Complex number

The plane of the picture represents the complex numbers, with points in the Mandelbrot set (a fractal) coloured blue. The points in the set are those which stay near the origin after repeated squaring and adding.

A complex number is a number comprising a real number and an imaginary number. It can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit with the property i ² = −1. The complex numbers contain the ordinary real numbers, but extend them by adding in extra numbers and correspondingly expanding the understanding of addition and multiplication. This is in order to form a closed field, where any polynomial equation has a root, including examples such as x² = −1.

Complex numbers were first conceived and defined by the Italian mathematician Gerolamo Cardano, who called them "fictitious", during his attempts to find solutions to cubic equations. The solution of a general cubic equation in radicals (without trigonometric functions) may require intermediate calculations containing the square roots of negative numbers, even when the final solutions are real numbers, a situation known as casus irreducibilis. This 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.

The rules for addition, subtraction, multiplication, and division of complex numbers were developed by the Italian mathematician Rafael Bombelli. 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.

Complex numbers are used in a number of fields, including: engineering, electromagnetism, quantum physics, applied mathematics, and chaos theory. When the underlying field of numbers for a mathematical construct is the field of complex numbers, the name usually reflects that fact. Examples are complex analysis, complex matrix, complex polynomial, and complex Lie algebra.

Definitions

Notation

The set of all complex numbers is usually denoted by C, or in blackboard bold by $\mathbb {C}$.

Although other notations can be used, complex numbers are usually written in the form

$a+bi\,$

where a and b are real numbers, and i is the imaginary unit, which has the property i ² = −1. The real number a is called the real part of the complex number, and the real number b is the imaginary part. For example, 3 + 2i is a complex number, with real part 3 and imaginary part 2. If z = a + bi, the real part a is denoted Re(z) or ℜ(z), and the imaginary part b is denoted Im(z) or ℑ(z).

Complex numbers (C) are regarded as an extension of real numbers (R) by considering every real number a complex number with an imaginary part of zero; that is, tthe real number a is identified with the complex number a + 0i. Complex numbers with a real part of zero (Re(z)=0) are called imaginary numbers; instead of writing 0 + bi, that imaginary number is usually denoted as just bi. If b equals 1, instead of using 0 + 1i or 1i, the number is denoted as i.

In some disciplines (in particular, electrical engineering, where i is a symbol for current), the imaginary unit i is instead written as j, so complex numbers are sometimes written as a + bj or a + jb.

Two complex numbers are said to be equal if and only if their real parts are equal and their imaginary parts are equal.

Formal development

In a rigorous setting, it is not acceptable to simply assume that there exists a number whose square is −1. There are various ways of defining C, building on the knowledge of real numbers. Firstly, write C for R², the set of ordered pairs of real numbers, and define operations on complex numbers in C according to

(a, b) + (c, d) = (a + c, b + d)

(a, b)·(c, d) = (a·c − b·d, b·c + a·d).

It is then just a matter of notation to express (a, b) as a + ib. This means we can associate the numbers (a, 0) with the real numbers, and write i = (0, 1). Since (0, 1)·(0, 1) = (−1, 0), we have found i by constructing it, not postulating it. Using these formal operations on R², it is easy to check that we satisfy the field axioms (associativity, commutativity, identity, inverses, distributivity). In particular, R is a subfield of C.

Though this low-level construction does accurately describe the structure of the complex numbers, the definitions seem arbitrary, so secondly C can be considered algebraically. In algebra (the theory of group-like structures), this explicit definition of operations in fact turns out to be the mechanism behind the idea of constructing the algebraic closure of the reals, that is, adding in some elements to R to make a new field, of which R is a subfield, where every non-constant polynomial has a root. Finally, yet another way of characterising C is in terms of its topological properties. Details of these are given below.

Operations

Complex numbers can be intuitively added, subtracted, multiplied, and divided by applying the associative, commutative and distributive laws of algebra, together with the equation i ² = −1. Start by defining the two basic operations:

$(a+bi)+(c+di):=(a+c)+(b+d)i$	[addition]
$(a+bi)(c+di):=(ac-bd)+(bc+ad)i.$	[multiplication]

The notation is well-designed, since proceeding to manipulate the symbols purely intuitively gives

$(a+bi)(c+di)=ac+bci+adi+bdi^{2}=(ac-bd)+(bc+ad)i.$

The fact that this agrees with the definition justifies this sort of manipulation, by indirectly deriving the distributive law, and so on.

Subtraction and division are found from these two by working backwards. In the case of subtraction, $u-w$ means "u plus the number which cancels out w (its additive inverse)", giving

$(a+bi)-(c+di)=(a-c)+(b-d)i$.

[subtraction]

Similarly for multiplication, the inverse of $c+di$ is found to be

$\left({\frac {c}{c^{2}+d^{2}}}\right)-\left({\frac {d}{c^{2}+d^{2}}}\right)i$

when c and d are not both zero (by solving for z in $z(c+di)=1$), which then yields

${\frac {a+bi}{c+di}}=\left({ac+bd \over c^{2}+d^{2}}\right)+\left({bc-ad \over c^{2}+d^{2}}\right)i.$

[division]

If the symbols are manipulated intuitively, so that

${\begin{aligned}{\frac {a+bi}{c+di}}&={\frac {(a+bi)(c-di)}{(c+di)(c-di)}}={\frac {ac+bci-adi+bd}{(c+di)(c-di)}}\\&=\left({ac+bd \over c^{2}+d^{2}}\right)+\left({bc-ad \over c^{2}+d^{2}}\right)i,\end{aligned}}$

the formula agrees with the one just derived. In fact, once the gaps are filled in, all the usual rules of simple arithmetic on real numbers can be shown to work when the symbols are appropriately swapped for complex numbers.

Finally, square (and higher) roots can now be defined. However, every non-zero number has two square roots, so there is an ambiguity. Where the radical symbol (√) is used, it is normally used to denote the root with smallest non-negative phase (see section below on polar representation).

Conjugation

Define the conjugate of $z=x+iy$ to be $x-iy$, written as ${\bar {z}}$ or $z*$. Both $z+{\bar {z}}$ and $z\cdot {\bar {z}}$ are real numbers. Conjugation distributes over all the algebraic operations and many derived functions, for example $\sin {\bar {z}}={\overline {\sin z}}$; this is as expected, since the labelling of one of the two roots of −1 as i is arbitrary, so many situations should exhibit symmetry about the real axis. It is important to note, however, that the function $f(z)={\bar {z}}$ is not complex-differentiable (see Holomorphic function).

Elementary functions

Main article: Elementary function

One of the most important functions on the complex numbers is perhaps the exponential function exp(z), also written e^z, defined in terms of the infinite series

$\exp(z):=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}=1+z+{\frac {z^{2}}{2\cdot 1}}+{\frac {z^{3}}{3\cdot 2\cdot 1}}+\cdots .$

The elementary functions are those which can be finitely built using exp and the arithmetic operations given above, as well as taking inverses; in particular, the inverse of the exponential function, the logarithm. The real-valued logarithm over the positive reals is well-defined, and the complex logarithm generalises this idea. The inverse of exp is shown to be

$\log(x+iy)={\tfrac {1}{2}}\ln(x^{2}+y^{2})+i\arg(x+iy),$

where arg is the argument defined below, and ln the real logarithm. As arg is a multivalued function, unique only up to a multiple of 2π, log is also multivalued. The principal value of log is often taken by restricting the imaginary part to the interval (−π,π].

The familiar trigonometric functions are composed of these, so they are also elementary. For example,

$\sin(z)={\frac {e^{iz}-e^{-iz}}{2i}}.$

Hyperbolic functions such as sinh are similarly constructed.

Exponentiation

Further information: Exponentiation

Raising numbers to positive integer powers is done using the operation of multiplication:

$z^{n}=\underbrace {z\cdot z\cdots z} _{n{\text{ factors}}}.\,$

Negative integer powers also are defined just as for real numbers, since 1/zⁿ is the only way of interpreting z⁻ⁿ such that the familiar rules of indices still work (z⁻ⁿ = z⁻ⁿ(zⁿ/zⁿ) = z⁻ⁿ⁺ⁿ/zⁿ = 1/zⁿ). Similar considerations show that we can define rational real powers just as for the reals, so z^1/n is the nth root of z. Roots are not unique, so it is already clear that complex powers are multivalued, thus careful treatment of powers is needed; for example (8^1/3)⁴ ≠ 16, as there are three cube roots of 8, so the given expression, often shortened to 8^4/3, is the simplest possible.

For arbitrary complex powers, the general meaning of z^ω must be multi-valued, since it is in the case of ω rational. To agree with the definitions so far, this suggests

$z^{\omega }:=\exp(\omega \log z),\,$

which is the general extension of exponentiation to the complex numbers.

The complex plane

Figure 1: A complex number plotted as a point (red) and position vector (blue) on an Argand diagram; $a+bi$ is the rectangular expression of the point.

A complex number can be viewed as a point or position vector in a two-dimensional Cartesian coordinate system called the complex plane or Argand diagram (see Pedoe 1988 and Solomentsev 2001), named after Jean-Robert Argand. The numbers are conventionally plotted using the real part as the horizontal component, and imaginary part as vertical (see Figure 1). These two values used to identify a given complex number are therefore called its Cartesian-, rectangular-, or algebraic form.

Geometric interpretation of the operations

The operations described algebraically above can be visualised using Argand diagrams.

	X = A + B: The sum of two points A and B of the complex plane is the point X = A + B such that the triangles with vertices 0, A, B, and X, B, A, are congruent. Thus the addition of two complex numbers is the same as vector addition of two vectors.
	X = AB: The product of two points A and B is the point X = AB such that the triangles with vertices 0, 1, A, and 0, B, X, are similar.
	X = A*: The complex conjugate of a point A is the point X = A* such that the triangles with vertices 0, 1, A, and 0, 1, X, are mirror images of each other.

These geometric interpretations allow problems of algebra to be translated into geometry. And, conversely, geometric problems can be examined algebraically. For example, the problem of the geometric construction of the 17-gon was by Gauss translated into the analysis of the algebraic equation x¹⁷ = 1 (see Heptadecagon).

Polar form

Further information: Polar coordinate system

Figure 2: The argument φ and modulus r locate a point on an Argand diagram; $r(\cos \phi +i\sin \phi )$ or $re^{i\phi }$ are polar expressions of the point.

The diagrams suggest various properties. Firstly, the distance of a point z from the origin (shown as r in Figure 2) is known as the modulus, absolute value, or magnitude, and written $|z|$. By Pythagoras' theorem,

$|x+iy|={\sqrt {x^{2}+y^{2}}}.$

In general, distances between complex numbers are given by the distance function $d(z,w)=|z-w|$, which turns the complex numbers into a metric space and introduces the ideas of limits and 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 ($|z+w|\leq |z|+|w|$ for all z, w).

Secondly, the argument or phase of a complex number $z=x+yi$ is the angle to the real axis (shown as φ in Figure 2), and is written as $\arg(z)$. As with the modulus, the argument can be found from the rectangular form $x+iy$:

$\varphi =\pm \arctan {\frac {y}{x}}$ (taking the sign appropriately so that $x+iy=\cos \phi +i\sin \phi$).

The value of φ can change by any multiple of 2π and still give the same angle (note that radians are being used). Hence, the arg function is sometimes considered as multivalued, but often the value is chosen to lie in the interval $(-\pi ,\pi ]$, or $[0,2\pi )$ (this is the principal value).

Together, these give another way of representing complex numbers, the polar form, as the combination of modulus and argument fully specify the position of a point on the plane (confirmed by recovering the original rectangular co-ordinates $(x,y)=(r\cos \varphi ,r\sin \varphi )$ from the polar pair (r,φ)). This can be notated in various ways, including

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

called trigonometric form, and sometimes abbreviated r cis φ, or using Euler's formula

$z=re^{i\varphi },$

which is called exponential form. In electronics it is common to use angle notation to represent a phasor with amplitude A and phase θ as

$A\angle \theta =Ae^{j\theta }.$

In angle notation θ may be in either radians or degrees. In electronics it is also common to use j instead of i.

Operations in polar form

Multiplication and division have simple formulas in polar form:

$(r_{1}e^{i\varphi _{1}})\cdot (r_{2}e^{i\varphi _{2}})=r_{1}r_{2}e^{i(\varphi _{1}+\varphi _{2})}$

and

${\frac {r_{1}\,e^{i\varphi _{1}}}{r_{2}\,e^{i\varphi _{2}}}}=\left({\frac {r_{1}}{r_{2}}}\right)\,e^{i(\varphi _{1}-\varphi _{2})}.$

This form demonstrates that multiplication can be visualised as a simultaneous stretching and rotation of one of the multiplicands, adding to its angle the phase of the other and scaling its length. For example, multiplying by i corresponds to a quarter-rotation counter-clockwise, from which it is clear why i ² = −1. In particular, multiplication by any number on the unit circle around the origin is a pure rotation. Division is the same, in reverse.

Exponentiation is also simple; with integer exponents:

$(r(\cos \varphi +i\sin \varphi ))^{n}=r^{n}\,(\cos n\varphi +i\sin n\varphi ).$

[De Moivre's formula]

Arbitrary complex exponents are discussed in Exponentiation.

Finally, polar forms are also useful for finding roots. Any complex number z satisfying zⁿ = c (for n a positive integer) is called an nth root of c. If c is non-zero, there are exactly n distinct nth roots of c (by the fundamental theorem of algebra). Let c = re ^iφ with r > 0; then the set of nth roots of c is

$\left\{{\sqrt[{n}]{r}}\,e^{i\left({\frac {\varphi +2k\pi }{n}}\right)}\mid k\in \{0,1,\ldots ,n-1\}\,\right\},$

where ${\sqrt[{n}]{r}}$ represents the usual (positive) nth root of the positive real number r. If c = 0, then the only nth root of c is 0 itself, which as nth root of 0 is considered to have multiplicity n, hence these do represent all the n roots. Note that the roots differ only by the rotations e^2kπi/n, the nth roots of unity, so all the roots of c lie on a circle about the origin.

Some properties

Matrix representation of complex numbers

While usually not useful, alternative representations of the complex field can give some insight into its nature. One particularly elegant representation interprets each complex number as a 2×2 matrix with real entries which stretches and rotates the points of the plane. Every such matrix has the form

${\begin{bmatrix}a&-b\\b&\;\;a\end{bmatrix}}$

where a and b are real numbers. The sum and product of two such matrices is again of this form, and the product operation on matrices of this form is commutative. Every non-zero matrix of this form is invertible, and its inverse is again of this form. Therefore, the matrices of this form are a field, isomorphic to the field of complex numbers. Every such matrix can be written as

${\begin{bmatrix}a&-b\\b&\;\;a\end{bmatrix}}=a{\begin{bmatrix}1&\;\;0\\0&\;\;1\end{bmatrix}}+b{\begin{bmatrix}0&-1\\1&\;\;0\end{bmatrix}}$

which suggests that we should identify the real number 1 with the identity matrix

${\begin{bmatrix}1&\;\;0\\0&\;\;1\end{bmatrix}},$

and the imaginary unit i with

${\begin{bmatrix}0&-1\\1&\;\;0\end{bmatrix}},$

a counter-clockwise rotation by 90 degrees. Note that the square of this latter matrix is indeed equal to the 2×2 matrix that represents −1.

More formally, this matrix representation is the regular representation of the complex numbers, thought of as an R-algebra (an R-vector space with a multiplication), with respect to the basis $1,i:$ the complex numbers are a 2-dimensional vector space over the real numbers, and multiplication by a complex number is a linear map (by distributivity) of the complex numbers to themselves, which is thus represented by a 2×2 matrix once a basis has been chosen. Thus this is not an ad hoc construction, but can be applied to any K-algebra over a field. For example, if the matrix elements are themselves complex numbers, the resulting algebra is that of the quaternions; stated alternatively, the quaternions are a 2-dimensional C-algebra, and hence their regular representation is as 2×2 complex matrices. Generalizing alternatively, this matrix representation is one way of expressing the Cayley–Dickson construction of algebras.

The square of the absolute value of a complex number expressed as a matrix is equal to the determinant of that matrix.

$|z|^{2}={\begin{vmatrix}a&-b\\b&a\end{vmatrix}}=(a^{2})-((-b)(b))=a^{2}+b^{2}.$

If the matrix is viewed as a transformation of the plane, then the transformation rotates points through an angle equal to the argument of the complex number and scales by a factor equal to the complex number's absolute value. The conjugate of the complex number z corresponds to the transformation which rotates through the same angle as z but in the opposite direction, and scales in the same manner as z; this can be represented by the transpose of the matrix corresponding to z. This is generalized in the polar decomposition of matrices.

It should also be noted that the two eigenvalues of the 2x2 matrix representing a complex number are the complex number itself and its conjugate.

While the above is a linear representation of 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: $J^{2}=-I.$ Then $\{z=aI+bJ:a,b\in R\}$ is also isomorphic to the field C, and gives an alternative complex structure on R². This is generalized by the notion of a linear complex structure.

Real vector space

C is a two-dimensional real vector space. Unlike the reals, the set of complex numbers cannot be totally ordered in any way that is compatible with its arithmetic operations: C cannot be turned into an ordered field. More generally, no field containing a square root of −1 can be ordered.

R-linear maps C → C have the general form

$f(z)=az+b{\overline {z}}$

with complex coefficients a and b. Only the first term is C-linear, and only the first term is holomorphic; the second term is real-differentiable, but does not satisfy the Cauchy-Riemann equations.

The function

$f(z)=az\,$

corresponds to rotations combined with scaling, while the function

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

corresponds to reflections combined with scaling.

Solutions of polynomial equations

A root of the polynomial p is a complex number z such that p(z) = 0. A surprising result in complex analysis is that all polynomials of degree n with real or complex coefficients have exactly n complex roots (counting multiple roots according to their multiplicity). This is known as the fundamental theorem of algebra, and it shows that the complex numbers are an algebraically closed field. Indeed, the complex numbers are the algebraic closure of the real numbers, as described below.

Construction and algebraic characterization

One construction of C is as a field extension of the field R of real numbers, in which a root of x²+1 is added. To construct this extension, begin with the polynomial ring R[x] of the real numbers in the variable x. Because the polynomial x²+1 is irreducible over R, the quotient ring R[x]/(x²+1) will be a field. This extension field will contain two square roots of -1; one of them is selected and denoted i. The set {1, i} will form a basis for the extension field over the reals, which means that each element of the extension field can be written in the form a+ b·i. Equivalently, elements of the extension field can be written as ordered pairs (a,b) of real numbers.

Although only roots of x²+1 were explicitly added, the resulting complex field is actually algebraically closed – every polynomial with coefficients in C factors into linear polynomials with coefficients in C. Because each field has only one algebraic closure, up to field isomorphism, the complex numbers can be characterized as the algebraic closure of the real numbers.

The field extension does yield the well-known complex plane, but it only characterizes it algebraically. The field C is characterized up to field isomorphism by the following three properties:

• it has characteristic 0
• its transcendence degree over the prime field is the cardinality of the continuum
• it is algebraically closed

One consequence of this characterization is that C contains many proper subfields which are isomorphic to C (the same is true of R, which contains many subfields isomorphic to itself). As described below, topological considerations are needed to distinguish these subfields from the fields C and R themselves.

Characterization as a topological field

As just noted, the algebraic characterization of C fails to capture some of its most important topological properties. These properties are key for the study of complex analysis, where the complex numbers are studied as a topological field.

The following properties characterize C as a topological field:

• C is a field.
• C contains a subset P of nonzero elements satisfying:
  • P is closed under addition, multiplication and taking inverses.
  • If x and y are distinct elements of P, then either x-y or y-x is in P
  • If S is any nonempty subset of P, then S+P=x+P for some x in C.
• C has a nontrivial involutive automorphism x→x*, fixing P and such that xx* is in P for any nonzero x in C.

Given a field with these properties, one can define a topology by taking the sets

• $B(x,p)=\{y|p-(y-x)(y-x)^{*}\in P\}$

as a base, where x ranges over the field and p ranges over P.

To see that these properties characterize C as a topological field, one notes that P ∪ {0} ∪ -P is an ordered Dedekind-complete field and thus can be identified with the real numbers R by a unique field isomorphism. The last property is easily seen to imply that the Galois group over the real numbers is of order two, completing the characterization.

Pontryagin has shown that the only connected locally compact topological fields are R and C. This gives another characterization of C as a topological field, since C can be distinguished from R by noting that the nonzero complex numbers are connected, while the nonzero real numbers are not.

Complex analysis

Further information: Complex analysis

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

Applications

Some applications of complex numbers are:

Control theory

In control theory, systems are often transformed from the time domain to the frequency domain using the Laplace transform. The system's poles and zeros 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 especially important whether the poles and zeros are in the left or right half planes, i.e. have real part greater than or less than zero. If a system has poles that are

• in the right half plane, it will be unstable,
• all in the left half plane, it will be 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 |z| of the corresponding z is the amplitude and the argument arg(z) the 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

$f(t)=ze^{i\omega t}\,$

where ω represents the angular frequency and the complex number z encodes the phase and amplitude as explained above.

In electrical engineering, the Fourier transform is used to analyze varying voltages and currents. The treatment of resistors, capacitors, and inductors 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 impedance. (Electrical engineers and some physicists use the letter j for the imaginary unit since i is typically reserved for varying currents and may come into conflict with i.) This approach is called phasor calculus. This use is also extended into digital signal processing and digital image processing, which utilize digital versions of Fourier analysis (and wavelet analysis) to transmit, compress, restore, and otherwise process digital audio signals, still images, and video signals.

Improper integrals

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

Quantum mechanics

The complex number field is relevant in the mathematical formulation of quantum mechanics, where complex Hilbert spaces 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 and general relativity, some formulas for the metric on spacetime become simpler if one takes the time variable to be imaginary. (This is no longer standard in classical relativity, but is used in an essential way in quantum field theory.) Complex numbers are essential to spinors, which are a generalization of the tensors used in relativity.

Applied mathematics

In differential equations, it is common to first find all complex roots r of the characteristic equation of a linear differential equation and then attempt to solve the system in terms of base functions of the form f(t) = e^rt.

Fluid dynamics

In fluid dynamics, complex functions are used to describe potential flow in two dimensions.

Fractals

Certain fractals are plotted in the complex plane, e.g. the Mandelbrot set and Julia sets.

History

The earliest fleeting reference to square roots of negative numbers perhaps occurred in the work of the Greek mathematician and inventor Heron of Alexandria in the 1st century AD, when, apparently inadvertently, he considered the volume of an impossible frustum of a pyramid, though negative numbers were not conceived in the Hellenistic world.

Complex numbers became more prominent in the 16th century, when closed formulas for the roots of cubic and quartic polynomials were discovered by Italian mathematicians (see Niccolo Fontana Tartaglia, Gerolamo Cardano). It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. For example, Tartaglia's cubic formula gives the following solution to the equation x³ − x = 0:

${\frac {1}{\sqrt {3}}}\left({\sqrt {-1}}^{1/3}+{\frac {1}{{\sqrt {-1}}^{1/3}}}\right).$

At first glance this looks like nonsense. However formal calculations with complex numbers show that the equation z³ = i has solutions –i, ${\scriptstyle {\frac {\sqrt {3}}{2}}}+{\scriptstyle {\frac {1}{2}}}i$ and ${\scriptstyle {\frac {-{\sqrt {3}}}{2}}}+{\scriptstyle {\frac {1}{2}}}i$. Substituting these in turn for ${\scriptstyle {\sqrt {-1}}^{1/3}}$ in Tartaglia's cubic formula and simplifying, one gets 0, 1 and −1 as the solutions of x³ – x = 0. Rafael Bombelli was the first to explicitly address these seemingly paradoxical solutions of cubic equations and developed the rules for complex arithmetic trying to resolve these issues.

This was doubly unsettling since not even negative numbers were considered to be on firm ground at the time. The term "imaginary" for these quantities was coined by René Descartes in 1637 and was meant to be derogatory (see imaginary number for a discussion of the "reality" of complex numbers). 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 positive real numbers a and b, and which was also used in complex number calculations with one of a, b positive and the other negative. The incorrect use of this identity (and the related identity $\scriptstyle 1/{\sqrt {a}}={\sqrt {1/a}}$) in the case when both a and b are negative even bedeviled Euler. This difficulty eventually led to the convention of using the special symbol i in 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 1730 Abraham de Moivre noted that the complicated identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be simply reexpressed by the following well-known formula which bears his name, de Moivre's formula:

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

In 1748 Leonhard Euler went further and obtained Euler's formula of complex analysis:

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

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 existence of complex numbers was not completely accepted until the geometrical interpretation (see above) had been described by Caspar Wessel in 1799; it was rediscovered several years later and popularized by Carl Friedrich Gauss, and as a result the theory of complex numbers received a notable expansion. The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in Wallis's De Algebra tractatus.

Wessel's memoir appeared in the Proceedings of the Copenhagen Academy for 1799, and is exceedingly clear and complete, even in comparison with modern works. He also considers the sphere, and gives a quaternion theory from which he develops a complete spherical trigonometry. In 1804 the Abbé Buée independently came upon the same idea which Wallis had suggested, that $\pm {\sqrt {-1}}$ should represent a unit line, and its negative, perpendicular to the real axis. Buée's paper was not published until 1806, in which year Jean-Robert Argand also issued a pamphlet on the same subject. It is to Argand's essay that the scientific foundation for the graphic representation of complex numbers is now generally referred. Nevertheless, in 1831 Gauss found the theory quite unknown, and in 1832 published his chief memoir on the subject, thus bringing it prominently before the mathematical world. Mention should also be made of an excellent little treatise by Mourey (1828), in which the foundations for the theory of directional numbers are scientifically laid. The general acceptance of the theory is not a little due to the labors of Augustin Louis Cauchy and Niels Henrik Abel, and especially the latter, who was the first to boldly use complex numbers with a success that is well known.

The common terms used in the theory are chiefly due to the founders. Argand called $\cos \phi +i\sin \phi$ the direction factor, and $r={\sqrt {a^{2}+b^{2}}}$ the modulus; Cauchy (1828) called $\cos \phi +i\sin \phi$ the reduced form (l'expression réduite); Gauss used i for ${\sqrt {-1}}$, introduced the term complex number for a + bi, and called a² + b² the norm.

The expression direction coefficient, often used for $\cos \phi +i\sin \phi$, is due to Hankel (1867), and absolute value, for modulus, is due to Weierstrass.

Following Cauchy and Gauss have come a number of contributors of high rank, of whom the following may be especially mentioned: Kummer (1844), Leopold Kronecker (1845), Scheffler (1845, 1851, 1880), Bellavitis (1835, 1852), Peacock (1845), and De Morgan (1849). Möbius must also be mentioned for his numerous memoirs on the geometric applications of complex numbers, and Dirichlet for the expansion of the theory to include primes, congruences, reciprocity, etc., as in the case of real numbers.

A complex ring or field is a set of complex numbers which is closed under addition, subtraction, and multiplication. Gauss studied complex numbers of the form a + bi, where a and b are integral, or rational (and i is one of the two roots of x² + 1 = 0). His student, Ferdinand Eisenstein, studied the type $a+b\omega$, where $\omega$ is a complex root of x³ − 1 = 0. Other such classes (called cyclotomic fields) of complex numbers are derived from the roots of unity x^k − 1 = 0 for higher values of k. This generalization is largely due to Kummer, who also invented ideal numbers, which were expressed as geometrical entities by Felix Klein in 1893. The general theory of fields was created by Évariste Galois, who studied the fields generated by the roots of any polynomial equation in one variable.

The late writers (from 1884) on the general theory include Weierstrass, Schwarz, Richard Dedekind, Otto Hölder, Henri Poincaré, Eduard Study, and Alexander MacFarlane.

See also

• Circular motion using complex numbers
• Complex base systems
• Complex geometry
• Complex plane
• Domain coloring
• Eisenstein integer
• Euler's identity
• Gaussian integer
• Hypercomplex number
• Local field
• Mandelbrot set
• Mathematical diagram
• Riemann sphere (extended complex plane)
• Split-complex number
• Square root of complex numbers
• Imaginary number/Imaginary unit

References

Mathematical references

• Ahlfors, Lars (1979), Complex analysis (3rd ed.), McGraw-Hill, ISBN 978-0070006577
• Conway, John B. (1986), Functions of One Complex Variable I, Springer, ISBN 0-387-90328-3
• Joshi, Kapil D. (1989), Foundations of Discrete Mathematics, New York: John Wiley & Sons, ISBN 978-0-470-21152-6
• Pedoe, Dan (1988), Geometry: A comprehensive course, Dover, ISBN 0-486-65812-0
• Solomentsev, E.D. (2001) [1994], "Complex number", Encyclopedia of Mathematics, EMS Press

Historical references

• Burton, David M. (1995), The History of Mathematics (3rd ed.), New York: McGraw-Hill, ISBN 978-0-07-009465-9
• Katz, Victor J. (2004), A History of Mathematics, Brief Version, Addison-Wesley, ISBN 978-0-321-16193-2
• Nahin, Paul J. (1998), An Imaginary Tale: The Story of ${\sqrt {-1}}$ (hardcover ed.), Princeton University Press, ISBN 0-691-02795-1

  A gentle introduction to the history of complex numbers and the beginnings of complex analysis.

• H.-D. Ebbinghaus ... (1991), Numbers (hardcover ed.), Springer, ISBN 0-387-97497-0 {{citation}}: Unknown parameter |authors= ignored (help)

  An advanced perspective on the historical development of the concept of number.

Further reading

• The Road to Reality: A Complete Guide to the Laws of the Universe, by Roger Penrose; Alfred A. Knopf, 2005; ISBN 0-679-45443-8. Chapters 4-7 in particular deal extensively (and enthusiastically) with complex numbers.
• Unknown Quantity: A Real and Imaginary History of Algebra, by John Derbyshire; Joseph Henry Press; ISBN 0-309-09657-X (hardcover 2006). A very readable history with emphasis on solving polynomial equations and the structures of modern algebra.
• Visual Complex Analysis, by Tristan Needham; Clarendon Press; ISBN 0-198-53447-7 (hardcover, 1997). History of complex numbers and complex analysis with compelling and useful visual interpretations.

External links

• Euler's work on Complex Roots of Polynomials at Convergence. MAA Mathematical Sciences Digital Library.
• John and Betty's Journey Through Complex Numbers
• MathWorld articles Complex number and Argand Diagram, and demonstration "Argand Diagram".
• Dimensions: a math film. Chapter 5 presents an introduction to complex arithmetic and stereographic projection. Chapter 6 discusses transformations of the complex plane, Julia sets, and the Mandelbrot set.