/books?id=voFsJ1EhCnYC&pg=PA139}}</ref>

The English mathematician G.H. Hardy remarked that Gauss was the first mathematician to use complex numbers in "a really confident and scientific way" although mathematicians such as {{wipe|Norway}} [[Biography:Niels Henrik Abel|Niels Henrik Abel]] and [[Biography:Carl Gustav Jacob Jacobi|Carl Gustav Jacob Jacobi]] were necessarily using them routinely before Gauss published his 1831 treatise.<ref>{{cite book |title=An Introduction to the Theory of Numbers |last1=Hardy |first1=G.H. |last2=Wright |first2=E.M. |year=2000 |orig-year=1938 |publisher=[[Organization:Oxford University Press|OUP Oxford]] |isbn= 978-0-19-921986-5 |page=189 (fourth edition)}}</ref>

[[Biography:Augustin-Louis Cauchy|Augustin-Louis Cauchy]] and [[Biography:Bernhard Riemann|Bernhard Riemann]] together brought the fundamental ideas of complex analysis to a high state of completion, commencing around 1825 in Cauchy's case.

The common terms used in the theory are chiefly due to the founders. Argand called {{math|cos ''φ'' + ''i'' sin ''φ''}} the ''direction factor'', and <math>r = \sqrt{a^2 + b^2}</math> the ''modulus'';{{efn| {{harvnb|Argand|1814|p=204}} defines the modulus of a complex number but he doesn't name it:<br/>''"Dans ce qui suit, les accens, indifféremment placés, seront employés pour indiquer la grandeur absolue des quantités qu'ils affectent; ainsi, si <math>a = m + n\sqrt{-1}</math>, <math>m</math> et <math>n</math> étant réels, on devra entendre que <math>a_'</math> ou <math>a' = \sqrt{m^2 + n^2}</math>."''<br/>[In what follows, accent marks, wherever they're placed, will be used to indicate the absolute size of the quantities to which they're assigned; thus if <math>a = m + n\sqrt{-1}</math>, <math>m</math> and <math>n</math> being real, one should understand that <math>a_'</math> or <math>a' = \sqrt{m^2 + n^2}</math>.]<br/>
{{harvnb|Argand|1814|p=208}} defines and names the ''module'' and the ''direction factor'' of a complex number:  ''"...&nbsp;<math>a = \sqrt{m^2 + n^2}</math> pourrait être appelé le ''module'' de <math>a + b  \sqrt{-1}</math>, et représenterait la ''grandeur absolue'' de la ligne <math>a + b  \sqrt{-1}</math>, tandis que l'autre facteur, dont le module est l'unité, en représenterait la direction."''<br/>[...&nbsp;<math>a = \sqrt{m^2 + n^2}</math> could be called the ''module'' of <math>a + b  \sqrt{-1}</math> and would represent the ''absolute size'' of the line <math>a + b  \sqrt{-1}\,,</math> (Argand represented complex numbers as vectors.) whereas the other factor [namely, <math>\tfrac{a}{\sqrt{a^2 + b^2}} + \tfrac{b}{\sqrt{a^2 + b^2}} \sqrt{-1} </math>], whose module is unity [1], would represent its direction.]}}<ref>{{cite web |author=Jeff Miller |date=Sep 21, 1999 |title=MODULUS |url=http://members.aol.com/jeff570/m.html|archive-url=https://web.archive.org/web/19991003034827/http://members.aol.com/jeff570/m.html |work=Earliest Known Uses of Some of the Words of Mathematics (M) |archive-date=1999-10-03 |url-status=usurped}}</ref> Cauchy (1821) called {{math|cos ''φ'' + ''i'' sin ''φ''}} the ''reduced form'' (l'expression réduite)<ref>{{cite book |last=Cauchy |first=Augustin-Louis |date=1821 |title=Cours d'analyse de l'École royale polytechnique |url=https://archive.org/details/coursdanalysede00caucgoog/page/n209/mode/2up |location=Paris, France |publisher=L'Imprimerie Royale |volume=1 |page=183 |language=fr }}</ref> and apparently introduced the term ''argument''; Gauss used {{math|''i''}} for <math>\sqrt{-1}</math>,{{efn| Gauss writes:<ref>{{harvnb|Gauss|1831|p=96}}</ref>''"Quemadmodum scilicet arithmetica sublimior in quaestionibus hactenus pertractatis inter solos numeros integros reales versatur, ita theoremata circa residua biquadratica tunc tantum in summa simplicitate ac genuina venustate resplendent, quando campus arithmeticae ad quantitates ''imaginarias'' extenditur, ita ut absque restrictione ipsius obiectum constituant numeri formae ''a + bi'', denotantibus ''i'', pro more quantitatem imaginariam <math>\sqrt{-1}</math>, atque ''a, b'' indefinite omnes numeros reales integros inter -<math>\infty</math> et +<math>\infty</math>."'' [Of course just as the higher arithmetic has been investigated so far in problems only among real integer numbers, so theorems regarding biquadratic residues then shine in greatest simplicity and genuine beauty, when the field of arithmetic is extended to ''imaginary'' quantities, so that, without restrictions on it, numbers of the form ''a + bi'' — ''i'' denoting by convention the imaginary quantity <math>\sqrt{-1}</math>, and the variables ''a, b'' [denoting] all real integer numbers between <math>-\infty</math> and <math>+\infty</math> — constitute an object.]}} introduced the term ''complex number'' for {{math|''a'' + ''bi''}},{{efn|Gauss:<ref>{{harvnb|Gauss|1831|p=96}}</ref>''"Tales numeros vocabimus numeros integros complexos, ita quidem, ut reales complexis non opponantur, sed tamquam species sub his contineri censeantur."'' [We will call such numbers [namely, numbers of the form ''a + bi'' ] "complex integer numbers", so that real [numbers] are regarded not as the opposite of complex [numbers] but [as] a type [of number that] is, so to speak, contained within them.]}} and called {{math|''a''<sup>2</sup> + ''b''<sup>2</sup>}} the ''norm''.{{efn|Gauss:<ref>{{harvnb|Gauss|1831|p=98}}</ref> ''"Productum numeri complexi per numerum ipsi conjunctum utriusque ''normam'' vocamus. Pro norma itaque numeri realis, ipsius quadratum habendum est."'' [We call a "norm" the product of a complex number [for example, ''a + ib'' ] with its conjugate [''a - ib'' ].  Therefore the square of a real number should be regarded as its norm.]}} The expression ''direction coefficient'', often used for {{math|cos ''φ'' + ''i'' sin ''φ''}}, is due to Hankel (1867),<ref>{{cite book |last=Hankel |first=Hermann |date=1867 |title=Vorlesungen über die complexen Zahlen und ihre Functionen |trans-title=Lectures About the Complex Numbers and Their Functions |url=https://books.google.com/books?id=754KAAAAYAAJ&pg=PA71 |location=Leipzig, [Germany] |publisher=Leopold Voss |volume=1 |page=71 |language=de }}  From p. 71:  ''"Wir werden den Factor (''cos'' φ + i ''sin'' φ) haüfig den ''Richtungscoefficienten'' nennen."'' (We will often call the factor (cos φ + i sin φ) the "coefficient of direction".)</ref> and ''absolute value,'' for ''modulus,'' is due to Weierstrass.

Later classical writers on the general theory include [[Biography:Richard Dedekind|Richard Dedekind]], [[Biography:Otto Hölder|Otto Hölder]], [[Biography:Felix Klein|Felix Klein]], [[Biography:Henri Poincaré|Henri Poincaré]], Hermann Schwarz, [[Biography:Karl Weierstrass|Karl Weierstrass]] and many others. Important work (including a systematization) in complex multivariate calculus has been started at beginning of the 20th century. Important results have been achieved by [[Biography:Wilhelm Wirtinger|Wilhelm Wirtinger]] in 1927.

==Relations and operations==

===Equality===
Complex numbers have a similar definition of equality to real numbers; two complex numbers {{math|''a''<sub>1</sub> + ''b''<sub>1</sub>''i''}} and {{math|''a''<sub>2</sub> + ''b''<sub>2</sub>''i''}} are equal [[If and only if|if and only if]] both their real and imaginary parts are equal, that is, if {{math|1=''a''<sub>1</sub> = ''a''<sub>2</sub>}} and {{math|1=''b''<sub>1</sub> = ''b''<sub>2</sub>}}. Nonzero complex numbers written in polar form are equal if and only if they have the same magnitude and their arguments differ by an integer multiple of {{math|2''π''}}.

===Ordering===
Unlike the real numbers, there is no natural ordering of the complex numbers. In particular, there is no linear ordering on the complex numbers that is compatible with addition and multiplication. Hence, the complex numbers do not have the structure of an ordered field. One explanation for this is that every non-trivial sum of squares in an [[Ordered field#nontrivialSquareSum|ordered field]] is nonzero, and {{math|1=''i''<sup>2</sup> + 1<sup>2</sup> = 0}} is a non-trivial sum of squares. Thus, complex numbers are naturally thought of as existing on a two-dimensional plane.

===Conjugate===
[[File:Complex conjugate picture.svg|right|thumb|upright=0.8|Geometric representation of {{mvar|z}} and its conjugate {{mvar|{{overline|z}}}} in the complex plane]]
The ''[[Complex conjugate|complex conjugate]]'' of the complex number {{math|1=''z'' = ''x'' + ''yi''}} is given by {{math|''x'' − ''yi''}}. It is denoted by either {{mvar|{{overline|z}}}} or {{math|''z''*}}.<ref>For the former notation, see {{harvnb|Apostol|1981|pp=15–16}}</ref> This [[Unary operation|unary operation]] on complex numbers cannot be expressed by applying only their basic operations addition, subtraction, multiplication and division.

Geometrically, {{mvar|{{overline|z}}}} is the [[Reflection symmetry|"reflection"]] of {{mvar|z}} about the real axis. Conjugating twice gives the original complex number
<math display=block>\overline{\overline{z}}=z,</math>

which makes this operation an [[Involution (mathematics)|involution]]. The reflection leaves both the real part and the magnitude of {{mvar|z}} unchanged, that is
<math display=block>\operatorname{Re}(\overline{z}) = \operatorname{Re}(z)\quad</math> and <math>\quad |\overline{z}| = |z|.</math>

The imaginary part and the argument of a complex number {{mvar|z}} change their sign under conjugation
<math display=block>\operatorname{Im}(\overline{z}) = -\operatorname{Im}(z)\quad \text{ and } \quad \operatorname{arg} \overline{z} \equiv -\operatorname{arg} z \pmod {2\pi}.</math>

For details on argument and magnitude, see the section on Polar form.

The product of a complex number {{math|''z'' {{=}} ''x'' + ''yi''}} and its conjugate is known as the ''absolute square''. It is always a non-negative real number and equals the square of the magnitude of each:
<math display=block>z\cdot \overline{z} = x^2 + y^2 = |z|^2 = |\overline{z}|^2.</math>

This property can be used to convert a fraction with a complex denominator to an equivalent fraction with a real denominator by expanding both numerator and denominator of the fraction by the conjugate of the given denominator. This process is sometimes called "[[Rationalisation (mathematics)|rationalization]]" of the denominator (although the denominator in the final expression might be an irrational real number), because it resembles the method to remove roots from simple expressions in a denominator.

The real and imaginary parts of a complex number {{mvar|z}} can be extracted using the conjugation:
<math display=block>\operatorname{Re}(z) = \dfrac{z+\overline{z}}{2},\quad \text{ and } \quad \operatorname{Im}(z) = \dfrac{z-\overline{z}}{2i}.</math>
Moreover, a complex number is real if and only if it equals its own conjugate.

Conjugation distributes over the basic complex arithmetic operations:
<math display=block>\begin{align}
    \overline{z\pm w} &= \overline{z} \pm \overline{w}, \\
  \overline{z\cdot w} &= \overline{z} \cdot \overline{w}, \\
       \overline{z/w} &= \overline{z}/\overline{w}.
\end{align}</math>

Conjugation is also employed in inversive geometry, a branch of geometry studying reflections more general than ones about a line. In the [[Engineering:Network analysis (electrical circuits)|network analysis of electrical circuits]], the complex conjugate is used in finding the equivalent impedance when the [[Engineering:Maximum power transfer theorem|maximum power transfer theorem]] is looked for.

===Addition and subtraction===
[[File:Vector Addition.svg|right|thumb|Addition of two complex numbers can be done geometrically by constructing a parallelogram.]]

Two complex numbers <math>a =x+yi</math> and <math>b =u+vi</math> are most easily [[Addition|added]] by separately adding their real and imaginary parts. That is to say:

<math display=block>a + b =(x+yi) + (u+vi) = (x+u) + (y+v)i.</math>
Similarly, [[Subtraction|subtraction]] can be performed as
<math display=block>a - b =(x+yi) - (u+vi) = (x-u) + (y-v)i.</math>

Multiplication of a complex number <math>a =x+yi</math> and a real number {{mvar|r}} can be done similarly by multiplying separately {{mvar|r}} and the real and imaginary parts of {{mvar|a}}:
<math display=block>ra=r(x+yi) = rx + ryi.</math>
In particular, subtraction can be done by negating the subtrahend (that is multiplying it with {{math|−1}}) and adding the result to the minuend:
<math display=block>a - b =a + (-1)\,b.</math>

Using the visualization of complex numbers in the complex plane, addition has the following geometric interpretation: the sum of two complex numbers {{mvar|a}} and {{mvar|b}}, interpreted as points in the complex plane, is the point obtained by building a [[Parallelogram|parallelogram]] from the three vertices {{mvar|O}}, and the points of the arrows labeled {{mvar|a}} and {{mvar|b}} (provided that they are not on a line). Equivalently, calling these points {{mvar|A}}, {{mvar|B}}, respectively and the fourth point of the parallelogram {{mvar|X}} the [[Triangle|triangle]]s {{mvar|OAB}} and {{mvar|XBA}} are [[Congruence (geometry)|congruent]].

===Multiplication and square{{anchor|Multiplication|Square}}===
The rules of  the [[Distributive property|distributive property]], the [[Commutative property|commutative properties]] (of addition and multiplication), and the defining property {{math|1=''i''{{sup|2}} = −1}} apply to complex numbers. It follows that 
<math display=block>(x+yi)\, (u+vi)= (xu - yv) + (xv + yu)i.</math>

In particular,
<math display=block>(x+yi)^2=x^2-y^2 + 2xyi.</math>

===Reciprocal and division===
Using the conjugate, the [[Multiplicative inverse|reciprocal]] of a nonzero complex number <math>z = x + yi</math> can be broken into real and imaginary components

<math display=block>
\frac{1}{z}
= \frac{\bar{z}}{z\bar{z}}
= \frac{\bar{z}}{|z|^2}
= \frac{x - yi}{x^2 + y^2}
= \frac{x}{x^2 + y^2} - \frac{y}{x^2 + y^2}i.</math>

This can be used to express a division of an arbitrary complex number <math>w = u + vi</math> by a non-zero complex number <math>z = x + yi</math> as
<math display=block>
\frac{w}{z}
= \frac{w\bar{z}}{|z|^2}
= \frac{(u + vi)(x - iy)}{x^2 + y^2}
= \frac{ux + vy}{x^2 + y^2} + \frac{vx - uy}{x^2 + y^2}i.
</math>

===Multiplication and division in polar form===
[[File:Complex multi.svg|right|thumb|Multiplication of {{math|2 + ''i''}} (blue triangle) and {{math|3 + ''i''}} (red triangle). The red triangle is rotated to match the vertex of the blue one (the adding of both angles in the terms ''φ''<sub>1</sub>+''φ''<sub>2</sub> in the equation) and stretched by the length of the [[Hypotenuse|hypotenuse]] of the blue triangle (the multiplication of both radiuses, as per term ''r''<sub>1</sub>''r''<sub>2</sub> in the equation).]]
Formulas for multiplication, division and exponentiation are simpler in polar form than the corresponding formulas in Cartesian coordinates. Given two complex numbers {{math|1=''z''<sub>1</sub> = ''r''<sub>1</sub>(cos ''φ''<sub>1</sub> + ''i'' sin ''φ''<sub>1</sub>)}} and {{math|1=''z''<sub>2</sub> = ''r''<sub>2</sub>(cos ''φ''<sub>2</sub> + ''i'' sin ''φ''<sub>2</sub>)}}, because of the trigonometric identities
<math display=block>\begin{alignat}{4}
  \cos a \cos b & - \sin a \sin b & {}={} & \cos(a + b) \\
  \cos a \sin b & + \sin a \cos b & {}={} & \sin(a + b) .
\end{alignat}</math>

we may derive

<math display=block>z_1 z_2 = r_1 r_2 (\cos(\varphi_1 + \varphi_2) + i \sin(\varphi_1 + \varphi_2)).</math>
In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. For example, multiplying by {{math|''i''}} corresponds to a quarter-[[Turn (geometry)|turn]] counter-clockwise, which gives back {{math|1=''i''<sup>2</sup> = −1}}. The picture at the right illustrates the multiplication of
<math display=block>(2+i)(3+i)=5+5i. </math>
Since the real and imaginary part of {{math|5 + 5''i''}} are equal, the argument of that number is 45 degrees, or {{math|''π''/4}} (in [[Radian|radian]]). On the other hand, it is also the sum of the angles at the origin of the red and blue triangles are arctan(1/3) and arctan(1/2), respectively. Thus, the formula
<math display=block>\frac{\pi}{4} = \arctan\left(\frac{1}{2}\right) + \arctan\left(\frac{1}{3}\right) </math>
holds. As the arctan function can be approximated highly efficiently, formulas like this – known as Machin-like formulas – are used for high-precision approximations of [[Pi|{{pi}}]].

Similarly, division is given by
<math display=block>\frac{z_1}{z_2} = \frac{r_1}{r_2} \left(\cos(\varphi_1 - \varphi_2) + i \sin(\varphi_1 - \varphi_2)\right).</math>

===Square root===
The square roots of {{math|''a'' + ''bi''}} (with {{math|''b'' ≠ 0}}) are <math> \pm (\gamma + \delta i)</math>, where

<math display=block>\gamma = \sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}}</math>

and

<math display=block>\delta = (\sgn b)\sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}},</math>

where {{math|sgn}} is the [[Sign function|signum]] function. This can be seen by squaring <math> \pm (\gamma + \delta i)</math> to obtain {{math|''a'' + ''bi''}}.<ref>{{cite book
|title=Handbook of mathematical functions with formulas, graphs, and mathematical tables
|first1=Milton
|last1=Abramowitz
|first2=Irene A.
|last2=Stegun
|publisher=Courier Dover Publications
|year=1964
|isbn=978-0-486-61272-0
|page=17
|url=https://books.google.com/books?id=MtU8uP7XMvoC
|access-date=16 February 2016
|archive-url=https://web.archive.org/web/20160423180235/https://books.google.com/books?id=MtU8uP7XMvoC
|archive-date=23 April 2016
|url-status=live
}}, [http://www.math.sfu.ca/~cbm/aands/page_17.htm Section 3.7.26, p.&nbsp;17] {{Webarchive|url=https://web.archive.org/web/20090910094533/http://www.math.sfu.ca/~cbm/aands/page_17.htm |date=10 September 2009 }}</ref><ref>{{cite book
|title=Classical Algebra: its nature, origins, and uses
|first1=Roger
|last1=Cooke
|publisher=John Wiley and Sons
|year=2008
|isbn=978-0-470-25952-8
|page=59
|url=https://books.google.com/books?id=lUcTsYopfhkC
|access-date=16 February 2016
|archive-url=https://web.archive.org/web/20160424023007/https://books.google.com/books?id=lUcTsYopfhkC
|archive-date=24 April 2016
|url-status=live
}}, [https://books.google.com/books?id=lUcTsYopfhkC&pg=PA59 Extract: page 59] {{Webarchive|url=https://web.archive.org/web/20160423183239/https://books.google.com/books?id=lUcTsYopfhkC&pg=PA59 |date=23 April 2016 }}</ref> Here <math>\sqrt{a^2 + b^2}</math> is called the [[Absolute value|modulus]] of {{math|''a'' + ''bi''}}, and the square root sign indicates the square root with non-negative real part, called the '''principal square root'''; also <math>\sqrt{a^2 + b^2}= \sqrt{z\overline{z}},</math> where {{math|''z'' {{=}} ''a'' + ''bi''}}.{{sfn|Ahlfors|1979|p=3}}

===Exponential function===
The [[Exponential function|exponential function]] <math>\exp \colon \Complex \to \Complex ; z \mapsto \exp z </math> can be defined for every complex number {{mvar|z}} by the [[Power series|power series]]
<math display=block>\exp z= \sum_{n=0}^\infty \frac {z^n}{n!},</math>
which has an infinite [[Radius of convergence|radius of convergence]].

The value at {{math|1}} of the exponential function is Euler's number
<math display=block>e = \exp 1 = \sum_{n=0}^\infty \frac1{n!}\approx 2.71828.</math>
If {{mvar|z}} is real, one has 
<math>\exp z=e^z.</math>
[[Analytic continuation]] allows extending this equality for every complex value of {{mvar|z}}, and thus to define the complex exponentiation with base {{mvar|e}} as
<math display=block>e^z=\exp z.</math>

====Functional equation====
The exponential function satisfies the [[Functional equation|functional equation]] <math>e^{z+t}=e^ze^t.</math>
This can be proved either by comparing the power series expansion of both members or by applying [[Analytic continuation|analytic continuation]] from the restriction of the equation to real arguments.

====Euler's formula====
[[Euler's formula]] states that, for any real number {{mvar|y}},
<math display=block>e^{iy} = \cos y + i\sin y .</math>

The functional equation implies thus that, if {{mvar|x}} and {{mvar|y}} are real, one has
<math display=block>e^{x+iy} = e^x(\cos y + i\sin y) = e^x \cos y + i e^x \sin y ,</math>
which is the decomposition of the exponential function into its real and imaginary parts.

===Complex logarithm===
In the real case, the [[Natural logarithm|natural logarithm]] can be defined as the [[Inverse function|inverse]] 
<math>\ln \colon \R^+ \to \R ; x \mapsto \ln x </math> of the exponential function. For extending this to the complex domain, one can start from Euler's formula. It implies that, if a complex number <math>z\in \Complex^\times</math> is written in polar form
<math display=block> z = r(\cos \varphi + i\sin \varphi )</math>
with <math>r, \varphi \in \R ,</math> then with
<math display=block> \ln z = \ln r + i \varphi </math>
as [[Complex logarithm|complex logarithm]] one has a proper inverse:
<math display=block> \exp \ln z = \exp(\ln r + i \varphi ) = r \exp i \varphi = r(\cos \varphi + i\sin \varphi ) = z .</math>

However, because cosine and sine are periodic functions, the addition of an integer multiple of {{math|2''π''}} to {{mvar|φ}} does not change {{mvar|z}}. For example, {{math|1=''e''{{sup|''iπ''}} = ''e''{{sup|3''iπ''}} = −1}} , so both {{mvar|iπ}} and {{math|3''iπ''}} are possible values for the natural logarithm of {{math|−1}}.

Therefore, if the complex logarithm is not to be defined as a [[Multivalued function|multivalued function]]
<math display=block> \ln z = \left\{ \ln r + i (\varphi + 2\pi k) \mid k \in \Z \right\},</math>
one has to use a branch cut and to restrict the [[Codomain|codomain]], resulting in the bijective function
<math display=block>\ln \colon \; \Complex^\times \; \to \; \; \; \R^+ + \; i \, \left(-\pi, \pi\right] .</math>

If <math>z \in \Complex \setminus \left( -\R_{\ge 0} \right)</math> is not a non-positive real number (a positive or a non-real number), the resulting [[Principal value|principal value]] of the complex logarithm is obtained with {{math|−''π'' < ''φ'' < ''π''}}. It is an [[Analytic function|analytic function]] outside the negative real numbers, but it cannot be prolongated to a function that is continuous at any negative real number <math>z \in -\R^+ </math>, where the principal value is {{math|1=ln ''z'' = ln(−''z'') + ''iπ''}}.{{efn|However for another inverse function of the complex exponential function (and not the above defined principal value), the branch cut could be taken at any other [[Line (geometry)#Ray|ray]] thru the origin.}}

===Exponentiation===
If {{math|''x'' > 0}} is real and {{mvar|z}} complex, the exponentiation is defined as 
<math display=block>x^z=e^{z\ln x},</math>
where {{math|ln}} denotes the natural logarithm.

It seems natural to extend this formula to complex values of {{mvar|x}}, but there are some difficulties resulting from the fact that the complex logarithm is not really a function, but a [[Multivalued function|multivalued function]].

It follows that if {{mvar|z}} is as above, and if {{mvar|t}} is another complex number, then the ''exponentiation'' is the multivalued function
<math display=block>z^t=\left\{e^{t(\ln r+ i (\varphi + 2\pi k))} \mid k\in \mathbb Z\right\}</math>

====Integer and fractional exponents====
If, in the preceding formula, {{mvar|t}} is an integer, then the sine and the cosine are independent of {{mvar|k}}. Thus, if the exponent {{mvar|n}} is an integer, then {{math|''z''{{sup|''n''}}}} is well defined, and the exponentiation formula simplifies to [[De Moivre's formula|de Moivre's formula]]:
<math display=block> z^{n}=(r(\cos \varphi + i\sin \varphi ))^n = r^n \, (\cos n\varphi + i \sin n \varphi).</math>

The {{mvar|n}} [[Nth root|{{mvar|n}}th roots]] of a complex number {{mvar|z}} are given by
<math display=block>z^{1/n} = \sqrt[n]r \left( \cos \left(\frac{\varphi+2k\pi}{n}\right) + i \sin \left(\frac{\varphi+2k\pi}{n}\right)\right)</math>
for {{math|0 ≤ ''k'' ≤ ''n'' − 1}}. (Here <math>\sqrt[n]r</math> is the usual (positive) {{mvar|n}}th root of the positive real number {{mvar|r}}.) Because sine and cosine are periodic, other integer values of {{mvar|k}} do not give other values.

While the {{mvar|n}}th root of a positive real number {{mvar|r}} is chosen to be the ''positive'' real number {{mvar|c}} satisfying {{math|1=''c''<sup>''n''</sup> = ''r''}}, there is no natural way of distinguishing one particular complex {{mvar|n}}th root of a complex number. Therefore, the {{mvar|n}}th root is a [[Multivalued function|{{mvar|n}}-valued function]] of {{mvar|z}}. This implies that, contrary to the case of positive real numbers, one has  
<math display=block>(z^n)^{1/n} \ne z,</math>
since the left-hand side consists of {{mvar|n}} values, and the right-hand side is a single value.

==Properties==

===Field structure===
The set <math>\Complex</math> of complex numbers is a [[Field (mathematics)|field]].{{sfn|Apostol|1981|pp=15–16}} Briefly, this means that the following facts hold: first, any two complex numbers can be added and multiplied to yield another complex number. Second, for any complex number {{mvar|z}}, its [[Additive inverse|additive inverse]] {{math|–''z''}} is also a complex number; and third, every nonzero complex number has a [[Multiplicative inverse|reciprocal]] complex number. Moreover, these operations satisfy a number of laws, for example the law of commutativity of addition and multiplication for any two complex numbers {{math|''z''<sub>1</sub>}} and {{math|''z''<sub>2</sub>}}:
<math display=block>\begin{align}
z_1 + z_2 & = z_2 + z_1 ,\\
z_1 z_2 & = z_2 z_1 .
\end{align}</math>
These two laws and the other requirements on a field can be proven by the formulas given above, using the fact that the real numbers themselves form a field.

Unlike the reals, <math>\Complex</math> is not an [[Ordered field|ordered field]], that is to say, it is not possible to define a relation {{math|''z''<sub>1</sub> < ''z''<sub>2</sub>}} that is compatible with the addition and multiplication. In fact, in any ordered field, the square of any element is necessarily positive, so {{math|1=''i''<sup>2</sup> = −1}} precludes the existence of an [[Total order|ordering]] on <math>\Complex.</math>{{sfn|Apostol|1981|p=25}}

When the underlying field for a mathematical topic or construct is the field of complex numbers, the topic's name is usually modified to reflect that fact. For example: [[Complex analysis|complex analysis]], complex [[Matrix (mathematics)|matrix]], complex [[Polynomial|polynomial]], and complex [[Lie algebra]].

===Solutions of polynomial equations===
Given any complex numbers (called [[Coefficient|coefficient]]s) {{math|''a''<sub>0</sub>, ..., ''a''<sub>''n''</sub>}}, the equation
<math display=block>a_n z^n + \dotsb + a_1 z + a_0 = 0</math>
has at least one complex solution ''z'', provided that at least one of the higher coefficients {{math|''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>}} is nonzero.<ref name="Bourbaki 1998 loc=§VIII.1"/> This is the statement of the [[Fundamental theorem of algebra|fundamental theorem of algebra]], of [[Biography:Carl Friedrich Gauss|Carl Friedrich Gauss]] and [[Biography:Jean le Rond d'Alembert|Jean le Rond d'Alembert]]. Because of this fact, <math>\Complex</math> is called an [[Algebraically closed field|algebraically closed field]]. This property does not hold for the [[Rational number|field of rational numbers]] <math>\Q</math> (the polynomial {{math|''x''<sup>2</sup> − 2}} does not have a rational root, since [[Square root of 2|{{math|√2}}]] is not a rational number) nor the real numbers <math>\R</math> (the polynomial {{math|''x''<sup>2</sup> + 4}} does not have a real root, since the square of {{mvar|x}} is positive for any real number {{mvar|x}}).

There are various proofs of this theorem, by either analytic methods such as [[Liouville's theorem (complex analysis)|Liouville's theorem]], or [[Topology|topological]] ones such as the [[Winding number|winding number]], or a proof combining [[Galois theory]] and the fact that any real polynomial of ''odd'' degree has at least one real root.

Because of this fact, theorems that hold ''for any algebraically closed field'' apply to <math>\Complex.</math> For example, any non-empty complex [[Square matrix|square matrix]] has at least one (complex) eigenvalue.

===Algebraic characterization===
The field <math>\Complex</math> has the following three properties:
* First, it has [[Characteristic (algebra)|characteristic]] 0. This means that {{math|1=1 + 1 + ⋯ + 1 ≠ 0}} for any number of summands (all of which equal one).
* Second, its [[Transcendence degree|transcendence degree]] over <math>\Q</math>, the prime field of <math>\Complex,</math> is the [[Cardinality of the continuum|cardinality of the continuum]].
* Third, it is algebraically closed (see above).
It can be shown that any field having these properties is isomorphic (as a field) to <math>\Complex.</math> For example, the algebraic closure of the field <math>\Q_p</math> of the [[P-adic number|{{mvar|p}}-adic number]] also satisfies these three properties, so these two fields are isomorphic (as fields, but not as topological fields).<ref>{{cite book
 | last = Marker | first = David
 | editor1-last = Marker | editor1-first = D.
 | editor2-last = Messmer | editor2-first = M.
 | editor3-last = Pillay | editor3-first = A.
 | contribution = Introduction to the Model Theory of Fields
 | contribution-url = https://projecteuclid.org/euclid.lnl/1235423155
 | isbn = 978-3-540-60741-0
 | mr = 1477154
 | pages = 1–37
 | publisher = Springer-Verlag | location = Berlin
 | series = Lecture Notes in Logic
 | title = Model theory of fields
 | volume = 5
 | year = 1996}}</ref> Also, <math>\Complex</math> is isomorphic to the field of complex [[Puiseux series]]. However, specifying an isomorphism requires the [[Axiom of choice|axiom of choice]]. Another consequence of this algebraic characterization is that <math>\Complex</math> contains many proper subfields that are isomorphic to <math>\Complex</math>.

===Characterization as a topological field===
The preceding characterization of <math>\Complex</math> describes only the algebraic aspects of <math>\Complex.</math> That is to say, the properties of nearness and continuity, which matter in areas such as [[Mathematical analysis|analysis]] and [[Topology|topology]], are not dealt with. The following description of <math>\Complex</math> as a [[Topological ring|topological field]] (that is, a field that is equipped with a [[Topological space|topology]], which allows the notion of convergence) does take into account the topological properties. <math>\Complex</math> contains a subset {{math|''P''}} (namely the set of positive real numbers) of nonzero elements satisfying the following three conditions:
* {{math|''P''}} is closed under addition, multiplication and taking inverses.
* If {{mvar|x}} and {{mvar|y}} are distinct elements of {{math|''P''}}, then either {{math|''x'' − ''y''}} or {{math|''y'' − ''x''}} is in {{math|''P''}}.
* If {{mvar|S}} is any nonempty subset of {{math|''P''}}, then {{math|1=''S'' + ''P'' = ''x'' + ''P''}} for some {{mvar|x}} in <math>\Complex.</math>
Moreover, <math>\Complex</math> has a nontrivial [[Involution (mathematics)|involutive]] [[Automorphism|automorphism]] {{math|''x'' ↦ ''x''*}} (namely the complex conjugation), such that {{math|''x x''*}} is in {{math|''P''}} for any nonzero {{mvar|x}} in <math>\Complex.</math>

Any field {{mvar|F}} with these properties can be endowed with a topology by taking the sets {{math|1= ''B''(''x'', ''p'') = { ''y'' {{!}} ''p'' − (''y'' − ''x'')(''y'' − ''x'')* ∈ ''P'' } }} as a [[Base (topology)|base]], where {{mvar|x}} ranges over the field and {{mvar|p}} ranges over {{math|''P''}}. With this topology {{mvar|F}} is isomorphic as a ''topological'' field to <math>\Complex.</math>

The only [[Connected space|connected]] locally compact [[Topological ring|topological fields]] are <math>\R</math> and <math>\Complex.</math> This gives another characterization of <math>\Complex</math> as a topological field, since <math>\Complex</math> can be distinguished from <math>\R</math> because the nonzero complex numbers are [[Connected space|connected]], while the nonzero real numbers are not.{{sfn|Bourbaki|1998|loc=§VIII.4}}

==Formal construction==
===Construction as ordered pairs===
[[Biography:William Rowan Hamilton|William Rowan Hamilton]] introduced the approach to define the set <math>\Complex</math> of complex numbers<ref>{{cite book|title=A Brief History of Numbers|first=Leo |last=Corry|publisher=Oxford University Press|year=2015|pages=215–16}}</ref> as the set <math>\mathbb{R}^2</math> of {{nowrap|[[Ordered pair|ordered pair]]s {{math|(''a'', ''b'')}}}} of real numbers, in which the following rules for addition and multiplication are imposed:{{sfn|Apostol|1981|pp=15–16}}

<math display=block>\begin{align}
(a, b) + (c, d) &= (a + c, b + d)\\
(a, b) \cdot (c, d) &= (ac - bd, bc + ad).
\end{align}</math>

It is then just a matter of notation to express {{math|(''a'', ''b'')}} as {{math|''a'' + ''bi''}}.

===Construction as a quotient field===
Though this low-level construction does accurately describe the structure of the complex numbers, the following equivalent definition reveals the algebraic nature of <math>\Complex</math> more immediately. This characterization relies on the notion of fields and polynomials. A field is a set endowed with addition, subtraction, multiplication and division operations that behave as is familiar from, say, rational numbers. For example, the distributive law
<math display=block>(x+y) z = xz + yz</math>
must hold for any three elements {{mvar|x}}, {{mvar|y}} and {{mvar|z}} of a field. The set <math>\R</math> of real numbers does form a field. A polynomial {{math|''p''(''X'')}} with real [[Coefficient|coefficient]]s is an expression of the form
<math display=block>a_nX^n+\dotsb+a_1X+a_0,</math>
where the {{math|''a''<sub>0</sub>, ..., ''a''<sub>''n''</sub>}} are real numbers. The usual addition and multiplication of polynomials endows the set <math>\R[X]</math> of all such polynomials with a [[Ring (mathematics)|ring]] structure. This ring is called the [[Polynomial ring|polynomial ring]] over the real numbers.

The set of complex numbers is defined as the [[Quotient ring|quotient ring]] <math>\R[X]/(X^2+1).</math><ref name="Bourbaki 1998 loc=§VIII.1"/> This extension field contains two square roots of {{math|−1}}, namely (the [[Coset|coset]]s of) {{math|''X''}} and {{math|−''X''}}, respectively. (The cosets of) {{math|1}} and {{math|''X''}} form a basis of <math>\mathbb{R}[X]/(X^2 + 1)</math> as a real [[Vector space|vector space]], which means that each element of the extension field can be uniquely written as a [[Linear combination|linear combination]] in these two elements. Equivalently, elements of the extension field can be written as ordered pairs {{math|(''a'', ''b'')}} of real numbers.  The quotient ring is a field, because {{math|''X''<sup>2</sup> + 1}} is [[Irreducible polynomial|irreducible]] over <math>\R,</math> so the ideal it generates is [[Maximal ideal|maximal]].

The formulas for addition and multiplication in the ring <math>\R[X],</math> modulo the relation {{math|''X''<sup>2</sup> {{=}} −1}}, correspond to the formulas for addition and multiplication of complex numbers defined as ordered pairs.  So the two definitions of the field <math>\Complex</math> are [[Isomorphism|isomorphic]] (as fields).

Accepting that <math>\Complex</math> is algebraically closed, since it is an algebraic extension of <math>\mathbb{R}</math> in this approach, <math>\Complex</math> is therefore the algebraic closure of <math>\R.</math>

===Matrix representation of complex numbers===<!-- .This section is linked from Cauchy-Riemann equations -->
Complex numbers {{math|''a'' + ''bi''}} can also be represented by {{math|2 × 2}} [[Matrix (mathematics)|matrices]] that have the form
<!-- 
This definition with the minus sign in the upper right corner matches the article [[Rotation matrix]]. Please do not change it.
-->
<math display=block>
\begin{pmatrix}
  a &   -b  \\
  b & \;\; a
\end{pmatrix}.
</math>
Here the entries {{mvar|a}} and {{mvar|b}} are real numbers. As the sum and product of two such matrices is again of this form, these matrices form a [[Subring|subring]] of the ring of {{math|2 × 2}} matrices.

A simple computation shows that the map
<math display=block>a+ib\mapsto \begin{pmatrix}
  a &   -b  \\
  b & \;\; a
\end{pmatrix}</math>
is a ring isomorphism from the field of complex numbers to the ring of these matrices, proving that these matrices form a field. This isomorphism associates the square of the absolute value of a complex number with the [[Determinant|determinant]] of the corresponding matrix, and the conjugate of a complex number with the [[Transpose|transpose]] of the matrix.

The geometric description of the multiplication of complex numbers can also be expressed in terms of [[Rotation matrix|rotation matrices]] by using this correspondence between complex numbers and such matrices. The action of the matrix on a vector {{math|(''x'', ''y'')}} corresponds to the multiplication of {{math|''x'' + ''iy''}} by {{math|''a'' + ''ib''}}. In particular, if the determinant is {{math|1}}, there is a real number {{mvar|t}} such that the matrix has the form 

<math display=block>\begin{pmatrix}
  \cos t & - \sin t   \\
  \sin t & \;\; \cos t
\end{pmatrix}.</math>
In this case, the action of the matrix on vectors and the multiplication by the complex number <math>\cos t+i\sin t</math> are both the [[Rotation (mathematics)|rotation]] of the angle {{mvar|t}}.

==Complex analysis==
[[File:Sin1z-cplot.svg|thumb|[[Domain coloring|Color wheel graph]] of {{math|sin(1/''z'')}}. White parts inside refer to numbers having large absolute values.]]
{{main|Complex analysis}}
<!--
[[File:Color complex plot.jpg|upright=0.8|right|thumb|[[Domain coloring]] plot of the function
<br /><math>f(x) = \tfrac{(x^2 - 1)(x - 2 - i)^2}{x^2 + 2 + 2 i}</math><br />
The hue represents the function argument, while the saturation and value represent the magnitude.]]

The absolute value has three important properties:

<math display=block> | z | \geq 0, \,</math> where <math> | z | = 0 \,</math> [[If and only if|if and only if]] <math> z = 0</math>

<math display=block> | z + w | \leq | z | + | w | \,</math> ([[Triangle inequality|triangle inequality]])

<math display=block> | z \cdot w | = | z | \cdot | w | </math>

for all complex numbers {{mvar|z}} and {{mvar|w}}. These imply that {{math|1={{!}}1{{!}} = 1}} and {{math|1={{!}}''z''/''w''{{!}} = {{!}}''z''{{!}}/{{!}}''w''{{!}}}}. By defining the '''distance''' function {{math|1=''d''(''z'', ''w'') = {{!}}''z'' − ''w''{{!}}}}, we turn the set of complex numbers into a [[Metric space|metric space]] and we can therefore talk about [[Limit (mathematics)|limits]] and [[Continuous function|continuity]].

In general, distances between complex numbers are given by the distance function {{math|1=''d''(''z'', ''w'') = {{!}}''z'' − ''w''{{!}}}}, which turns the complex numbers into a [[Metric space|metric space]] and introduces the ideas of [[Limit (mathematics)|limits]] and [[Continuous function|continuity]]. All of the standard properties of two dimensional space therefore hold for the complex numbers, including important properties of the modulus such as non-negativity, and the [[Triangle inequality|triangle inequality]] (<math>| z + w | \leq | z | + | w |</math> for all {{mvar|z}} and {{mvar|w}}).

-->

The study of functions of a complex variable is known as [[Complex analysis|complex analysis]] and has enormous practical use in [[Applied mathematics|applied mathematics]] as well as in other branches of mathematics. Often, the most natural proofs for statements in [[Real analysis|real analysis]] or even [[Number theory|number theory]] employ techniques from complex analysis (see [[Prime number theorem|prime number theorem]] for an example). Unlike real functions, which are commonly represented as two-dimensional graphs, complex functions have four-dimensional graphs and may usefully be illustrated by color-coding a three-dimensional graph to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.

===Complex exponential and related functions===
The notions of [[Convergent series|convergent series]] and [[Continuous function|continuous function]]s in (real) analysis have natural analogs in complex analysis. A sequence <!--(''a''<sub>''n''</sub>)<sub>''n'' ≥ 0</sub>--> of complex numbers is said to converge if and only if its real and imaginary parts do. This is equivalent to the [[(ε, δ)-definition of limit|(ε, δ)-definition of limit]]s, where the absolute value of real numbers is replaced by the one of complex numbers. From a more abstract point of view, <math>\mathbb{C}</math>, endowed with the [[Metric (mathematics)|metric]]
<math display=block>\operatorname{d}(z_1, z_2) = |z_1 - z_2|</math>
is a complete [[Metric space|metric space]], which notably includes the [[Triangle inequality|triangle inequality]]
<math display=block>|z_1 + z_2| \le |z_1| + |z_2|</math>
for any two complex numbers {{math|''z''<sub>1</sub>}} and {{math|''z''<sub>2</sub>}}.

Like in real analysis, this notion of convergence is used to construct a number of [[Elementary function|elementary function]]s: the ''[[Exponential function|exponential function]]'' {{math|exp ''z''}}, also written {{math|''e''<sup>''z''</sup>}}, is defined as the infinite series
<math display=block>\exp z:= 1+z+\frac{z^2}{2\cdot 1}+\frac{z^3}{3\cdot 2\cdot 1}+\cdots = \sum_{n=0}^{\infty} \frac{z^n}{n!}. </math>

The series defining the real trigonometric functions [[Sine|sine]] and [[Cosine|cosine]], as well as the [[Hyperbolic functions|hyperbolic functions]] sinh and cosh, also carry over to complex arguments without change. For the other trigonometric and hyperbolic functions, such as tangent, things are slightly more complicated, as the defining series do not converge for all complex values. Therefore, one must define them either in terms of sine, cosine and exponential, or, equivalently, by using the method of [[Analytic continuation|analytic continuation]].

''[[Euler's formula]]'' states:
<math display=block>\exp(i\varphi) = \cos \varphi + i\sin \varphi </math>
for any real number {{mvar|φ}}, in particular
<math display=block>\exp(i \pi) = -1 </math>, which is [[Euler's identity]].
Unlike in the situation of real numbers, there is an [[Infinite set|infinitude]] of complex solutions {{mvar|z}} of the equation
<math display=block>\exp z = w </math>
for any complex number {{math|''w'' ≠ 0}}. It can be shown that any such solution {{mvar|z}} – called [[Complex logarithm|complex logarithm]] of {{mvar|w}} – satisfies
<math display=block>\log w = \ln|w| + i\arg w, </math>
where arg is the argument defined above, and ln the (real) [[Natural logarithm|natural logarithm]]. As arg is a [[Multivalued function|multivalued function]], unique only up to a multiple of {{math|2''π''}}, log is also multivalued. The [[Principal value|principal value]] of log is often taken by restricting the imaginary part to the [[Interval (mathematics)|interval]] {{open-closed|−''π'', ''π''}}.

Complex [[Exponentiation|exponentiation]] {{math|''z''<sup>''ω''</sup>}} is defined as
<math display=block>z^\omega = \exp(\omega \ln z), </math>
and is multi-valued, except when {{mvar|ω}} is an integer. For {{math|1=''ω'' = 1 / ''n''}}, for some natural number {{mvar|n}}, this recovers the non-uniqueness of {{mvar|n}}th roots mentioned above.

Complex numbers, unlike real numbers, do not in general satisfy the unmodified power and logarithm identities, particularly when naïvely treated as single-valued functions; see [[Exponentiation#Failure of power and logarithm identities|failure of power and logarithm identities]]. For example, they do not satisfy
<math display=block>a^{bc} = \left(a^b\right)^c.</math>
Both sides of the equation are multivalued by the definition of complex exponentiation given here, and the values on the left are a subset of those on the right.

===Holomorphic functions===
A function ''f'': <math>\mathbb{C}</math> → <math>\mathbb{C}</math> is said to be [[Holomorphic function|holomorphic]] at a point if it is complex differentiable in an open neighborhood of that point. A necessary (but not sufficient) condition for ''f'' to be holomorphic is that it satisfies the [[Cauchy–Riemann equations]]. For example, any <math>\mathbb{R}</math>-linear map <math>\mathbb{C}</math> → <math>\mathbb{C}</math> can be written in the form
<math display=block>f(z)=az+b\overline{z}</math>
with complex coefficients {{mvar|a}} and {{mvar|b}}. This map is holomorphic [[If and only if|if and only if]] {{math|1=''b'' = 0}}. The function <math>\overline z</math> is real-differentiable, but does not satisfy the [[Cauchy–Riemann equations]].

Complex analysis shows some features not apparent in real analysis. For example, any two holomorphic functions {{mvar|f}} and {{mvar|g}} that agree on an arbitrarily small open subset of <math>\mathbb{C}</math> necessarily agree everywhere. [[Meromorphic function]]s, functions that can locally be written as {{math|''f''(''z'')/(''z'' − ''z''<sub>0</sub>)<sup>''n''</sup>}} with a holomorphic function {{mvar|f}}, still share some of the features of holomorphic functions. Other functions have [[Essential singularity|essential singularities]], such as {{math|sin(1/''z'')}} at {{math|1=''z'' = 0}}.

==Applications==
Complex numbers have applications in many scientific areas, including [[Signal processing|signal processing]], [[Control theory|control theory]], [[Physics:Electromagnetism|electromagnetism]], [[Physics:Fluid dynamics|fluid dynamics]], [[Physics:Quantum mechanics|quantum mechanics]], [[Earth:Cartography|cartography]], and [[Vibration#Vibration analysis|vibration analysis]]. Some of these applications are described below.

===Geometry===
====Shapes====
Three [[Collinearity|non-collinear]] points <math>u, v, w</math> in the plane determine the [[Shape#Similarity classes|shape]] of the triangle <math>\{u, v, w\}</math>. Locating the points in the complex plane, this shape of a triangle may be expressed by complex arithmetic as
<math display=block>S(u, v, w) = \frac {u - w}{u - v}. </math>
The shape <math>S</math> of a triangle will remain the same, when the complex plane is transformed by translation or dilation (by an [[Affine transformation|affine transformation]]), corresponding to the intuitive notion of shape, and describing [[Similarity (geometry)|similarity]]. Thus each triangle <math>\{u, v, w\}</math> is in a [[Shape#Similarity classes|similarity class]] of triangles with the same shape.<ref>{{cite journal |last=Lester |first=J.A. |title=Triangles I: Shapes |journal=[[Aequationes Mathematicae]] |volume=52 |pages=30–54 |year=1994 |doi=10.1007/BF01818325 |s2cid=121095307}}</ref>

====Fractal geometry====
[[File:Mandelset hires.png|right|thumb|The Mandelbrot set with the real and imaginary axes labeled.]]
The [[Mandelbrot set]] is a popular example of a fractal formed on the complex plane. It is defined by plotting every location <math>c</math> where iterating the sequence <math>f_c(z)=z^2+c</math> does not diverge when [[Iteration|iterated]] infinitely. Similarly, [[Julia set]]s have the same rules, except where <math>c</math> remains constant.

====Triangles====
Every triangle has a unique [[Steiner inellipse]] – an [[Ellipse|ellipse]] inside the triangle and tangent to the midpoints of the three sides of the triangle. The [[Focus (geometry)|foci]] of a triangle's Steiner inellipse can be found as follows, according to [[Marden's theorem]]:<ref>{{cite journal |last1=Kalman|first1=Dan|title=An Elementary Proof of Marden's Theorem |url=http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3338&pf=1 |journal=[[American Mathematical Monthly]] |volume=115 |issue=4 |pages=330–38 |year=2008a |doi=10.1080/00029890.2008.11920532 |s2cid=13222698 |issn=0002-9890 |access-date=1 January 2012 |archive-url=https://web.archive.org/web/20120308104622/http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3338&pf=1 |archive-date=8 March 2012|url-status=live}}</ref><ref>{{cite journal |last1=Kalman |first1=Dan |title=The Most Marvelous Theorem in Mathematics |url=http://mathdl.maa.org/mathDL/4/?pa=content&sa=viewDocument&nodeId=1663 |journal=Journal of Online Mathematics and Its Applications |year=2008b |access-date=1 January 2012|archive-url=https://web.archive.org/web/20120208014954/http://mathdl.maa.org/mathDL/4/?pa=content&sa=viewDocument&nodeId=1663 |archive-date=8 February 2012 |url-status=live}}</ref> Denote the triangle's vertices in the complex plane as {{math|1=''a'' = ''x''<sub>''A''</sub> + ''y''<sub>''A''</sub>''i''}}, {{math|1=''b'' = ''x''<sub>''B''</sub> + ''y''<sub>''B''</sub>''i''}}, and {{math|1=''c'' = ''x''<sub>''C''</sub> + ''y''<sub>''C''</sub>''i''}}. Write the [[Cubic equation|cubic equation]] <math>(x-a)(x-b)(x-c)=0</math>, take its derivative, and equate the (quadratic) derivative to zero. Marden's theorem says that the solutions of this equation are the complex numbers denoting the locations of the two foci of the Steiner inellipse.

===Algebraic number theory===
[[File:Pentagon construct.gif|right|thumb|Construction of a regular pentagon using straightedge and compass.]]
As mentioned above, any nonconstant polynomial equation (in complex coefficients) has a solution in <math>\mathbb{C}</math>. ''[[Philosophy:Argumentum a fortiori|A fortiori]]'', the same is true if the equation has rational coefficients. The roots of such equations are called [[Algebraic number|algebraic number]]s – they are a principal object of study in [[Algebraic number theory|algebraic number theory]]. Compared to <math>\overline{\mathbb{Q}}</math>, the algebraic closure of <math>\mathbb{Q}</math>, which also contains all algebraic numbers, <math>\mathbb{C}</math> has the advantage of being easily understandable in geometric terms. In this way, algebraic methods can be used to study geometric questions and vice versa. With algebraic methods, more specifically applying the machinery of [[Field theory (mathematics)|field theory]] to the number field containing [[Root of unity|roots of unity]], it can be shown that it is not possible to construct a regular [[Nonagon|nonagon]] using only compass and straightedge – a purely geometric problem.

Another example is the [[Gaussian integer]]s; that is, numbers of the form {{math|''x'' + ''iy''}}, where {{mvar|x}} and {{mvar|y}} are integers, which can be used to classify [[Fermat's theorem on sums of two squares|sums of squares]].

===Analytic number theory===
{{main|Analytic number theory}}
Analytic number theory studies numbers, often integers or rationals, by taking advantage of the fact that they can be regarded as complex numbers, in which analytic methods can be used. This is done by encoding number-theoretic information in complex-valued functions. For example, the [[Riemann zeta function]] {{math|ζ(''s'')}} is related to the distribution of [[Prime number|prime number]]s.

===Improper integrals===
In applied fields, complex numbers are often used to compute certain real-valued [[Improper integral|improper integral]]s, by means of complex-valued functions. Several methods exist to do this; see methods of contour integration.

===Dynamic equations===
In [[Differential equation|differential equation]]s, it is common to first find all complex roots {{mvar|r}} of the [[Linear differential equation#Homogeneous equation with constant coefficients|characteristic equation]] of a [[Linear differential equation|linear differential equation]] or equation system and then attempt to solve the system in terms of base functions of the form {{math|1=''f''(''t'') = ''e''<sup>''rt''</sup>}}. Likewise, in difference equations, the complex roots {{mvar|r}} of the characteristic equation of the difference equation system are used, to attempt to solve the system in terms of base functions of the form {{math|1=''f''(''t'') = ''r''<sup>''t''</sup>}}.

=== Linear algebra ===
[[Eigendecomposition of a matrix|Eigendecomposition]] is a useful tool for computing matrix powers and [[Matrix exponential|matrix exponential]]s. However, it often requires the use of complex numbers, even if the matrix is real (for example, a [[Rotation matrix|rotation matrix]]).

Complex numbers often generalize concepts originally conceived in the real numbers. For example, the [[Conjugate transpose|conjugate transpose]] generalizes the [[Transpose|transpose]], [[Hermitian matrix|hermitian matrices]] generalize [[Symmetric matrix|symmetric matrices]], and [[Unitary matrix|unitary matrices]] generalize [[Orthogonal matrix|orthogonal matrices]].

===In applied mathematics===

====Control theory====

In [[Control theory|control theory]], systems are often transformed from the [[Time domain|time domain]] to the complex [[Frequency domain|frequency domain]] using the [[Laplace transform]]. The system's [[Zeros and poles|zeros and poles]] are then analyzed in the ''complex plane''. The [[Root locus|root locus]], [[Nyquist plot]], and [[Nichols plot]] techniques all make use of the complex plane.

In the root locus method, it is important whether zeros and poles are in the left or right half planes, that is, have real part greater than or less than zero. If a linear, time-invariant (LTI) system has poles that are

* in the right half plane, it will be unstable,
* all in the left half plane, it will be [[BIBO stability|stable]],
* on the imaginary axis, it will have [[Marginal stability|marginal stability]].

If a system has zeros in the right half plane, it is a nonminimum phase system.

====Signal analysis====
Complex numbers are used in signal analysis and other fields for a convenient description for periodically varying signals. For given real functions representing actual physical quantities, often in terms of sines and cosines, corresponding complex functions are considered of which the real parts are the original quantities. For a [[Physics:Sine wave|sine wave]] of a given [[Physics:Frequency|frequency]], the absolute value {{math|{{!}}''z''{{!}}}} of the corresponding {{mvar|z}} is the [[Physics:Amplitude|amplitude]] and the [[Argument (complex analysis)|argument]] {{math|arg ''z''}} is the [[Physics:Phase (waves)|phase]].

If [[Fourier analysis]] is employed to write a given real-valued signal as a sum of periodic functions, these periodic functions are often written as complex-valued functions of the form

<math display=block>x(t) = \operatorname{Re} \{X( t ) \} </math>

and

<math display=block>X( t ) = A e^{i\omega t} = a e^{ i \phi } e^{i\omega t} = a e^{i (\omega t + \phi) } </math>

where ω represents the [[Physics:Angular frequency|angular frequency]] and the complex number ''A'' encodes the phase and amplitude as explained above.

This use is also extended into [[Digital signal processing|digital signal processing]] and [[Digital image processing|digital image processing]], which use digital versions of Fourier analysis (and [[Wavelet|wavelet]] analysis) to transmit, [[Data compression|compress]], restore, and otherwise process [[Digital data|digital]] [[Physics:Sound|audio]] signals, still images, and [[Engineering:Video|video]] signals.

Another example, relevant to the two side bands of [[Physics:Amplitude modulation|amplitude modulation]] of AM radio, is:

<math display=block>\begin{align}
  \cos((\omega + \alpha)t) + \cos\left((\omega - \alpha)t\right)
    & = \operatorname{Re}\left(e^{i(\omega + \alpha)t} + e^{i(\omega - \alpha)t}\right) \\
    & = \operatorname{Re}\left(\left(e^{i\alpha t} + e^{-i\alpha t}\right) \cdot e^{i\omega t}\right) \\
    & = \operatorname{Re}\left(2\cos(\alpha t) \cdot e^{i\omega t}\right) \\
    & = 2 \cos(\alpha t) \cdot \operatorname{Re}\left(e^{i\omega t}\right) \\
    & = 2 \cos(\alpha t) \cdot \cos\left(\omega t\right).
\end{align}</math>

===In physics===
====Electromagnetism and electrical engineering====
{{Main|Physics:Alternating current}}
In [[Electrical engineering|electrical engineering]], the [[Fourier transform]] is used to analyze varying [[Physics:Voltage|voltage]]s and [[Physics:Electric current|currents]]. The treatment of [[Engineering:Resistor|resistor]]s, [[Engineering:Capacitor|capacitor]]s, and [[Engineering:Inductor|inductor]]s can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the [[Physics:Electrical impedance|impedance]]. This approach is called [[Phasor|phasor]] calculus.

In electrical engineering, the imaginary unit is denoted by {{mvar|j}}, to avoid confusion with {{mvar|I}}, which is generally in use to denote [[Physics:Electric current|electric current]], or, more particularly, {{mvar|i}}, which is generally in use to denote instantaneous electric current.

Since the [[Physics:Voltage|voltage]] in an AC circuit is oscillating, it can be represented as

<math display=block> V(t) = V_0 e^{j \omega t} = V_0 \left (\cos\omega t + j \sin\omega t \right ),</math>

To obtain the measurable quantity, the real part is taken:

<math display=block> v(t) = \operatorname{Re}(V) = \operatorname{Re}\left [ V_0 e^{j \omega t} \right ] = V_0 \cos \omega t.</math>

The complex-valued signal {{math|''V''(''t'')}} is called the [[Analytic signal|analytic]] representation of the real-valued, measurable signal {{math|''v''(''t'')}}.
<ref>{{cite book |last1=Grant |first1=I.S. |title=Electromagnetism |year=2008|edition=2 |publisher=Manchester Physics Series |isbn=978-0-471-92712-9 |last2=Phillips |first2=W.R.}}</ref>

====Fluid dynamics====
In [[Physics:Fluid dynamics|fluid dynamics]], complex functions are used to describe potential flow in two dimensions.

====Quantum mechanics====
The complex number field is intrinsic to the mathematical formulations of quantum mechanics, where complex [[Hilbert space]]s provide the context for one such formulation that is convenient and perhaps most standard. The original foundation formulas of quantum mechanics – the [[Physics:Schrödinger equation|Schrödinger equation]] and Heisenberg's [[Physics:Matrix mechanics|matrix mechanics]] – make use of complex numbers.

====Relativity====
In [[Physics:Special relativity|special]] and [[Physics:General relativity|general relativity]], some formulas for the metric on [[Physics:Spacetime|spacetime]] become simpler if one takes the time component of the spacetime continuum to be imaginary. (This approach is no longer standard in classical relativity, but is [[Physics:Wick rotation|used in an essential way]] in [[Physics:Quantum field theory|quantum field theory]].) Complex numbers are essential to [[Physics:Spinor|spinor]]s, which are a generalization of the [[Tensor|tensor]]s used in relativity.

==Generalizations and related notions==
[[File:Cayley_Q8_quaternion_multiplication_graph.svg|thumb|upright=1.05|link={{filepath:Cayley_Q8_quaternion_multiplication_graph.svg}}|Cayley Q8 quaternion graph showing cycles of multiplication by {{color|#e00000|'''i'''}}, {{green|'''j'''}} and {{blue|'''k'''}}]]
The process of extending the field <math>\mathbb R</math> of reals to <math>\mathbb C</math> is known as the [[Cayley–Dickson construction]]. It can be carried further to higher dimensions, yielding the [[Quaternion|quaternion]]s <math>\mathbb H</math> and [[Octonion|octonion]]s <math>\mathbb{O}</math> which (as a real vector space) are of dimension&nbsp;4 and 8, respectively.
In this context the complex numbers have been called the '''binarions'''.<ref>{{cite book |first=Kevin |last=McCrimmon |year=2004 |title=A Taste of Jordan Algebras |page=64 |series=Universitext |publisher=Springer |isbn=0-387-95447-3}} {{mr|id=2014924}}</ref>

Just as by applying the construction to reals the property of [[Ordered field|ordering]] is lost, properties familiar from real and complex numbers vanish with each extension. The [[Quaternion|quaternion]]s lose commutativity, that is, {{math|''x''·''y'' ≠ ''y''·''x''}} for some quaternions {{math|''x'', ''y''}}, and the multiplication of [[Octonion|octonion]]s, additionally to not being commutative, fails to be associative: {{math|(''x''·''y'')·''z'' ≠ ''x''·(''y''·''z'')}} for some octonions {{math|''x'', ''y'', ''z''}}.

Reals, complex numbers, quaternions and octonions are all normed division algebras over <math>\mathbb R</math>. By [[Hurwitz's theorem (normed division algebras)|Hurwitz's theorem]] they are the only ones; the [[Sedenion|sedenion]]s, the next step in the Cayley–Dickson construction, fail to have this structure.

The Cayley–Dickson construction is closely related to the [[Regular representation|regular representation]] of <math>\mathbb C,</math> thought of as an <math>\mathbb R</math>-[[Algebra (ring theory)|algebra]] (an <math>\mathbb{R}</math>-vector space with a multiplication), with respect to the basis {{math|(1, ''i'')}}. This means the following: the <math>\mathbb R</math>-linear map
<math display=block>\begin{align}
   \mathbb{C} &\rightarrow \mathbb{C} \\
   z &\mapsto wz
 \end{align}</math>
for some fixed complex number {{mvar|w}} can be represented by a {{math|2 × 2}} matrix (once a basis has been chosen). With respect to the basis {{math|(1, ''i'')}}, this matrix is
<math display=block>\begin{pmatrix}
  \operatorname{Re}(w) & -\operatorname{Im}(w) \\
  \operatorname{Im}(w) &  \operatorname{Re}(w)
 \end{pmatrix},</math>
that is, the one mentioned in the section on matrix representation of complex numbers above. While this is a linear representation of <math>\mathbb C</math> in the 2 × 2 real matrices, it is not the only one. Any matrix
<math display=block>J = \begin{pmatrix}p & q \\ r & -p \end{pmatrix}, \quad p^2 + qr + 1 = 0</math>
has the property that its square is the negative of the identity matrix: {{math|1=''J''<sup>2</sup> = −''I''}}. Then
<math display=block>\{ z = a I + b J : a,b \in \mathbb{R} \}</math>
is also isomorphic to the field <math>\mathbb C,</math> and gives an alternative complex structure on <math>\mathbb R^2.</math> This is generalized by the notion of a [[Linear complex structure|linear complex structure]].

[[Hypercomplex number]]s also generalize <math>\mathbb R,</math> <math>\mathbb C,</math> <math>\mathbb H,</math> and <math>\mathbb{O}.</math> For example, this notion contains the [[Split-complex number|split-complex number]]s, which are elements of the ring <math>\mathbb R[x]/(x^2-1)</math> (as opposed to <math>\mathbb R[x]/(x^2+1)</math> for complex numbers). In this ring, the equation {{math|1=''a''<sup>2</sup> = 1}} has four solutions.

The field <math>\mathbb R</math> is the completion of <math>\mathbb Q,</math> the field of [[Rational number|rational number]]s, with respect to the usual [[Absolute value|absolute value]] [[Metric (mathematics)|metric]]. Other choices of [[Metric (mathematics)|metrics]] on <math>\mathbb Q</math> lead to the fields <math>\mathbb Q_p</math> of [[P-adic number|{{mvar|p}}-adic numbers]] (for any [[Prime number|prime number]] {{mvar|p}}), which are thereby analogous to <math>\mathbb{R}</math>. There are no other nontrivial ways of completing <math>\mathbb Q</math> than <math>\mathbb R</math> and <math>\mathbb Q_p,</math> by [[Ostrowski's theorem]]. The algebraic closures <math>\overline {\mathbb{Q}_p}</math> of <math>\mathbb Q_p</math> still carry a norm, but (unlike <math>\mathbb C</math>) are not complete with respect to it. The completion <math>\mathbb{C}_p</math> of <math>\overline {\mathbb{Q}_p}</math> turns out to be algebraically closed. By analogy, the field is called {{mvar|p}}-adic complex numbers.

The fields <math>\mathbb R,</math> <math>\mathbb Q_p,</math> and their finite field extensions, including <math>\mathbb C,</math> are called [[Local field|local field]]s.

==See also==
* [[Algebraic surface]]
* [[Physics:Circular motion#Using complex numbers|Circular motion using complex numbers]]
* [[Complex-base system]]
* [[Complex geometry]]
* [[Dual-complex number]]
* [[Eisenstein integer]]
* [[Geometric algebra#Unit pseudoscalars|Geometric algebra]] (which includes the complex plane as the 2-dimensional [[Physics:Spinor#Two dimensions|spinor]] subspace <math>\mathcal{G}_2^+</math>)
* Unit complex number

==Notes==
{{notelist}}

==References==
{{reflist|refs=
<ref name="Campbell_1911">{{cite journal |title=Cisoidal oscillations |author-first=George Ashley |author-last=Campbell |journal=Proceedings of the American Institute of Electrical Engineers |publisher=[[Organization:American Institute of Electrical Engineers|American Institute of Electrical Engineers]] |volume=XXX |issue=1–6 |date=April 1911 |doi=10.1109/PAIEE.1911.6659711 |s2cid=51647814 |pages=789–824 [Fig. 13 on p. 810] |url=https://ia800708.us.archive.org/view_archive.php?archive=/28/items/crossref-pre-1923-scholarly-works/10.1109%252Fpaiee.1910.6660428.zip&file=10.1109%252Fpaiee.1911.6659711.pdf |access-date=2023-06-24 |quote-page=789 |quote=The use of ''i'' (or Greek ''ı'') for the imaginary symbol is nearly universal in mathematical work, which is a very strong reason for retaining it in the applications of mathematics in electrical engineering. Aside, however, from the matter of established conventions and facility of reference to mathematical literature, the substitution of the symbol ''j'' is objectionable because of the vector terminology with which it has become associated in engineering literature, and also because of the confusion resulting from the divided practice of engineering writers, some using ''j'' for +''i'' and others using ''j'' for &minus;''i''.}}</ref>
<ref name="Brown-Churchill_1996">{{cite book |author-last1=Brown |author-first1=James Ward |author-last2=Churchill |author-first2=Ruel V. |title=Complex variables and applications |date=1996 |publisher=McGraw-Hill |location=New York, USA |isbn=978-0-07-912147-9 |edition=6 |page=2 |quote-page=2 |quote=In electrical engineering, the letter ''j'' is used instead of ''i''.}}</ref>
}}

===Works cited===
{{refbegin}}
* {{cite book |last=Ahlfors |first=Lars |year=1979 |title=Complex analysis |edition=3rd |publisher=McGraw-Hill |url=https://archive.org/details/lars-ahlfors-complex-analysis-third-edition-mcgraw-hill-science_engineering_math-1979/page/n1/mode/2up |url-access=registration |isbn=978-0-07-000657-7}}
* {{cite book |last=Apostol |first=Tom |year=1981 |title=Mathematical analysis |publisher=Addison-Wesley}}
* {{cite journal |last=Argand |date=1814 |title=Reflexions sur la nouvelle théorie des imaginaires, suives d'une application à la demonstration d'un theorème d'analise |journal=Annales de mathématiques pures et appliquées |volume=5 |pages=197–209 |url=https://babel.hathitrust.org/cgi/pt?id=uc1.$c126479&view=1up&seq=209 |trans-title=Reflections on the new theory of complex numbers, followed by an application to the proof of a theorem of analysis |language=fr}}
* {{cite journal |last=Gauss |first=C. F. |date= 1831 |title=Theoria residuorum biquadraticorum. Commentatio secunda. |trans-title=Theory of biquadratic residues. Second memoir. |url=https://babel.hathitrust.org/cgi/pt?id=mdp.39015073697180&view=1up&seq=292 |journal=Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores |volume=7 |pages=89–148 |language=la |}}
* {{springer |id=c/c024140 |title=Complex number |year=2001|first=E.D. |last=Solomentsev}}
{{refend}}

==Further reading==
{{EB1911 poster|Number/Complex Numbers}}

* {{cite book |last=Penrose |first=Roger |year= 2005 |title=The Road to Reality: A complete guide to the laws of the universe |publisher=Alfred A. Knopf |isbn=978-0-679-45443-4 |url=https://archive.org/details/roadtorealitycom00penr_0}}
* {{cite book |last=Derbyshire |first=John |year=2006 |title=Unknown Quantity: A real and imaginary history of algebra |publisher=Joseph Henry Press |isbn=978-0-309-09657-7 |url=https://archive.org/details/isbn_9780309096577}}
* {{cite book |last=Needham |first=Tristan |year=1997 |title=Visual Complex Analysis |publisher=Clarendon Press |isbn=978-0-19-853447-1}}

===Mathematical===
{{refbegin}}
* {{cite book |ref=none |last=Ahlfors |first=Lars |title=Complex analysis |publisher=McGraw-Hill |year=1979 |edition=3rd |isbn=978-0-07-000657-7}}
* {{cite book |ref=none |last=Conway |first=John B. |title=Functions of One Complex Variable I |year=1986 |publisher=Springer |isbn=978-0-387-90328-6}}
* {{cite book |ref=none |last1=Joshi |first1=Kapil D. |title=Foundations of Discrete Mathematics |publisher={{wipe|John Wiley & Sons}} |location=New York |isbn=978-0-470-21152-6 |year=1989}}
* {{cite book |ref=none |last=Pedoe |first=Dan |title=Geometry: A comprehensive course |publisher=Dover |year=1988 |isbn=978-0-486-65812-4}}
* {{cite book |ref=none |last1=Press |first1=W.H. |last2=Teukolsky |first2=S.A. |last3=Vetterling |first3=W.T. |last4=Flannery |first4=B.P. |year=2007 |title=Numerical Recipes: The art of scientific computing |edition=3rd |publisher=Cambridge University Press |location=New York |isbn=978-0-521-88068-8 |chapter=Section 5.5 Complex Arithmetic |chapter-url=http://apps.nrbook.com/empanel/index.html?pg=225 |access-date=9 August 2011 |archive-date=13 March 2020 |archive-url=https://web.archive.org/web/20200313111530/http://apps.nrbook.com/empanel/index.html?pg=225 |url-status=dead }}
* {{springer|ref=none|id=c/c024140|title=Complex number|year=2001|first=E.D.|last=Solomentsev}}
{{refend}}

===Historical===
{{refbegin}}
* {{cite book |ref=none |last=Bourbaki |first=Nicolas |title= Elements of the history of mathematics |chapter= Foundations of mathematics § logic: set theory |publisher= Springer |year= 1998}}
* {{cite book |ref=none |last1=Burton |first1=David M. |title=The History of Mathematics |publisher=McGraw-Hill |location=New York |edition= 3rd |isbn=978-0-07-009465-9 |year=1995}}
* {{cite book |ref=none |last1=Katz |first1=Victor J. |title=A History of Mathematics, Brief Version |publisher=Addison-Wesley |isbn=978-0-321-16193-2 |year=2004}}
* {{cite book |ref=none |title=An Imaginary Tale: The Story of <math>\scriptstyle\sqrt{-1}</math> |first=Paul J. |last=Nahin |publisher=Princeton University Press |isbn=978-0-691-02795-1 |year=1998}} — A gentle introduction to the history of complex numbers and the beginnings of complex analysis.
* {{cite book |ref=none |first1=H. D. |last1= Ebbinghaus |first2=H. |last2= Hermes |first3=F. |last3=Hirzebruch |first4=M. |last4=Koecher |first5=K. |last5= Mainzer |first6=J. |last6= Neukirch |first7=A. |last7=Prestel |first8=R. |last8=Remmert |title=Numbers |publisher=Springer |isbn=978-0-387-97497-2 |edition=hardcover |year=1991}} — An advanced perspective on the historical development of the concept of number.
{{refend}}

{{Complex numbers}}
{{Number systems}}


{{DEFAULTSORT:Complex Number}}
[[Category:Composition algebras]]
[[Category:Complex numbers| ]]

{{Sourceattribution|Complex number}}