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In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for one of the square roots of negative one (−1). The real number a is called the real part of the complex number, and the real number b is the imaginary part. When the imaginary part b is 0, then the complex number is identified with the real number a. A complex number then consists of a pair of real numbers.


Complex numbers can be added, subtracted, multiplied, and divided, just like real numbers; but they have additional elegant properties. For example, every polynomial algebraic equation has a complex number as a solution, not just some, as in the real numbers.

The real numbers are exactly equivalent (isomorphic) to the subset of complex numbers with imaginary part equal to zero: a + 0i, and it is usual to regard the two sets as identical.



Notation and operations[]

The set of all complex numbers is usually denoted by C, or in blackboard bold by . It includes the real numbers because every real number can be regarded as complex: a = a + 0i.


(a + bi) − (c + di) = (ac) + (bd)i
(a + bi)(c + di) = ac + bci + adi + bd i 2 = (acbd) + (bc+ad)i

Division of complex numbers can also be defined (see below). Thus the set of complex numbers forms a field which, in contrast to the real numbers, is algebraically closed.


The complex number field[]





In C, we have:



The complex plane[]



The Cartesian coordinates of the complex number are the real part x and the imaginary part y, while the circular coordinates are r = |z|, called the absolute value or modulus, and φ = arg(z), called the complex argument of z (mod-arg form). Together with Euler's formula we have

Additionally the notation r cis φ is sometimes used.


By simple trigonometric identities, we see that

and that


Multiplication with i corresponds to a counter clockwise rotation by 90 degrees ( radians). The geometric content of the equation i2 = −1 is that a sequence of two 90 degree rotations results in a 180 degree ( radians) rotation. Even the fact (−1) · (−1) = +1 from arithmetic can be understood geometrically as the combination of two 180 degree turns.

Absolute value, conjugation and distance[]





The complex conjugate of the complex number z = a + ib is defined to be a - ib, written as or . As seen in the figure, is the "reflection" of z about the real axis. The following can be checked:

  if and only if z is real
  if z is non-zero.


That conjugation commutes with all the algebraic operations (and many functions; e.g. ) is rooted in the ambiguity in choice of i (−1 has two square roots); note, however, that conjugation is not differentiable (see holomorphic).

Complex number division[]


Matrix representation of complex numbers[]




and the imaginary unit i with




Geometric interpretation of the operations on complex numbers[]

The point X is the sum of A and B.

Choose a point in the plane which will be the origin, . Given two points A and B in the plane, their sum is the point X in the plane such that the triangles with vertices 0, A, B and X, B, A are similar.

The point X is the product of A and B.

Choose in addition a point in the plane different from zero, which will be the unity, 1. Given two points A and B in the plane, their product is the point X in the plane such that the triangles with vertices 0, 1, A, and 0, B, X are similar.

The point X is the complex conjugate of A.

Given a point A in the plane, its complex conjugate is a point X in the plane such that the triangles with vertices 0, 1, A and 0, 1, X are mirror image of each other.

Some properties[]

Real vector space[]

C is a two-dimensional real vector space. Unlike the reals, complex numbers cannot be ordered in any way that is compatible with its arithmetic operations: C cannot be turned into an ordered field.

R-linear maps CC have the general form

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

The function

corresponds to rotations combined with scaling, while the function

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 most striking result 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 shows that the complex numbers are an algebraically closed field.

Indeed, the complex number field is the algebraic closure of the real number field, and Cauchy constructed complex numbers in this way. It can be identified as the quotient ring of the polynomial ring R[X] by the ideal generated by the polynomial X2 + 1:

This is indeed a field because X2 + 1 is irreducible, hence generating a maximal ideal, in R[X]. The image of X in this quotient ring becomes the imaginary unit i.

Algebraic characterization[]

The field C is (up to field isomorphism) characterized by the following three facts:

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

Consequently, C contains many proper subfields which are isomorphic to C. Another consequence of this characterization is that the Galois group of C over the rational numbers is enormous, with cardinality equal to that of the power set of the continuum.

Characterization as a topological field[]

As noted above, the algebraic characterization of C fails to capture some of its most important properties. These properties, which underpin the foundations of complex analysis, arise from the topology of C. 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
  • C has a nontrivial involutive automorphism x->x*, fixing P and such that xx* is in P for any nonzero x in C.

Given these properties, one can then define a topology on C by taking the sets

as a base, where x ranges over C, 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 the nonzero complex numbers are connected whereas the nonzero real numbers are not.

Complex analysis[]

For more details on this topic, see 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.


Control theory[]



  • 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 be marginally stable.


Signal analysis[]

Complex numbers are used in signal analysis and other fields as a convenient description for periodically varying signals. The absolute value |z| is interpreted as the amplitude and the argument arg(z) as the phase of a sine wave of given frequency.

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 the real part of complex valued functions of the form


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 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, the use of complex analysis is 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 also of utmost importance in quantum mechanics since the underlying theory is built on (infinite dimensional) Hilbert spaces over C.



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) = ert.

Fluid dynamics[]

In fluid dynamics, complex functions are used to describe potential flow in 2d.


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


The earliest fleeting reference to square roots of negative numbers occurred in the work of the Greek mathematician and inventor Heron of Alexandria in the 1st century AD, when he considered the volume of an impossible frustum of a pyramid. They became more prominent when in the 16th century closed formulas for the roots of third and fourth degree 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 :

At first glance this looks like nonsense. However formal calculations with complex numbers show that the equation has solutions −i, and . Substituting these in turn for into the cubic formula and simplifying, one gets 0, 1 and -1 as the solutions of



and to Euler (1748) Euler's formula of complex analysis:

The existence of complex numbers was not completely accepted until the geometrical interpretation (see below) 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 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 the direction factor, and the modulus; Cauchy (1828) called the reduced form (l'expression réduite); Gauss used i for , introduced the term complex number for , and called the norm.


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 , where a and b are integral, or rational (and i is one of the two roots of ). His student, Ferdinand Eisenstein, studied the type , where is a complex root of . Other such classes (called cyclotomic fields) of complex numbers are derived from the roots of unity for higher values of . 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

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

The formally correct definition using pairs of real numbers was given in the 19th century.

See also[]

  • Circular_motion#Using_complex_numbers
  • Complex geometry
  • De Moivre's formula
  • Euler's identity
  • Hypercomplex number
  • Leonhard Euler
  • Local field
  • Mandelbrot Set
  • Phasor (physics)
  • Phasor (electronics)
  • Quaternion
  • Riemann sphere (extended complex plane)
  • Split-complex number

Further reading[]

  • An Imaginary Tale: The Story of , by Paul J. Nahin; Princeton University Press; ISBN 0691027951 (hardcover, 1998). A gentle introduction to the history of complex numbers and the beginnings of complex analysis.
  • Numbers, by H.-D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer, J. Neukirch, A. Prestel, R. Remmert; Springer; ISBN 0-387-97497 (hardcover, 1991). An advanced perspective on the historical development of the concept of number.
  • The Road to Reality: A Complete Guide to the Laws of the Universe, by Roger Penrose; Alfred A. Knopf, 2005; ISBN 0679454438. Chapters 4-7 in particular deal extensively (and enthusiastically) with complex numbers.

External links[]
