Complex number

A complex number is a number composed of two parts - a real component and an imaginary component, of the form <math>a + bi</math>, where a and b are real numbers and <math>i^2 = -1</math>.

Whereas the real numbers can be represented as all the possible points on an infinitely extended number line, to represent all the complex numbers requires the use of a two dimensional coordinate system, usually with the real components on the horizontal axis (the abscissa) and the imaginary components on the vertical axis (the ordinate). This representation is known as the Argand diagram.

The complex numbers form an algebraically closed field but do not permit a non-trivial ordering that is preserved under operations. They are the algebraic closure of the real numbers.

Many functions used in real analysis can be extended in to complex numbers using Taylor series. This is the subject of complex analysis.

Polar notation

The complex number <math>a+bi</math> can also be written in the form <math>\rho e^{i\theta}</math>, where

<math>\rho^2=a^2+b^2</math> is the magnitude squared

<math>\theta=\arctan\frac{b}{a}</math> is the phase

If a line is drawn on the Argand diagram from the origin to a given complex number, the length of that line will be <math>\rho</math> and the angle it makes to the real (horizontal) axis will be <math>\theta</math>. This leads to a straight-forward geometric interpretation for multiplication by a complex number: multiplying a complex number by <math>e^{i\theta}</math> is equivalent to an anticlockwise rotation through an angle <math>\theta</math> in the Argand diagram.

In popular culture

In Yvgeny Zamyatin's satirical novel We, the narrator's psychological distress at contemplating the concept of complex numbers becomes a metaphor for the limitations of totalitarian systems of thought.