Editing Complex number

{{Short description|Number with a real and an imaginary part}}
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{{Use dmy dates|date=June 2020|cs1-dates = l}}

[[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. ''Re'' is the real axis, ''Im'' is the imaginary axis, and {{mvar|i}} is the "imaginary unit", that satisfies {{math|1=''i''<sup>2</sup> = −1}}.]]

In mathematics, a '''complex number''' is an element of a [[number system]] that extends the [[real number]]s with a specific element denoted {{mvar|i}}, called the [[imaginary unit]] and satisfying the 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]] by [[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 have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in 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 |author-link=Nicolas Bourbaki |year=1998 |title=Elements of the History of Mathematics |chapter=Foundations of Mathematics § Logic: Set theory |pages=18–24 |publisher=Springer}}
</ref><ref>"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.", {{harvnb|Penrose|2005|loc=pp.72–73 |url=https://books.google.com/books?id=VWTNCwAAQBAJ&pg=PA73}}.</ref>

Complex numbers allow solutions to all [[polynomial equation]]s, even those that have no solutions in real numbers. More precisely, the [[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, because 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> along with the [[associative law|associative]], [[commutative law|commutative]], and [[distributive law]]s. Every nonzero complex number has a [[multiplicative inverse]]. This makes the complex numbers a [[field (mathematics)|field]] with the real numbers as a subfield. Because of these properties, {{tmath|1=a + bi = a + ib}}, and which form is written depends upon convention and style considerations.

The complex numbers also form a [[real vector space]] of [[Two-dimensional space|dimension two]], with <math>\{1,i\}</math> as a [[standard basis]]. This standard basis makes the complex numbers a [[Cartesian plane]], called the complex plane. This allows a geometric interpretation of the complex numbers and their operations, and conversely some geometric objects and operations can be expressed in terms of complex numbers. For example, the real numbers form the [[real line]], which is pictured as the horizontal axis of the complex plane, while real multiples of <math>i</math> are the vertical axis. A complex number can also be defined by its geometric [[Polar coordinate system|polar coordinates]]: the radius is called the [[absolute value]] of the complex number, while the angle from the positive real axis is called the argument of the complex number. The complex numbers of absolute value one form the [[unit circle]]. Adding a fixed complex number to all complex numbers defines a [[translation (geometry)|translation]] in the complex plane, and multiplying by a fixed complex number is a [[similarity (geometry)|similarity]] centered at the origin (dilating by the absolute value, and rotating by the argument). The operation of [[complex conjugation]] is the [[reflection symmetry]] with respect to the real axis. 

The complex numbers form a rich structure that is simultaneously an [[algebraically closed field]], a [[commutative algebra (structure)|commutative algebra]] over the reals, and a [[Euclidean vector space]] of dimension two. 
{{TOC limit|3}}

==Definition and basic operations==

[[File:Complex_numbers_intheplane.svg|right|thumb|Various complex numbers depicted in the complex plane.]]

A complex number is an expression of the form {{math|1=''a'' + ''bi''}}, where {{mvar|a}} and {{mvar|b}} are real numbers, and {{math|''i''}} is an abstract symbol, the so-called imaginary unit, whose meaning will be explained further below. 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>

For a complex number {{math|''a'' + ''bi''}}, the real number {{mvar|a}} is called its ''real part'', and the real number {{mvar|b}} (not the complex number {{math|''bi''}}) is its ''imaginary part''.<ref>{{cite book |last1=Spiegel |first1=M.R. |title=Complex Variables |last2=Lipschutz |first2=S. |last3=Schiller |first3=J.J. |last4=Spellman |first4=D. |date=14 April 2009 |publisher=McGraw Hill |isbn=978-0-07-161569-3 |edition=2nd |series=Schaum's Outline Series}}</ref><ref>{{harvnb|Aufmann|Barker|Nation|loc=p. 66, Chapter P|2007}}</ref> The real part of a complex number {{mvar|z}} is denoted {{math|Re(''z'')}}, <math>\mathcal{Re}(z)</math>, or <math>\mathfrak{R}(z)</math>; the imaginary part is {{math|Im(''z'')}}, <math>\mathcal{Im}(z)</math>, or <math>\mathfrak{I}(z)</math>: for example, <math display="inline">  \operatorname{Re}(2 + 3i) = 2  </math>, <math>   \operatorname{Im}(2 + 3i) = 3  </math>.

A complex number {{mvar|z}} can be identified with the [[ordered pair]] of real numbers <math>(\Re (z),\Im (z))</math>, which may be interpreted as coordinates of a point in a Euclidean plane with standard coordinates, which is then called the ''[[complex plane]]'' or ''[[Argand diagram]].''<ref>{{cite book |last=Pedoe |first=Dan |author-link=Daniel Pedoe |title=Geometry: A comprehensive course |publisher=Dover |year=1988 |isbn=978-0-486-65812-4}}</ref><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>{{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".}} The horizontal axis is generally used to display the real part, with increasing values to the right, and the imaginary part marks the vertical axis, with increasing values upwards. 

[[File:Complex number illustration.svg|thumb|right|A complex number {{mvar|z}}, as a point (black) and its [[vector (geometric)|position vector]] (blue).]]
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 {{math|''bi''}} is a complex number {{math|0 + ''bi''}}, whose real part is zero. It is common to write {{math|1=''a'' + 0''i'' = ''a''}}, {{math|1=0 + ''bi'' = ''bi''}}, and {{math|1=''a'' + (−''b'')''i'' = ''a'' − ''bi''}}; for example, {{math|1=3 + (−4)''i'' = 3 − 4''i''}}.

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

In some disciplines such as electromagnetism and electrical engineering, {{mvar|j}} is used instead of {{mvar|i}}, as {{mvar|i}} frequently represents electric current,<ref name="Campbell_1911" /><ref name="Brown-Churchill_1996" /> and complex numbers are written as {{math|''a'' + ''bj''}} or {{math|''a'' + ''jb''}}.

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

The addition can be geometrically visualized as follows: 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]] 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]]s {{mvar|OAB}} and {{mvar|XBA}} are [[Congruence (geometry)|congruent]].

===Multiplication{{anchor|Multiplication|Square}}===
The product of two complex numbers is computed as follows:
:<math>(a+bi) \cdot (c+di) = ac - bd + (ad+bc)i.</math>
For example, <math>(3+2i)(4-i) = 3 \cdot 4 - (2 \cdot (-1)) + (3 \cdot (-1) + 2 \cdot 4)i = 14 +5i.</math> 
In particular, this includes as a special case the fundamental formula
:<math>i^2 = i \cdot i = -1.</math>
This formula distinguishes the complex number ''i'' from any real number, since the square of any (negative or positive) real number is always a non-negative real number.

With this definition of multiplication and addition, familiar rules for the arithmetic of rational or real numbers continue to hold for complex numbers. More precisely, the [[distributive property]], the [[commutative property|commutative properties]] (of addition and multiplication) hold. Therefore, the complex numbers form an algebraic structure known as a [[field (mathematics)|''field'']], the same way as the rational or real numbers do.{{sfn|Apostol|1981|pp=15–16}}

===Complex conjugate, absolute value, argument and division===

[[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]]'' of the complex number {{math|1=''z'' = ''x'' + ''yi''}} is defined as
<math>\overline z = x-yi.</math><ref>{{harvnb|Apostol|1981|pp=15–16}}</ref> It is also denoted by some authors by <math>z^*</math>. Geometrically, {{mvar|{{overline|z}}}} is the [[reflection symmetry|"reflection"]] of {{mvar|z}} about the real axis. Conjugating twice gives the original complex number: <math>\overline{\overline{z}}=z.</math> A complex number is real if and only if it equals its own conjugate. The [[unary operation]] of taking the complex conjugate of a complex number cannot be expressed by applying only the basic operations of addition, subtraction, multiplication and division. 

[[File:Complex number illustration modarg.svg|right|thumb|Argument {{mvar|φ}} and modulus {{mvar|r}} locate a point in the complex plane.]]
For any complex number {{math|1=''z'' = ''x'' + ''yi''}} , the product 
:<math>z \cdot \overline z = (x+iy)(x-iy) = x^2 + y^2</math>
is a ''non-negative real'' number. This allows to define the ''[[absolute value]]'' (or ''modulus'' or ''magnitude'') of ''z'' to be the square root{{sfn|Apostol|1981|p=18}}
<math display="block">|z|=\sqrt{x^2+y^2}.</math>
By [[Pythagoras' theorem]], <math>|z|</math> is the distance from the origin to the point representing the complex number ''z'' in the complex plane. In particular, the [[unit circle|circle of radius one]] around the origin consists precisely of the numbers ''z'' such that <math>|z| = 1 </math>. If <math>  z = x = x + 0i  </math> is a real number, then <math>  |z|= |x| </math>: its absolute value as a complex number and as a real number are equal. 

Using the conjugate, the [[multiplicative inverse|reciprocal]] of a nonzero complex number <math>z = x + yi</math> can be computed to be 

<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>
More generally, the division of an arbitrary complex number <math>w = u + vi</math> by a non-zero complex number <math>z = x + yi</math> equals
<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>
This process is sometimes called "[[rationalisation (mathematics)|rationalization]]" of the denominator (although the denominator in the final expression may be an irrational real number), because it resembles the method to remove roots from simple expressions in a denominator.<ref>{{cite book |title=Numerical Linear Algebra with Applications: Using MATLAB and Octave |author1=William Ford |edition=reprinted |publisher=Academic Press |year=2014 |isbn=978-0-12-394784-0 |page=570 |url=https://books.google.com/books?id=OODs2mkOOqAC}} [https://books.google.com/books?id=OODs2mkOOqAC&pg=PA570 Extract of page 570]</ref><ref>{{cite book |title=Precalculus with Calculus Previews: Expanded Volume |author1=Dennis Zill |author2=Jacqueline Dewar |edition=revised |publisher=Jones & Bartlett Learning |year=2011 |isbn=978-0-7637-6631-3 |page=37 |url=https://books.google.com/books?id=TLgjLBeY55YC}} [https://books.google.com/books?id=TLgjLBeY55YC&pg=PA37 Extract of page 37]</ref>

The ''[[argument (complex analysis)|argument]]'' of {{mvar|z}} (sometimes called the "phase" {{mvar|φ}})<ref name=":2" /> is the angle of the [[radius]] {{mvar|Oz}} with the positive real axis, and is written as {{math|arg ''z''}}, expressed in [[radian]]s in this article. The angle is defined only up to adding integer multiples of <math>  2\pi </math>, since a rotation by <math>2\pi</math> (or 360°) around the origin leaves all points in the complex plane unchanged. One possible choice to uniquely specify the argument is to require it to be within the interval <math>  (-\pi,\pi]  </math>, which is referred to as the [[principal value]].<ref>Other authors, including {{harvnb|Ebbinghaus|Hermes|Hirzebruch|Koecher|Mainzer|Neukirch|Prestel|Remmert|1991|loc=§6.1}}, chose the argument to be in the interval <math>[0, 2\pi)</math>.</ref>
The argument can be computed from the rectangular form {{mvar|x + yi}} by means of the [[arctan]] (inverse tangent) function.<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>

===Polar form{{anchor|Polar form}}===
{{Main|Polar coordinate system}}
{{Redirect|Polar form|the higher-dimensional analogue|Polar decomposition}}

[[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]] of the blue triangle (the multiplication of both radiuses, as per term ''r''<sub>1</sub>''r''<sub>2</sub> in the equation).]]

For any complex number ''z'', with absolute value <math>r = |z|</math> and argument <math>\varphi</math>, the equation 
:<math>z=r(\cos\varphi +i\sin\varphi) </math>
holds. This identity is referred to as the polar form of ''z''. It is sometimes abbreviated as <math display="inline"> z = r \operatorname\mathrm{cis} \varphi </math>. 
In electronics, one represents a [[Phasor (sine waves)|phasor]] with amplitude {{mvar|r}} and phase {{mvar|φ}} in [[angle notation]]:<ref>
{{cite book |last1=Nilsson |first1=James William |title=Electric circuits |last2=Riedel |first2=Susan A. |publisher=Prentice Hall |year=2008 |isbn=978-0-13-198925-2 |edition=8th |page=338 |chapter=Chapter 9 |chapter-url=https://books.google.com/books?id=sxmM8RFL99wC&pg=PA338}}
</ref><math display="block">z = r \angle \varphi . </math>

If two complex numbers are given in polar form, i.e., {{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>)}}, the product and division can be computed as 
<math display=block>z_1 z_2 = r_1 r_2 (\cos(\varphi_1 + \varphi_2) + i \sin(\varphi_1 + \varphi_2)).</math>
<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), \text{if }z_2 \ne 0.</math>
(These are a consequence of the [[trigonometric identities]] for the sine and cosine function.)
In other words, the absolute values are ''multiplied'' and the arguments are ''added'' to yield the polar form of the product. The picture at the right illustrates the multiplication of
<math display=block>(2+i)(3+i)=5+5i. </math>
Because 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]]). 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 formula]]s – are used for high-precision approximations of [[pi|{{pi}}]]:<ref>{{cite book |title=Modular Forms: A Classical And Computational Introduction |author1=Lloyd James Peter Kilford |edition= 2nd|publisher=World Scientific Publishing Company |year=2015 |isbn=978-1-78326-547-3 |page=112 |url=https://books.google.com/books?id=qDk8DQAAQBAJ}} [https://books.google.com/books?id=qDk8DQAAQBAJ&pg=PA112 Extract of page 112]</ref>
<math display=block>\frac{\pi}{4} = 4 \arctan\left(\frac{1}{5}\right) - \arctan\left(\frac{1}{239}\right) </math>

===Powers and roots===
{{see also|Square root#Square roots of negative and complex numbers|l1=Square roots of negative and complex numbers}}
The ''n''-th power of a complex number can be computed using [[de Moivre's formula]], which is obtained by repeatedly applying the above formula for the product:
<math display=block> z^{n}=\underbrace{z \cdot \dots \cdot z}_{n \text{ factors}} = (r(\cos \varphi + i\sin \varphi ))^n = r^n \, (\cos n\varphi + i \sin n \varphi).</math>
For example, the first few powers of the imaginary unit ''i'' are <math>i, i^2 = -1, i^3 = -i, i^4 = 1, i^5 = i, \dots</math>.

{{Visualisation complex number roots|1=upright=1.35}}
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. For any <math>z \ne 0</math>, there are, in particular ''n'' distinct complex ''n''-th roots. For example, there are 4 fourth roots of 1, namely
:<math>z_1 = 1, z_2 = i, z_3 = -1, z_4 = -i.</math>
In general there is ''no'' natural way of distinguishing one particular complex {{mvar|n}}th root of a complex number. (This is in contrast to the roots of a positive real number ''x'', which has a unique positive real ''n''-th root, which is therefore commonly referred to as ''the'' ''n''-th root of ''x''.) One refers to this situation by saying that the {{mvar|n}}th root is a [[multivalued function|{{mvar|n}}-valued function]] of {{mvar|z}}. 

===Fundamental theorem of algebra===
The [[fundamental theorem of algebra]], of [[Carl Friedrich Gauss]] and [[Jean le Rond d'Alembert]], states that for any complex numbers (called [[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">{{harvnb|Bourbaki|1998|loc=§VIII.1}}</ref> 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, because {{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, because the square of {{mvar|x}} is positive for any real number {{mvar|x}}). 

Because of this fact, <math>\Complex</math> is called an [[algebraically closed field]]. It is a cornerstone of various applications of complex numbers, as is detailed further below. 
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]], or a proof combining [[Galois theory]] and the fact that any real polynomial of ''odd'' degree has at least one real root.

==History==
{{See also|Negative number#History}}
The solution in [[nth root|radicals]] (without [[trigonometric functions]]) of a general [[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]]'' ("irreducible case"). This conundrum led Italian mathematician [[Gerolamo Cardano]] to conceive of complex numbers in around 1545 in his ''[[Ars Magna (Cardano book)|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 being "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. Because 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, 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]], 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 [[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 [[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]]s of [[negative number]]s can perhaps be said to occur in the work of the Greek mathematician [[Hero of Alexandria]] in the 1st century [[AD]], where in his ''[[Hero of Alexandria#Bibliography|Stereometrica]]'' he considered, apparently in error, the volume of an impossible [[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 the negative value 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]]s for the roots of [[Cubic equation|cubic]] and [[Quartic equation|quartic]] [[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 [[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 |author-link=René Descartes |year=1954 |orig-year=1637 |publisher=[[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 [[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.<ref>{{cite book |title=Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers |author1=Joseph Mazur |edition=reprinted |publisher=Princeton University Press |year=2016 |isbn=978-0-691-17337-5 |page=138 |url=https://books.google.com/books?id=O3CYDwAAQBAJ}} [https://books.google.com/books?id=O3CYDwAAQBAJ&pg=PA138 Extract of page 138]</ref><ref>{{cite book |title=Mathematical Fallacies and Paradoxes |author1=Bryan Bunch |edition=reprinted, revised |publisher=Courier Corporation |year=2012 |isbn=978-0-486-13793-3 |page=32 |url=https://books.google.com/books?id=jUTCAgAAQBAJ}} [https://books.google.com/books?id=jUTCAgAAQBAJ&pg=PA32 Extract of page 32]</ref> 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 [[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]]:

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

[[File:Circle_cos_sin.gif |thumb |upright=1.5 |Euler's formula relates the complex exponential function of an imaginary argument, which can be thought of as describing [[uniform circular motion]] in the complex plane, to the cosine and sine functions, geometrically its projections onto the real and imaginary axes, respectively.]]
In 1748, Euler went further and obtained [[Euler's formula]] of [[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">e ^{i\theta } = \cos \theta + i\sin \theta </math>

by formally manipulating complex [[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 was first described by [[Denmark|Danish]]–[[Norway|Norwegian]] [[mathematician]] [[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=561 |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 [[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 [[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> [[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}}

<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> [[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> [[John Warren (mathematician)|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> [[Jacques Frédéric Français|Français]] and his brother, [[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 Norwegian [[Niels Henrik Abel]] and [[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=[[Oxford University Press|OUP Oxford]] |isbn= 978-0-19-921986-5 |page=189 (fourth edition)}}</ref>

[[Augustin-Louis Cauchy]] and [[Bernhard Riemann]] together brought the fundamental ideas of [[#Complex analysis|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 [[Richard Dedekind]], [[Otto Hölder]], [[Felix Klein]], [[Henri Poincaré]], [[Hermann Schwarz]], [[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 [[Wilhelm Wirtinger]] in 1927.

==Abstract algebraic aspects==
While the above low-level definitions, including the addition and multiplication, accurately describe the complex numbers, there are other, equivalent approaches that reveal the abstract algebraic structure of the complex numbers more immediately.

===Construction as a quotient field===
One approach to <math>\C</math> is via [[polynomial]]s, i.e., expressions of the form
<math display=block>p(X) = a_nX^n+\dotsb+a_1X+a_0,</math>
where the [[coefficient]]s {{math|''a''<sub>0</sub>, ..., ''a''<sub>''n''</sub>}} are real numbers. The set of all such polynomials is denoted by <math>\R[X]</math>. Since sums and products of polynomials are again polynomials, this set <math>\R[X]</math> forms a [[commutative ring]], called the [[polynomial ring]] (over the reals). To every such polynomial ''p'', one may assign the complex number <math>p(i) = a_n i^n + \dotsb + a_1 i + a_0</math>, i.e., the value obtained by setting <math>X = i</math>. This defines a function
:<math>\R[X] \to \C</math> 
This function is [[surjective]] since every complex number can be obtained in such a way: the evaluation of a [[linear polynomial]] <math>a+bX</math> at <math>X = i</math> is <math>a+bi</math>. However, the evaluation of polynomial <math>X^2 + 1</math> at ''i'' is 0, since <math>i^2 + 1 = 0.</math> This polynomial is [[irreducible polynomial|irreducible]], i.e., cannot be written as a product of two linear polynomials. Basic facts of [[abstract algebra]] then imply that the [[Kernel (algebra)|kernel]] of the above map is an [[ideal (ring theory)|ideal]] generated by this polynomial, and that the quotient by this ideal is a field, and that there is an [[isomorphism]] 
:<math>\R[X] / (X^2 + 1) \stackrel \cong \to \C</math>
between the quotient ring and <math>\C</math>. Some authors take this as the definition of <math>\C</math>.<ref>{{harvnb|Bourbaki|1998|loc=§VIII.1}}</ref>

Accepting that <math>\Complex</math> is algebraically closed, because 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]] 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]] of the corresponding matrix, and the conjugate of a complex number with the [[transpose]] of the matrix.

The [[polar form]] representation of complex numbers explicitly gives these matrices as scaled [[rotation matrix|rotation matrices]].
<math display=block>r (\cos \theta + i \sin \theta)\mapsto \begin{pmatrix}
  r \cos \theta &   -r \sin \theta  \\
  r \sin \theta & \;\; r \cos \theta
\end{pmatrix}</math>
In particular, the case of {{math|1=''r'' = 1}}, which is <math>|a + ib| = \sqrt{a^2+b^2} = 1</math>, gives (unscaled) rotation matrices.

==Complex analysis==

{{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 [[Lightness (color)|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]] <math> z = 0</math>

<math display=block> | z + w | \leq | z | + | w | \,</math> ([[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]] 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]] 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]] (<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]]'' and has enormous practical use in [[applied mathematics]] as well as in other branches of mathematics. Often, the most natural proofs for statements in [[real analysis]] or even [[number theory]] employ techniques from complex analysis (see [[prime number theorem]] for an example). 

[[File:Complex-plot.png|right|thumb|A [[domain coloring]] graph of the function
{{math|{{sfrac|(''z''<sup>2</sup> − 1)(''z'' − 2 − ''i'')<sup>2</sup>|''z''<sup>2</sup> + 2 + 2''i''}}}}. Darker spots mark moduli near zero, brighter spots are farther away from the origin. The color encodes the argument. The function has zeros for {{math|±1, (2 + ''i'')}} and [[pole (complex analysis)|poles]] at <math>\pm \sqrt{{-2-2i}}.</math>]]
Unlike real functions, which are commonly represented as two-dimensional graphs, [[complex function]]s have four-dimensional graphs and may usefully be illustrated by color-coding a [[graph of a function of two variables|three-dimensional graph]] to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.

===Convergence===
[[File:ComplexPowers.svg|right|thumb|Illustration of the behavior of the sequence <math>z^n</math> for three different values of ''z'' (all having the same argument): for <math>|z|<1</math> the sequence converges to 0 (inner spiral), while it diverges for <math>|z|>1</math> (outer spiral).]]
The notions of [[convergent series]] and [[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 [[convergent sequence|converge]] if and only if its real and imaginary parts do. This is equivalent to the [[(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]], which notably includes the [[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>}}.

===Complex exponential===
[[File:ComplexExpMapping.svg|thumb|right|Illustration of the complex exponential function mapping the complex plane, ''w'' = exp ⁡(''z''). The left plane shows a square mesh with mesh size 1, with the three complex numbers 0, 1, and ''i'' highlighted. The two rectangles (in magenta and green) are mapped to circular segments, while the lines parallel to the ''x''-axis are mapped to rays emanating from, but not containing the origin. Lines parallel to the ''y''-axis are mapped to circles.]]
Like in real analysis, this notion of convergence is used to construct a number of [[elementary function]]s: the ''[[exponential function]]'' {{math|exp ''z''}}, also written {{math|''e''<sup>''z''</sup>}}, is defined as the [[infinite series]], which can be shown to [[radius of convergence|converge]] for any ''z'':
<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>
For example, <math>\exp (1)</math> is [[E (mathematical constant)|Euler's number]] <math>e \approx 2.718</math>. ''[[Euler's formula]]'' states:
<math display=block>\exp(i\varphi) = \cos \varphi + i\sin \varphi </math>
for any real number {{mvar|φ}}. This formula is a quick consequence of general basic facts about convergent power series and the definitions of the involved functions as power series. As a special case, this includes [[Euler's identity]]
<math display=block>\exp(i \pi) = -1. </math>

===Complex logarithm===
{{main|Complex logarithm}}
[[File:ComplexExpStrips.svg|right|thumb|The exponential function maps complex numbers ''z'' differing by a multiple of <math>2\pi i</math> to the same complex number ''w''.]]
For any positive real number ''t'', there is a unique real number ''x'' such that <math>\exp(x) = t</math>. This leads to the definition of the [[natural logarithm]] as the [[inverse function|inverse]] 
<math>\ln \colon \R^+ \to \R ; x \mapsto \ln x </math> of the exponential function. The situation is different for complex numbers, since
:<math>\exp(z+2\pi i) = \exp z \exp (2 \pi i) = \exp z</math>
by the functional equation and Euler's identity. 
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 complex logarithm of {{math|−1}}. 

In general, given any non-zero complex number ''w'', any number ''z'' solving the equation
:<math>\exp z = w</math>
is called a [[complex logarithm]] of {{mvar|w}}, denoted <math>\log w</math>. It can be shown that these numbers satisfy
<math display=block>z = \log w = \ln|w| + i\arg w, </math>
where <math>\arg</math> is the [[arg (mathematics)|argument]] defined [[#Polar form|above]], and <math>\ln</math> the (real) [[natural logarithm]]. As arg is a [[multivalued function]], unique only up to a multiple of {{math|2''π''}}, log is also multivalued. The [[principal value]] of log is often taken by restricting the imaginary part to the [[interval (mathematics)|interval]] {{open-closed|−''π'', ''π''}}. This leads to the complex logarithm being a [[bijective]] function taking values in the strip <math>\R^+ + \; i \, \left(-\pi, \pi\right]</math> (that is denoted <math>S_0</math> in the above illustration)
<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]] of the complex logarithm is obtained with {{math|−''π'' < ''φ'' < ''π''}}. It is an [[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.}}

Complex [[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. If {{math|''z'' > 0}} is real (and {{mvar|ω}} an arbitrary complex number), one has a preferred choice of <math>\ln x</math>, the real logarithm, which can be used to define a preferred exponential function.

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.

===Complex sine and cosine===
The series defining the real trigonometric functions [[sine|{{math|sin}}]] and [[cosine|{{math|cos}}]], as well as the [[hyperbolic functions]] {{math|sinh}} and {{math|cosh}}, also carry over to complex arguments without change. For the other trigonometric and hyperbolic functions, such as [[tangent (function)|{{math|tan}}]], 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]].

The value of a trigonometric or hyperbolic function of a complex number can be expressed in terms of those functions evaluated on real numbers, via angle-addition formulas.  For {{math|1=''z'' = ''x'' + ''iy''}},

<math display=block>\sin{z} = \sin{x} \cosh{y} + i \cos{x} \sinh{y}</math>

<math display=block>\cos{z} = \cos{x} \cosh{y} - i \sin{x} \sinh{y}</math>

<math display=block>\tan{z} = \frac{\tan{x} + i \tanh{y}}{1 - i \tan{x} \tanh{y}}</math>

<math display=block>\cot{z} = -\frac{1 + i \cot{x} \coth{y}}{\cot{x} -i \coth{y}}</math>

<math display=block>\sinh{z} = \sinh{x} \cos{y} + i \cosh{x} \sin{y}</math>

<math display=block>\cosh{z} = \cosh{x} \cos{y} + i \sinh{x} \sin{y}</math>

<math display=block>\tanh{z} = \frac{\tanh{x} + i \tan{y}}{1 + i \tanh{x} \tan{y}}</math>

<math display=block>\coth{z} = \frac{1 - i \coth{x} \cot{y}}{\coth{x} - i \cot{y}}</math>

Where these expressions are not well defined, because a trigonometric or hyperbolic function evaluates to infinity or there is division by zero, they are nonetheless correct as [[Limit (mathematics)|limit]]s.

===Holomorphic functions===
[[File:Sin1z-cplot.svg|thumb|Color wheel graph of the function {{math|sin(1/''z'')}} that is holomorphic except at ''z'' = 0, which is an essential singularity of this function. White parts inside refer to numbers having large absolute values.]]
A function <math>f: \mathbb{C}</math> → <math>\mathbb{C}</math> is called [[Holomorphic function|holomorphic]] or ''complex differentiable'' at a point <math>z_0</math> if the limit
:<math>\lim_{z \to z_0} {f(z) - f(z_0) \over z - z_0 }</math>
exists (in which case it is denoted by <math>f'(z_0)</math>). This mimics the definition for real differentiable functions, except that all quantities are complex numbers. Loosely speaking, the freedom of approaching <math>z_0</math> in different directions imposes a much stronger condition than being (real) differentiable. For example, the function
:<math>f(z) = \overline z</math>
is differentiable as a function <math>\R^2 \to \R^2</math>, but is ''not'' complex differentiable.
A real differentiable function is complex differentiable [[if and only if]] it satisfies the [[Cauchy–Riemann equations]], which are sometimes abbreviated as
:<math>\frac{\partial f}{\partial \overline z} = 0.</math>

Complex analysis shows some features not apparent in real analysis. For example, the [[identity theorem]] asserts that two holomorphic functions {{mvar|f}} and {{mvar|g}} agree if they agree on an arbitrarily small [[open subset]] of <math>\mathbb{C}</math>. [[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]], [[control theory]], [[electromagnetism]], [[fluid dynamics]], [[quantum mechanics]], [[cartography]], and [[Vibration#Vibration analysis|vibration analysis]]. Some of these applications are described below.

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

===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]]), 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 (stability theory)|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]] 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]] <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 [[compass and straightedge constructions|using straightedge and compass]].]]
As mentioned above, any nonconstant polynomial equation (in complex coefficients) has a solution in <math>\mathbb{C}</math>. ''[[Argumentum a fortiori|A fortiori]]'', the same is true if the equation has rational coefficients. The roots of such equations are called [[algebraic number]]s – they are a principal object of study in [[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]] [[compass and straightedge constructions|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]]s.

===Improper integrals===
In applied fields, complex numbers are often used to compute certain real-valued [[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]]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]] 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 ===
Since <math>\C</math> is algebraically closed, any non-empty complex [[square matrix]] has at least one (complex) [[eigenvalue]]. By comparison, real matrices do not always have real eigenvalues, for example [[rotation matrix|rotation matrices]] (for rotations of the plane for angles other than 0° or 180°) leave no direction fixed, and therefore do not have any ''real'' eigenvalue. The existence of (complex) eigenvalues, and the ensuing existence of [[Eigendecomposition of a matrix|eigendecomposition]] is a useful tool for computing matrix powers and [[matrix exponential]]s. 

Complex numbers often generalize concepts originally conceived in the real numbers. For example, the [[conjugate transpose]] generalizes the [[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====
{{see also|Complex plane#Use in control theory}}

In [[control theory]], systems are often transformed from the [[time domain]] to the complex [[frequency domain]] using the [[Laplace transform]]. The system's [[zeros and poles]] are then analyzed in the ''complex plane''. The [[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]].

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 [[sine wave]] of a given [[frequency]], the absolute value {{math|{{!}}''z''{{!}}}} of the corresponding {{mvar|z}} is the [[amplitude]] and the [[Argument (complex analysis)|argument]] {{math|arg ''z''}} is the [[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 [[angular frequency]] and the complex number ''A'' encodes the phase and amplitude as explained above.

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

Another example, relevant to the two side bands of [[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|Alternating current}}
In [[electrical engineering]], the [[Fourier transform]] is used to analyze varying [[electric current]]s and [[voltage]]s. The treatment of [[resistor]]s, [[capacitor]]s, and [[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 [[Electrical impedance|impedance]]. This approach is called [[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 electric current, or, more particularly, {{mvar|i}}, which is generally in use to denote instantaneous electric current.

Because the 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 [[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 [[Schrödinger equation]] and Heisenberg's [[matrix mechanics]] – make use of complex numbers.

====Relativity====
In [[special relativity]] and [[general relativity]], some formulas for the metric on [[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 [[Wick rotation|used in an essential way]] in [[quantum field theory]].) Complex numbers are essential to [[spinor]]s, which are a generalization of the [[tensor]]s used in relativity.

==Characterizations, generalizations and related notions==

===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]] over <math>\Q</math>, the [[prime field]] of <math>\Complex,</math> is the [[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]]. 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 [[neighborhood (topology)|nearness]] and [[continuity (topology)|continuity]], which matter in areas such as [[Mathematical analysis|analysis]] and [[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]] {{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, because <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}}

===Other number systems===
{{main|Cayley–Dickson construction|Quaternion|Octonion}}

{| class="wikitable"
|+ Number systems
|-
!
! rational numbers <math>\Q</math>
! real numbers <math>\R</math>
! complex numbers <math>\C</math>
! quaternions <math>\mathbb H</math>
! octonions <math>\mathbb O</math>
! sedenions <math>\mathbb S</math>
|-
! [[complete metric space|complete]]
| {{no}} || {{yes}} || {{yes}} || {{yes}} || {{yes}} || {{yes}}
|-
! [[dimension (vector space)|dimension]] as an <math>\R</math>-vector space
| [does not apply] || 1 || 2 || 4 || 8 || 16 
|-
! [[ordered field|ordered]] 
| {{yes}} || {{yes}} || {{no}} || {{no}} || {{no}} || {{no}}
|-
! multiplication commutative {{nowrap|1=(<math>xy=yx</math>)}}
| {{yes}} || {{yes}} || {{yes}} || {{no}} || {{no}} || {{no}}
|-
! multiplication associative {{nowrap|1=(<math>(xy)z=x(yz)</math>)}}
| {{yes}} || {{yes}} || {{yes}} || {{yes}} || {{no}} || {{no}}
|-
! [[normed division algebra]] {{nowrap|1=(over <math>\R</math>)}}
| [does not apply] || {{yes}} || {{yes}} || {{yes}} || {{yes}} || {{no}}
|}

The process of extending the field <math>\mathbb R</math> of reals to <math>\mathbb C</math> is an instance of the ''Cayley–Dickson construction''. Applying this construction iteratively to <math>\C</math> then yields the [[quaternion]]s, the [[octonion]]s,<ref>{{cite book |first=Kevin |last=McCrimmon |authorlink=Kevin McCrimmon|year=2004 |title=A Taste of Jordan Algebras |page=64 |series=Universitext |publisher=Springer |isbn=0-387-95447-3}} {{mr|id=2014924}}</ref> the [[sedenion]]s, and the [[trigintaduonion]]s. This construction turns out to diminish the structural properties of the involved number systems. 

Unlike the reals, <math>\Complex</math> is not an [[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}} Passing from <math>\C</math> to the quaternions <math>\mathbb H</math> loses commutativity, while the octonions (additionally to not being commutative) fail to be associative. The reals, complex numbers, quaternions and octonions are all [[normed division algebra]]s over <math>\mathbb R</math>. By [[Hurwitz's theorem (normed division algebras)|Hurwitz's theorem]] they are the only ones; the [[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]] 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]].

[[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]]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]]s, with respect to the usual [[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]] {{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]]s.

==See also==
{{Commons category|Complex numbers}}
{{wikiversity|Complex Numbers}}
{{wikibooks|Calculus/Complex numbers}}
{{EB1911 poster|Number/Complex Numbers}}
* [[Analytic continuation]]
* [[Circular motion#Using complex numbers|Circular motion using complex numbers]]
* [[Complex-base system]]
* [[Complex coordinate space]]
* [[Complex geometry]]
* [[Geometry of numbers]]
* [[Dual-complex number]]
* [[Eisenstein integer]]
* [[Geometric algebra#Unit pseudoscalars|Geometric algebra]] (which includes the complex plane as the 2-dimensional [[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-link=George Ashley Campbell |author-first=George Ashley |author-last=Campbell |journal=[[Proceedings of the American Institute of Electrical Engineers]] |publisher=[[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>
}}

{{refbegin}}

* {{cite book |last=Ahlfors |first=Lars |author-link=Lars Ahlfors |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}}
* {{Citation
| last1 = Andreescu
| first1 = Titu
| last2 = Andrica
| first2 = Dorin
| title = Complex Numbers from A to ... Z
| edition = Second
| publisher = Springer
| location = New York
| year = 2014
| isbn = 978-0-8176-8414-3
| doi = 10.1007/978-0-8176-8415-0
}}
* {{cite book |last=Apostol |first=Tom |author-link=Tom Apostol |year=1981 |title=Mathematical analysis |publisher=Addison-Wesley}}
* {{cite book |last1=Aufmann |first1=Richard N. |title=College Algebra and Trigonometry |last2=Barker |first2=Vernon C. |last3=Nation |first3=Richard D. |publisher=Cengage Learning |year=2007 |isbn=978-0-618-82515-8 |edition=6 |url=https://books.google.com/books?id=g5j-cT-vg_wC&pg=PA66}}
* {{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 |last=Derbyshire |first=John |author-link=John Derbyshire |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 |ref=none |last1=Joshi |first1=Kapil D. |title=Foundations of Discrete Mathematics |publisher=[[John Wiley & Sons]] |location=New York |isbn=978-0-470-21152-6 |year=1989}}
* {{cite book |last=Needham |first=Tristan |year=1997 |title=Visual Complex Analysis |publisher=Clarendon Press |isbn=978-0-19-853447-1}}
* {{cite book |ref=none |last=Pedoe |first=Dan |author-link=Daniel Pedoe |title=Geometry: A comprehensive course |publisher=Dover |year=1988 |isbn=978-0-486-65812-4}}
* {{cite book |last=Penrose |first=Roger |author-link=Roger Penrose |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 |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 |id=c/c024140 |title=Complex number |year=2001|first=E.D. |last=Solomentsev}}
{{refend}}


===Historical references===
{{refbegin}}
* {{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 book |ref=none |last=Bourbaki |first=Nicolas |author-link=Nicolas Bourbaki |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 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=283 |journal=Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores |volume=7 |pages=89–148 |language=la |author-link= Carl Friedrich Gauss}}
* {{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 |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}}

{{Authority control}}

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