Complex number

Inmathematics,acomplex numberis an element of anumber systemthat extends thereal numberswith a specific element denotedi,called theimaginary unitand satisfying theequation$i^{2}=-1$;every complex number can be expressed in the form$a+bi$,whereaandbare real numbers. Because no real number satisfies the above equation,iwas called animaginary numberbyRené Descartes.For the complex number$a+bi$,ais called thereal part,andbis called theimaginary part.The set of complex numbers is denoted by either of the symbols$\mathbb {C}$orC.Despite the historical nomenclature, "imaginary" complex numbers have amathematicalexistence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world.

A complex number can be visually represented as a pair of numbers(a, b)forming a vector on a diagram called anArgand diagram,representing thecomplex plane.Reis the real axis,Imis the imaginary axis, andiis the "imaginary unit",that satisfiesi²= −1.

Complex numbers allow solutions to allpolynomial equations,even those that have no solutions in real numbers. More precisely, thefundamental theorem of algebraasserts that every non-constant polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation $(x+1)^{2}=-9$ has no real solution, because the square of a real number cannot be negative, but has the two nonreal complex solutions$-1+3i$and$-1-3i$.

Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule$i^{2}=-1$along with theassociative,commutative,anddistributive laws.Every nonzero complex number has amultiplicative inverse.This makes the complex numbers afieldwith the real numbers as a subfield.

The complex numbers also form areal vector spaceofdimension two,with$\{1,i\}$as astandard basis.This standard basis makes the complex numbers aCartesian plane,called thecomplex 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 thereal line,which is pictured as the horizontal axis of the complex plane, while real multiples of$i$are the vertical axis. A complex number can also be defined by its geometricpolar coordinates:the radius is called theabsolute valueof 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 theunit circle.Adding a fixed complex number to all complex numbers defines atranslationin the complex plane, and multiplying by a fixed complex number is asimilaritycentered at the origin (dilating by the absolute value, and rotating by the argument). The operation ofcomplex conjugationis thereflection symmetrywith respect to the real axis.

The complex numbers form a rich structure that is simultaneously analgebraically closed field,acommutative algebraover the reals, and aEuclidean vector spaceof dimension two.

Definition and basic operations

Various complex numbers depicted in the complex plane.

A complex number is an expressionof the forma+bi,whereaandbarereal numbers,andiis an abstract symbol, the so-calledimaginary unit,whose meaning will be explained further below. For example,2 + 3iis a complex number.

For a complex numbera+bi,the real numberais called itsreal part,and the real numberb(not the complex numberbi) is itsimaginary part.The real part of a complex numberzis denotedRe(z),${\mathcal {Re}}(z)$,or${\mathfrak {R}}(z)$;the imaginary part isIm(z),${\mathcal {Im}}(z)$,or${\mathfrak {I}}(z)$:for example,${\textstyle \operatorname {Re} (2+3i)=2}$,$\operatorname {Im} (2+3i)=3$.

A complex numberzcan be identified with theordered pairof real numbers$(\Re (z),\Im (z))$,which may be interpreted as coordinates of a point in a Euclidean plane with standard coordinates, which is then called thecomplex planeorArgand diagram,.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.

A complex numberz,as a point (black) and itsposition vector(blue).

A real numberacan be regarded as a complex numbera+ 0i,whose imaginary part is 0. A purelyimaginary numberbiis a complex number0 +bi,whose real part is zero. As with polynomials, it is common to writea+ 0i=a,0 +bi=bi,anda+ (−b)i=a−bi;for example,3 + (−4)i= 3 − 4i.

Thesetof all complex numbers is denoted by$\mathbb {C}$(blackboard bold) orC(upright bold).

In some disciplines such aselectromagnetismandelectrical engineering,jis used instead ofi,asifrequently representselectric current,and complex numbers are written asa+bjora+jb.

Addition and subtraction

Addition of two complex numbers can be done geometrically by constructing a parallelogram.

Two complex numbers$a=x+yi$and$b=u+vi$areaddedby separately adding their real and imaginary parts. That is to say:

$$a+b=(x+yi)+(u+vi)=(x+u)+(y+v)i.$$ Similarly,subtractioncan be performed as $$a-b=(x+yi)-(u+vi)=(x-u)+(y-v)i.$$

The addition can be geometrically visualized as follows: the sum of two complex numbersaandb,interpreted as points in the complex plane, is the point obtained by building aparallelogramfrom the three verticesO,and the points of the arrows labeledaandb(provided that they are not on a line). Equivalently, calling these pointsA,B,respectively and the fourth point of the parallelogramXthetrianglesOABandXBAarecongruent.

Multiplication

The product of two complex numbers is computed as follows:

$(a+bi)\cdot (c+di)=ac-bd+(ad+bc)i.$

For example,$(3+2i)(4-i)=3\cdot 4-(2\cdot (-1))+(3\cdot (-1)+2\cdot 4)i=14+5i.$ In particular, this includes as a special case the fundamental formula

$i^{2}=i\cdot i=-1.$

This formula distinguishes the complex numberifrom 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, thedistributive property,thecommutative properties(of addition and multiplication) hold. Therefore, the complex numbers form an algebraic structure known as afield,the same way as the rational or real numbers do.

Complex conjugate, absolute value and argument

Geometric representation ofzand its conjugatezin the complex plane.

Thecomplex conjugateof the complex numberz=x+yiis defined as ${\overline {z}}=x-yi.$It is also denoted by some authors by$z^{*}$.Geometrically,zis the"reflection"ofzabout the real axis. Conjugating twice gives the original complex number:${\overline {\overline {z}}}=z.$A complex number is real if and only if it equals its own conjugate. Theunary operationof taking the complex conjugate of a complex number cannot be expressed by applying only their basic operations addition, subtraction, multiplication and division.

Argumentφand modulusrlocate a point in the complex plane.

For any complex numberz=x+yi,the product

$z\cdot {\overline {z}}=(x+iy)(x-iy)=x^{2}+y^{2}$

is anon-negative realnumber. This allows to define theabsolute value(ormodulusormagnitude) ofzto be thesquare root $$|z|={\sqrt {x^{2}+y^{2}}}.$$ ByPythagoras' theorem,$|z|$is the distance from the origin to the point representing the complex numberzin the complex plane. In particular, thecircle of radius onearound the origin consists precisely of the numberszsuch that$|z|=1$.If$z=x=x+0i$is a real number, then$|z|=|x|$:its absolute value as a complex number and as a real number are equal.

Using the conjugate, thereciprocalof a nonzero complex number$z=x+yi$can be computed to be

$${\frac {1}{z}}={\frac {\bar {z}}{z{\bar {z}}}}={\frac {\bar {z}}{|z|^{2}}}={\frac {x-yi}{x^{2}+y^{2}}}={\frac {x}{x^{2}+y^{2}}}-{\frac {y}{x^{2}+y^{2}}}i.$$ More generally, the division of an arbitrary complex number$w=u+vi$by a non-zero complex number$z=x+yi$equals $${\frac {w}{z}}={\frac {w{\bar {z}}}{|z|^{2}}}={\frac {(u+vi)(x-iy)}{x^{2}+y^{2}}}={\frac {ux+vy}{x^{2}+y^{2}}}+{\frac {vx-uy}{x^{2}+y^{2}}}i.$$ This process is sometimes called "rationalization"of the denominator (although the denominator in the final expression might be an irrational real number), because it resembles the method to remove roots from simple expressions in a denominator.

Theargumentofz(sometimes called the "phase"φ)is the angle of theradiusOzwith the positive real axis, and is written asargz,expressed inradiansin this article. The angle is defined only up to adding integer multiples of$2\pi$,since a rotation by$2\pi$(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$(-\pi,\pi ]$,which is referred to as theprincipal value. The argument can be computed from the rectangular formx + yiby means of thearctan(inverse tangent) function.

Polar form

Main article:Polar coordinate system

"Polar form" redirects here. For the higher-dimensional analogue, seePolar decomposition.

Multiplication of2 +i(blue triangle) and3 +i(red triangle). The red triangle is rotated to match the vertex of the blue one (the adding of both angles in the termsφ₁+φ₂in the equation) and stretched by the length of thehypotenuseof the blue triangle (the multiplication of both radiuses, as per termr₁r₂in the equation).

For any complex numberz,with absolute value$r=|z|$and argument$\varphi$,the equation

$z=r(\cos \varphi +i\sin \varphi )$

holds. This identity is referred to as the polar form ofz.It is sometimes abbreviated as${\textstyle z=r\operatorname {\mathrm {cis} } \varphi }$. Inelectronics,one represents aphasorwith amplituderand phaseφinangle notation:$$z=r\angle \varphi.$$

If two complex numbers are given in polar form, i.e.,z₁=r₁(cos φ₁+i sin φ₁)andz₂=r₂(cos φ₂+i sin φ₂),the product and division can be computed as $$z_{1}z_{2}=r_{1}r_{2}(\cos(\varphi _{1}+\varphi _{2})+i\sin(\varphi _{1}+\varphi _{2})).$$ $${\frac {z_{1}}{z_{2}}}={\frac {r_{1}}{r_{2}}}\left(\cos(\varphi _{1}-\varphi _{2})+i\sin(\varphi _{1}-\varphi _{2})\right),{\text{if }}z_{2}\neq 0.$$ (These are a consequence of thetrigonometric identitiesfor the sine and cosine function.) In other words, the absolute values aremultipliedand the arguments areaddedto yield the polar form of the product. The picture at the right illustrates the multiplication of $$(2+i)(3+i)=5+5i.$$ Because the real and imaginary part of5 + 5iare equal, the argument of that number is 45 degrees, orπ/4(inradian). On the other hand, it is also the sum of the angles at the origin of the red and blue triangles arearctan(1/3) and arctan(1/2), respectively. Thus, the formula $${\frac {\pi }{4}}=\arctan \left({\frac {1}{2}}\right)+\arctan \left({\frac {1}{3}}\right)$$ holds. As thearctanfunction can be approximated highly efficiently, formulas like this – known asMachin-like formulas– are used for high-precision approximations ofπ.

Powers and roots

See also:Square roots of negative and complex numbers

Then-th power of a complex number can be computed usingde Moivre's formula,which is obtained by repeatedly applying the above formula for the product: $$z^{n}=\underbrace {z\cdot \dots \cdot z} _{n{\text{ factors}}}=(r(\cos \varphi +i\sin \varphi ))^{n}=r^{n}\,(\cos n\varphi +i\sin n\varphi ).$$ For example, the first few powers of the imaginary unitiare$i,i^{2}=-1,i^{3}=-i,i^{4}=1,i^{5}=i,\dots$.

Geometric representation of the 2nd to 6th roots of a complex numberz,in polar formre^iφ wherer= |z |andφ= argz.Ifzis real,φ= 0orπ.Principal roots are shown in black.

Thennth rootsof a complex numberzare given by $$z^{1/n}={\sqrt[{n}]{r}}\left(\cos \left({\frac {\varphi +2k\pi }{n}}\right)+i\sin \left({\frac {\varphi +2k\pi }{n}}\right)\right)$$ for0 ≤k≤n− 1.(Here${\sqrt[{n}]{r}}$is the usual (positive)nth root of the positive real numberr.) Because sine and cosine are periodic, other integer values ofkdo not give other values. For any$z\neq 0$,there are, in particularndistinct complexn-th roots. For example, there are 4 fourth roots of 1, namely

$z_{1}=1,z_{2}=i,z_{3}=-1,z_{4}=-i.$

In general there isnonatural way of distinguishing one particular complexnth root of a complex number. (This is in contrast to the roots of a positive real numberx,which has a unique positive realn-th root, which is therefore commonly referred to asthen-th root ofx.) One refers to this situation by saying that thenth root is an-valued functionofz.

Fundamental theorem of algebra

Thefundamental theorem of algebra,ofCarl Friedrich GaussandJean le Rond d'Alembert,states that for any complex numbers (calledcoefficients)a₀, ..., a_n,the equation $$a_{n}z^{n}+\dotsb +a_{1}z+a_{0}=0$$ has at least one complex solutionz,provided that at least one of the higher coefficientsa₁, ..., a_nis nonzero.This property does not hold for thefield of rational numbers$\mathbb {Q}$(the polynomialx²− 2does not have a rational root, because√2is not a rational number) nor the real numbers$\mathbb {R}$(the polynomialx²+ 4does not have a real root, because the square ofxis positive for any real numberx).

Because of this fact,$\mathbb {C}$is called analgebraically 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 asLiouville's theorem,ortopologicalones such as thewinding number,or a proof combiningGalois theoryand the fact that any real polynomial ofodddegree has at least one real root.

History

See also:Negative number § History

The solution inradicals(withouttrigonometric functions) of a generalcubic equation,when all three of its roots are real numbers, contains the square roots ofnegative numbers,a situation that cannot be rectified by factoring aided by therational root test,if the cubic isirreducible;this is the so-calledcasus irreducibilis( "irreducible case" ). This conundrum led Italian mathematicianGerolamo Cardanoto conceive of complex numbers in around 1545 in hisArs Magna,though his understanding was rudimentary; moreover, he later described complex numbers as being "as subtle as they are useless".Cardano did use imaginary numbers, but described using them as "mental torture."This was prior to the use of the graphical complex plane. Cardano and other Italian mathematicians, notablyScipione 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.

Work on the problem of general polynomials ultimately led to thefundamental theorem of algebra,which shows that with complex numbers, a solution exists to everypolynomial equationof degree one or higher. Complex numbers thus form analgebraically closed field,where any polynomial equation has aroot.

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 mathematicianRafael Bombelli.A more abstract formalism for the complex numbers was further developed by the Irish mathematicianWilliam Rowan Hamilton,who extended this abstraction to the theory ofquaternions.

The earliest fleeting reference tosquare rootsofnegative numberscan perhaps be said to occur in the work of theGreek mathematicianHero of Alexandriain the 1st centuryAD,where in hisStereometricahe considered, apparently in error, the volume of an impossiblefrustumof apyramidto arrive at the term${\sqrt {81-144}}$in his calculations, which today would simplify to${\sqrt {-63}}=3i{\sqrt {7}}$.Negative quantities were not conceived of inHellenistic mathematicsand Hero merely replaced it by its positive${\sqrt {144-81}}=3{\sqrt {7}}.$

The impetus to study complex numbers as a topic in itself first arose in the 16th century whenalgebraic solutionsfor the roots ofcubicandquarticpolynomialswere discovered by Italian mathematicians (Niccolò Fontana TartagliaandGerolamo Cardano). It was soon realized (but proved much later)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 numbersis unavoidablewhen all three roots are real and distinct.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 byRené Descartesin 1637, who was at pains to stress their unreal nature:

... 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.
[... 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${\sqrt {-1}}^{2}={\sqrt {-1}}{\sqrt {-1}}=-1$seemed to be capriciously inconsistent with the algebraic identity${\sqrt {a}}{\sqrt {b}}={\sqrt {ab}}$,which is valid for non-negative real numbersaandb,and which was also used in complex number calculations with one ofa,bpositive and the other negative. The incorrect use of this identity in the case when bothaandbare negative, and the related identity${\textstyle {\frac {1}{\sqrt {a}}}={\sqrt {\frac {1}{a}}}}$,even bedeviledLeonhard Euler.This difficulty eventually led to the convention of using the special symboliin place of${\sqrt {-1}}$to guard against this mistake.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 1730Abraham de Moivrenoted 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 followingde Moivre's formula:

$$(\cos \theta +i\sin \theta )^{n}=\cos n\theta +i\sin n\theta.$$

Euler's formula relates the complex exponential function of an imaginary argument, which can be thought of as describinguniform circular motionin 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 obtainedEuler's formulaofcomplex analysis:

$$e^{i\theta }=\cos \theta +i\sin \theta$$

by formally manipulating complexpower seriesand observed that this formula could be used to reduce any trigonometric identity to much simpler exponential identities.

The idea of a complex number as a point in the complex plane (above) was first described byDanish–NorwegianmathematicianCaspar Wesselin 1799,although it had been anticipated as early as 1685 inWallis'sA Treatise of Algebra.

Wessel's memoir appeared in the Proceedings of theCopenhagen Academybut went largely unnoticed. In 1806Jean-Robert Argandindependently issued a pamphlet on complex numbers and provided a rigorous proof of thefundamental theorem of algebra.Carl Friedrich Gausshad earlier published an essentiallytopologicalproof of the theorem in 1797 but expressed his doubts at the time about "the true metaphysics of the square root of −1".It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane,largely establishing modern notation and terminology:

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,${\sqrt {-1}}$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.

In the beginning of the 19th century, other mathematicians discovered independently the geometrical representation of the complex numbers: Buée,Mourey,Warren,Françaisand his brother,Bellavitis.

The English mathematicianG.H. Hardyremarked that Gauss was the first mathematician to use complex numbers in "a really confident and scientific way" although mathematicians such asNorwegianNiels Henrik AbelandCarl Gustav Jacob Jacobiwere necessarily using them routinely before Gauss published his 1831 treatise.

Augustin-Louis CauchyandBernhard Riemanntogether brought the fundamental ideas ofcomplex analysisto 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 calledcosφ+isinφthedirection factor,and$r={\sqrt {a^{2}+b^{2}}}$themodulus;Cauchy (1821) calledcosφ+isinφthereduced form(l'expression réduite)and apparently introduced the termargument;Gauss usedifor${\sqrt {-1}}$,introduced the termcomplex numberfora+bi,and calleda²+b²thenorm.The expressiondirection coefficient,often used forcosφ+isinφ,is due to Hankel (1867),andabsolute value,formodulus,is due to Weierstrass.

Later classical writers on the general theory includeRichard Dedekind,Otto Hölder,Felix Klein,Henri Poincaré,Hermann Schwarz,Karl Weierstrassand 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 byWilhelm Wirtingerin 1927.

Abstract algebraic aspects

While the above low-level definitions, including the addition and multiplication, accurately describes 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$\mathbb {C}$is viapolynomials,i.e., expressions of the form $$p(X)=a_{n}X^{n}+\dotsb +a_{1}X+a_{0},$$ where thecoefficientsa₀,..., a_nare real numbers. The set of all such polynomials is denoted by$\mathbb {R} [X]$.Since sums and products of polynomials are again polynomials, this set$\mathbb {R} [X]$forms acommutative ring,called thepolynomial ring(over the reals). To every such polynomialp,one may assign the complex number$p(i)=a_{n}i^{n}+\dotsb +a_{1}i+a_{0}$,i.e., the value obtained by setting$X=i$.This defines a function

$\mathbb {R} [X]\to \mathbb {C}$

This function issurjectivesince every complex number can be obtained in such a way: the evaluation of alinear polynomial$a+bX$at$X=i$is$a+bi$.However, the evaluation of polynomial$X^{2}+1$atiis 0, since$i^{2}+1=0.$This polynomial isirreducible,i.e., cannot be written as a product of two linear polynomials. Basic facts ofabstract algebrathen imply that thekernelof the above map is anidealgenerated by this polynomial, and that the quotient by this ideal is a field, and that there is anisomorphism

$\mathbb {R} [X]/(X^{2}+1){\stackrel {\cong }{\to }}\mathbb {C}$

between the quotient ring and$\mathbb {C}$.Some authors take this as the definition of$\mathbb {C}$.

Accepting that$\mathbb {C}$is algebraically closed, because it is analgebraic extensionof$\mathbb {R}$in this approach,$\mathbb {C}$is therefore thealgebraic closureof$\mathbb {R}.$

Matrix representation of complex numbers

Complex numbersa+bican also be represented by2 × 2matricesthat have the form $${\begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}}.$$ Here the entriesaandbare real numbers. As the sum and product of two such matrices is again of this form, these matrices form asubringof the ring of2 × 2matrices.

A simple computation shows that the map $$a+ib\mapsto {\begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}}$$ is aring isomorphismfrom 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 thedeterminantof the corresponding matrix, and the conjugate of a complex number with thetransposeof the matrix.

The geometric description of the multiplication of complex numbers can also be expressed in terms ofrotation matricesby using this correspondence between complex numbers and such matrices. The action of the matrix on a vector(x,y)corresponds to the multiplication ofx+iybya+ib.In particular, if the determinant is1,there is a real numbertsuch that the matrix has the form

$${\begin{pmatrix}\cos t&-\sin t\\\sin t&\;\;\cos t\end{pmatrix}}.$$ In this case, the action of the matrix on vectors and the multiplication by the complex number$\cos t+i\sin t$are both therotationof the anglet.

Complex analysis

Main article:Complex analysis

The study of functions of a complex variable is known ascomplex analysisand has enormous practical use inapplied mathematicsas well as in other branches of mathematics. Often, the most natural proofs for statements inreal analysisor evennumber theoryemploy techniques from complex analysis (seeprime number theoremfor an example).

Adomain coloringgraph of the function⁠(z²− 1)(z− 2 −i)²/z²+ 2 + 2i⁠.Darker spots mark moduli near zero, brighter spots are farther away from the origin. The color encodes the argument. The function has zeros for±1, (2 +i)andpolesat$\pm {\sqrt {-2-2i}}.$

Unlike real functions, which are commonly represented as two-dimensional graphs,complex functionshave four-dimensional graphs and may usefully be illustrated by color-coding athree-dimensional graphto suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.

Convergence

Illustration of the behavior of the sequence$z^{n}$for three different values ofz(all having the same argument): for$|z|<1$the sequence converges to 0 (inner spiral), while it diverges for$|z|>1$(outer spiral).

The notions ofconvergent seriesandcontinuous functionsin (real) analysis have natural analogs in complex analysis. A sequence of complex numbers is said toconvergeif and only if its real and imaginary parts do. This is equivalent to the(ε, δ)-definition of limits,where the absolute value of real numbers is replaced by the one of complex numbers. From a more abstract point of view,$\mathbb {C}$,endowed with themetric $$\operatorname {d} (z_{1},z_{2})=|z_{1}-z_{2}|$$ is a completemetric space,which notably includes thetriangle inequality $$|z_{1}+z_{2}|\leq |z_{1}|+|z_{2}|$$ for any two complex numbersz₁andz₂.

Complex exponential

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, andihighlighted. The two rectangles (in magenta and green) are mapped to circular segments, while the lines parallel to thex-axis are mapped to rays emanating from, but not containing the origin. Lines parallel to they-axis are mapped to circles.

Like in real analysis, this notion of convergence is used to construct a number ofelementary functions:theexponential functionexpz,also writtene^z,is defined as theinfinite series,which can be shown toconvergefor anyz: $$\exp z:=1+z+{\frac {z^{2}}{2\cdot 1}}+{\frac {z^{3}}{3\cdot 2\cdot 1}}+\cdots =\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}.$$ For example,$\exp(1)$isEuler's number$e\approx 2.718$.Euler's formulastates: $$\exp(i\varphi )=\cos \varphi +i\sin \varphi$$ for any real numberφ.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 includesEuler's identity $$\exp(i\pi )=-1.$$

Complex logarithm

The exponential function maps complex numberszdiffering by a multiple of$2\pi i$to the same complex numberw.

For any positive real numbert,there is a unique real numberxsuch that$\exp(x)=t$.This leads to the definition of thenatural logarithmas theinverse $\ln \colon \mathbb {R} ^{+}\to \mathbb {R};x\mapsto \ln x$of the exponential function. The situation is different for complex numbers, since

$\exp(z+2\pi i)=\exp z\exp(2\pi i)=\exp z$

by the functional equation and Euler's identity. For example,e^iπ=e^3iπ= −1,so bothiπand3iπare possible values for the complex logarithm of−1.

In general, given any non-zero complex numberw,any numberzsolving the equation

$\exp z=w$

is called acomplex logarithmofw,denoted$\log w$.It can be shown that these numbers satisfy $$z=\log w=\ln |w|+i\arg w,$$ where arg is theargumentdefinedabove,and ln the (real)natural logarithm.As arg is amultivalued function,unique only up to a multiple of2π,log is also multivalued. Theprincipal valueof log is often taken by restricting the imaginary part to theinterval(−π,π].This leads to the complex logarithm being abijectivefunction taking values in the strip$\mathbb {R} ^{+}+\;i\,\left(-\pi,\pi \right]$(that is denoted$S_{0}$in the above illustration) $$\ln \colon \;\mathbb {C} ^{\times }\;\to \;\;\;\mathbb {R} ^{+}+\;i\,\left(-\pi,\pi \right].$$

If$z\in \mathbb {C} \setminus \left(-\mathbb {R} _{\geq 0}\right)$is not a non-positive real number (a positive or a non-real number), the resultingprincipal valueof the complex logarithm is obtained with−π<φ<π.It is ananalytic functionoutside the negative real numbers, but it cannot be prolongated to a function that is continuous at any negative real number$z\in -\mathbb {R} ^{+}$,where the principal value islnz= ln(−z) +iπ.

Complexexponentiationz^ωis defined as $$z^{\omega }=\exp(\omega \ln z),$$ and is multi-valued, except whenωis an integer. Forω= 1 /n,for some natural numbern,this recovers the non-uniqueness ofnth roots mentioned above. Ifz> 0is real (andωan arbitrary complex number), one has a preferred choice of$\ln x$,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; seefailure of power and logarithm identities.For example, they do not satisfy $$a^{bc}=\left(a^{b}\right)^{c}.$$ 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 functionssineandcosine,as well as thehyperbolic functionssinh and cosh, also carry over to complex arguments without change. For the other trigonometric and hyperbolic functions, such astangent,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 ofanalytic continuation.

Holomorphic functions

Color wheel graph of the functionsin(1/z)that is holomorphic except atz= 0, which is an essential singularity of this function. White parts inside refer to numbers having large absolute values.

A function$f:\mathbb {C}$→$\mathbb {C}$is calledholomorphicorcomplex differentiableat a point$z_{0}$if the limit

$\lim _{z\to z_{0}}{f(z)-f(z_{0}) \over z-z_{0}}$

exists (in which case it is denoted by$f'(z_{0})$). This mimics the definition for real differentiable functions, except that all quantities are complex numbers. Loosely speaking, the freedom of approaching$z_{0}$in different directions imposes a much stronger condition than being (real) differentiable. For example, the function

$f(z)={\overline {z}}$

is differentiable as a function$\mathbb {R} ^{2}\to \mathbb {R} ^{2}$,but isnotcomplex differentiable. A real differentiable function is complex differentiableif and only ifit satisfies theCauchy–Riemann equations,which are sometimes abbreviated as

${\frac {\partial f}{\partial {\overline {z}}}}=0.$

Complex analysis shows some features not apparent in real analysis. For example, theidentity theoremasserts that two holomorphic functionsfandgagree if they agree on an arbitrarily smallopen subsetof$\mathbb {C}$.Meromorphic functions,functions that can locally be written asf(z)/(z−z₀)ⁿwith a holomorphic functionf,still share some of the features of holomorphic functions. Other functions haveessential singularities,such assin(1/z)atz= 0.

Applications

Complex numbers have applications in many scientific areas, includingsignal processing,control theory,electromagnetism,fluid dynamics,quantum mechanics,cartography,andvibration analysis.Some of these applications are described below.

Complex conjugation is also employed ininversive geometry,a branch of geometry studying reflections more general than ones about a line. In thenetwork analysis of electrical circuits,the complex conjugate is used in finding the equivalent impedance when themaximum power transfer theoremis looked for.

Geometry

Shapes

Threenon-collinearpoints$u,v,w$in the plane determine theshapeof the triangle$\{u,v,w\}$.Locating the points in the complex plane, this shape of a triangle may be expressed by complex arithmetic as $$S(u,v,w)={\frac {u-w}{u-v}}.$$ The shape$S$of a triangle will remain the same, when the complex plane is transformed by translation or dilation (by anaffine transformation), corresponding to the intuitive notion of shape, and describingsimilarity.Thus each triangle$\{u,v,w\}$is in asimilarity classof triangles with the same shape.

Fractal geometry

The Mandelbrot set with the real and imaginary axes labeled.

TheMandelbrot setis a popular example of a fractal formed on the complex plane. It is defined by plotting every location$c$where iterating the sequence$f_{c}(z)=z^{2}+c$does notdivergewheniteratedinfinitely. Similarly,Julia setshave the same rules, except where$c$remains constant.

Triangles

Every triangle has a uniqueSteiner inellipse– anellipseinside the triangle and tangent to the midpoints of the three sides of the triangle. Thefociof a triangle's Steiner inellipse can be found as follows, according toMarden's theorem:Denote the triangle's vertices in the complex plane asa=x_A+y_Ai,b=x_B+y_Bi,andc=x_C+y_Ci.Write thecubic equation$(x-a)(x-b)(x-c)=0$,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

Construction of a regular pentagonusing straightedge and compass.

As mentioned above, any nonconstant polynomial equation (in complex coefficients) has a solution in$\mathbb {C}$.A fortiori,the same is true if the equation has rational coefficients. The roots of such equations are calledalgebraic numbers– they are a principal object of study inalgebraic number theory.Compared to${\overline {\mathbb {Q} }}$,the algebraic closure of$\mathbb {Q}$,which also contains all algebraic numbers,$\mathbb {C}$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 offield theoryto thenumber fieldcontainingroots of unity,it can be shown that it is not possible to construct a regularnonagonusing only compass and straightedge– a purely geometric problem.

Another example is theGaussian integers;that is, numbers of the formx+iy,wherexandyare integers, which can be used to classifysums of squares.

Analytic number theory

Main article: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, theRiemann zeta functionζ(s)is related to the distribution ofprime numbers.

Improper integrals

In applied fields, complex numbers are often used to compute certain real-valuedimproper integrals,by means of complex-valued functions. Several methods exist to do this; seemethods of contour integration.

Dynamic equations

Indifferential equations,it is common to first find all complex rootsrof thecharacteristic equationof alinear differential equationor equation system and then attempt to solve the system in terms of base functions of the formf(t) =e^rt.Likewise, indifference equations,the complex rootsrof the characteristic equation of the difference equation system are used, to attempt to solve the system in terms of base functions of the formf(t) =r^t.

Linear algebra

Since$\mathbb {C}$is algebraically closed, any non-empty complexsquare matrixhas at least one (complex)eigenvalue.By comparison, real matrices do not always have real eigenvalues, for examplerotation matrices(for rotations of the plane for angles other than 0° or 180°) leave no direction fixed, and therefore do not have anyrealeigenvalue. The existence of (complex) eigenvalues, and the ensuing existence ofeigendecompositionis a useful tool for computing matrix powers andmatrix exponentials.

Complex numbers often generalize concepts originally conceived in the real numbers. For example, theconjugate transposegeneralizes thetranspose,hermitian matricesgeneralizesymmetric matrices,andunitary matricesgeneralizeorthogonal matrices.

In applied mathematics

Control theory

See also:Complex plane § Use in control theory

Incontrol theory,systems are often transformed from thetime domainto the complexfrequency domainusing theLaplace transform.The system'szeros and polesare then analyzed in thecomplex plane.Theroot locus,Nyquist plot,andNichols plottechniques 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 beunstable,
• all in the left half plane, it will bestable,
• on the imaginary axis, it will havemarginal stability.

If a system has zeros in the right half plane, it is anonminimum phasesystem.

Signal analysis

Complex numbers are used insignal analysisand 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 asine waveof a givenfrequency,the absolute value|z|of the correspondingzis theamplitudeand theargumentargzis thephase.

IfFourier analysisis 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

$$x(t)=\operatorname {Re} \{X(t)\}$$

and

$$X(t)=Ae^{i\omega t}=ae^{i\phi }e^{i\omega t}=ae^{i(\omega t+\phi )}$$

where ω represents theangular frequencyand the complex numberAencodes the phase and amplitude as explained above.

This use is also extended intodigital signal processinganddigital image processing,which use digital versions of Fourier analysis (andwaveletanalysis) to transmit,compress,restore, and otherwise processdigitalaudiosignals, still images, andvideosignals.

Another example, relevant to the two side bands ofamplitude modulationof AM radio, is:

$${\begin{aligned}\cos((\omega +\alpha )t)+\cos \left((\omega -\alpha )t\right)&=\operatorname {Re} \left(e^{i(\omega +\alpha )t}+e^{i(\omega -\alpha )t}\right)\\&=\operatorname {Re} \left(\left(e^{i\alpha t}+e^{-i\alpha t}\right)\cdot e^{i\omega t}\right)\\&=\operatorname {Re} \left(2\cos(\alpha t)\cdot e^{i\omega t}\right)\\&=2\cos(\alpha t)\cdot \operatorname {Re} \left(e^{i\omega t}\right)\\&=2\cos(\alpha t)\cdot \cos \left(\omega t\right).\end{aligned}}$$

In physics

Electromagnetism and electrical engineering

Main article:Alternating current

Inelectrical engineering,theFourier transformis used to analyze varyingvoltagesandcurrents.The treatment ofresistors,capacitors,andinductorscan then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called theimpedance.This approach is calledphasorcalculus.

In electrical engineering, the imaginary unit is denoted byj,to avoid confusion withI,which is generally in use to denoteelectric current,or, more particularly,i,which is generally in use to denote instantaneous electric current.

Because thevoltagein an ACcircuitis oscillating, it can be represented as

$$V(t)=V_{0}e^{j\omega t}=V_{0}\left(\cos \omega t+j\sin \omega t\right),$$

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

$$v(t)=\operatorname {Re} (V)=\operatorname {Re} \left[V_{0}e^{j\omega t}\right]=V_{0}\cos \omega t.$$

The complex-valued signalV(t)is called theanalyticrepresentation of the real-valued, measurable signalv(t).

Fluid dynamics

Influid dynamics,complex functions are used to describepotential flow in two dimensions.

Quantum mechanics

The complex number field is intrinsic to themathematical formulations of quantum mechanics,where complexHilbert spacesprovide the context for one such formulation that is convenient and perhaps most standard. The original foundation formulas of quantum mechanics – theSchrödinger equationand Heisenberg'smatrix mechanics– make use of complex numbers.

Relativity

Inspecialandgeneral relativity,some formulas for the metric onspacetimebecome simpler if one takes the time component of the spacetime continuum to be imaginary. (This approach is no longer standard in classical relativity, but isused in an essential wayinquantum field theory.) Complex numbers are essential tospinors,which are a generalization of thetensorsused in relativity.

Characterizations, generalizations and related notions

Algebraic characterization

The field$\mathbb {C}$has the following three properties:

• First, it hascharacteristic0. This means that1 + 1 + ⋯ + 1 ≠ 0for any number of summands (all of which equal one).
• Second, itstranscendence degreeover$\mathbb {Q}$,theprime fieldof$\mathbb {C},$is thecardinality of the continuum.
• Third, it isalgebraically closed(see above).

It can be shown that any field having these properties isisomorphic(as a field) to$\mathbb {C}.$For example, thealgebraic closureof the field$\mathbb {Q} _{p}$of thep-adic numberalso satisfies these three properties, so these two fields are isomorphic (as fields, but not as topological fields).Also,$\mathbb {C}$is isomorphic to the field of complexPuiseux series.However, specifying an isomorphism requires theaxiom of choice.Another consequence of this algebraic characterization is that$\mathbb {C}$contains many proper subfields that are isomorphic to$\mathbb {C}$.

Characterization as a topological field

The preceding characterization of$\mathbb {C}$describes only the algebraic aspects of$\mathbb {C}.$That is to say, the properties ofnearnessandcontinuity,which matter in areas such asanalysisandtopology,are not dealt with. The following description of$\mathbb {C}$as atopological field(that is, a field that is equipped with atopology,which allows the notion of convergence) does take into account the topological properties.$\mathbb {C}$contains a subsetP(namely the set of positive real numbers) of nonzero elements satisfying the following three conditions:

• Pis closed under addition, multiplication and taking inverses.
• Ifxandyare distinct elements ofP,then eitherx−yory−xis inP.
• IfSis any nonempty subset ofP,thenS+P=x+Pfor somexin$\mathbb {C}.$

Moreover,$\mathbb {C}$has a nontrivialinvolutiveautomorphismx↦x*(namely the complex conjugation), such thatx x*is inPfor any nonzeroxin$\mathbb {C}.$

Any fieldFwith these properties can be endowed with a topology by taking the setsB(x, p) = { y|p− (y−x)(y−x)* ∈P } as abase,wherexranges over the field andpranges overP.With this topologyFis isomorphic as atopologicalfield to$\mathbb {C}.$

The onlyconnectedlocally compacttopological fieldsare$\mathbb {R}$and$\mathbb {C}.$This gives another characterization of$\mathbb {C}$as a topological field, because$\mathbb {C}$can be distinguished from$\mathbb {R}$because the nonzero complex numbers areconnected,while the nonzero real numbers are not.

Other number systems

Main articles:Cayley–Dickson construction,Quaternion,andOctonion

Number systems

	rational numbers$\mathbb {Q}$	real numbers$\mathbb {R}$	complex numbers$\mathbb {C}$	quaternions$\mathbb {H}$	octonions$\mathbb {O}$	sedenions$\mathbb {S}$
complete	No	Yes	Yes	Yes	Yes	Yes
dimensionas an$\mathbb {R}$-vector space	[does not apply]	1	2	4	8	16
ordered	Yes	Yes	No	No	No	No
multiplication commutative($xy=yx$)	Yes	Yes	Yes	No	No	No
multiplication associative($(xy)z=x(yz)$)	Yes	Yes	Yes	Yes	No	No
normed division algebra(over$\mathbb {R}$)	[does not apply]	Yes	Yes	Yes	Yes	No

The process of extending the field$\mathbb {R}$of reals to$\mathbb {C}$is an instance of theCayley–Dickson construction.Applying this construction iteratively to$\mathbb {C}$then yields thequaternions,theoctonions,thesedenions,and thetrigintaduonions.This construction turns out to diminish the structural properties of the involved number systems.

Unlike the reals,$\mathbb {C}$is not anordered field,that is to say, it is not possible to define a relationz₁<z₂that is compatible with the addition and multiplication. In fact, in any ordered field, the square of any element is necessarily positive, soi²= −1precludes the existence of anorderingon$\mathbb {C}.$Passing from$\mathbb {C}$to the quaternions$\mathbb {H}$loses commutativity, while the octonions (additionally to not being commutative) fail to be associative. The reals, complex numbers, quaternions and octonions are allnormed division algebrasover$\mathbb {R}$.ByHurwitz's theoremthey are the only ones; thesedenions,the next step in the Cayley–Dickson construction, fail to have this structure.

The Cayley–Dickson construction is closely related to theregular representationof$\mathbb {C},$thought of as an$\mathbb {R}$-algebra(an$\mathbb {R}$-vector space with a multiplication), with respect to the basis(1, i).This means the following: the$\mathbb {R}$-linear map $${\begin{aligned}\mathbb {C} &\rightarrow \mathbb {C} \\z&\mapsto wz\end{aligned}}$$ for some fixed complex numberwcan be represented by a2 × 2matrix (once a basis has been chosen). With respect to the basis(1, i),this matrix is $${\begin{pmatrix}\operatorname {Re} (w)&-\operatorname {Im} (w)\\\operatorname {Im} (w)&\operatorname {Re} (w)\end{pmatrix}},$$ that is, the one mentioned in the section on matrix representation of complex numbers above. While this is alinear representationof$\mathbb {C}$in the 2 × 2 real matrices, it is not the only one. Any matrix $$J={\begin{pmatrix}p&q\\r&-p\end{pmatrix}},\quad p^{2}+qr+1=0$$ has the property that its square is the negative of the identity matrix:J²= −I.Then $$\{z=aI+bJ:a,b\in \mathbb {R} \}$$ is also isomorphic to the field$\mathbb {C},$and gives an alternative complex structure on$\mathbb {R} ^{2}.$This is generalized by the notion of alinear complex structure.

Hypercomplex numbersalso generalize$\mathbb {R},$$\mathbb {C},$$\mathbb {H},$and$\mathbb {O}.$For example, this notion contains thesplit-complex numbers,which are elements of the ring$\mathbb {R} [x]/(x^{2}-1)$(as opposed to$\mathbb {R} [x]/(x^{2}+1)$for complex numbers). In this ring, the equationa²= 1has four solutions.

The field$\mathbb {R}$is the completion of$\mathbb {Q},$the field ofrational numbers,with respect to the usualabsolute valuemetric.Other choices ofmetricson$\mathbb {Q}$lead to the fields$\mathbb {Q} _{p}$ofp-adic numbers(for anyprime numberp), which are thereby analogous to$\mathbb {R}$.There are no other nontrivial ways of completing$\mathbb {Q}$than$\mathbb {R}$and$\mathbb {Q} _{p},$byOstrowski's theorem.The algebraic closures${\overline {\mathbb {Q} _{p}}}$of$\mathbb {Q} _{p}$still carry a norm, but (unlike$\mathbb {C}$) are not complete with respect to it. The completion$\mathbb {C} _{p}$of${\overline {\mathbb {Q} _{p}}}$turns out to be algebraically closed. By analogy, the field is calledp-adic complex numbers.

The fields$\mathbb {R},$$\mathbb {Q} _{p},$and their finite field extensions, including$\mathbb {C},$are calledlocal fields.

See also

• Analytic continuation
• Circular motion using complex numbers
• Complex-base system
• Complex coordinate space
• Complex geometry
• Geometry of numbers
• Dual-complex number
• Eisenstein integer
• Geometric algebra(which includes the complex plane as the 2-dimensionalspinorsubspace${\mathcal {G}}_{2}^{+}$)
• Unit complex number

References

• Ahlfors, Lars(1979).Complex analysis(3rd ed.). McGraw-Hill.ISBN978-0-07-000657-7.
• Andreescu, Titu; Andrica, Dorin (2014),Complex Numbers from A to... Z(Second ed.), New York: Springer,doi:10.1007/978-0-8176-8415-0,ISBN978-0-8176-8414-3
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• Penrose, Roger(2005).The Road to Reality: A complete guide to the laws of the universe.Alfred A. Knopf.ISBN978-0-679-45443-4.
• Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. (2007)."Section 5.5 Complex Arithmetic".Numerical Recipes: The art of scientific computing(3rd ed.). New York: Cambridge University Press.ISBN978-0-521-88068-8.Archived fromthe originalon 13 March 2020.Retrieved9 August2011.
• Solomentsev, E.D. (2001) [1994],"Complex number",Encyclopedia of Mathematics,EMS Press

Historical references

• Argand (1814)."Reflexions sur la nouvelle théorie des imaginaires, suives d'une application à la demonstration d'un theorème d'analise"[Reflections on the new theory of complex numbers, followed by an application to the proof of a theorem of analysis].Annales de mathématiques pures et appliquées(in French).5:197–209.
• Bourbaki, Nicolas(1998). "Foundations of mathematics § logic: set theory".Elements of the history of mathematics.Springer.
• Burton, David M. (1995).The History of Mathematics(3rd ed.). New York:McGraw-Hill.ISBN978-0-07-009465-9.
• Gauss, C. F.(1831)."Theoria residuorum biquadraticorum. Commentatio secunda"[Theory of biquadratic residues. Second memoir.].Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores(in Latin).7:89–148.
• Katz, Victor J. (2004).A History of Mathematics, Brief Version.Addison-Wesley.ISBN978-0-321-16193-2.
• Nahin, Paul J. (1998).An Imaginary Tale: The Story of$\scriptstyle {\sqrt {-1}}$.Princeton University Press.ISBN978-0-691-02795-1.— A gentle introduction to the history of complex numbers and the beginnings of complex analysis.
• Ebbinghaus, H. D.; Hermes, H.; Hirzebruch, F.; Koecher, M.; Mainzer, K.; Neukirch, J.; Prestel, A.; Remmert, R. (1991).Numbers(hardcover ed.). Springer.ISBN978-0-387-97497-2.— An advanced perspective on the historical development of the concept of number.