Definition: complex number from Philip's Encyclopedia

Number of the form a + bi, where i = √-1, and a and b are real numbers. In order to obtain a solution to the equation x2 + 1 = 0, we need to introduce a new number i such that i2 = -1. The solutions to similar equations then give rise to a set of numbers of the general form a + bi. These are known as the complex numbers. Since b can be equal to zero, the set of complex numbers includes the real numbers.

Summary Article: complex number
From The Hutchinson Unabridged Encyclopedia with Atlas and Weather Guide

In mathematics, a number written in the form a + ib, where a and b are real numbers and i is the square root of −1 (that is, i2 = −1); i used to be known as the ‘imaginary’ part of the complex number. Some equations in algebra, such as those of the form

x2 + 5 = 0

cannot be solved without recourse to complex numbers, because the real numbers do not include square roots of negative numbers.

The sum of two or more complex numbers is obtained by adding separately their real and imaginary parts, for example:

(a + bi) + (c + di) = (a + c) + (b + d)i

Complex numbers can be represented graphically on an Argand diagram, which uses rectangular Cartesian coordinates in which the x-axis represents the real part of the number and the y-axis the imaginary part. Thus the number z = a + ib is plotted as the point (a, b). Complex numbers have applications in various areas of science, such as the theory of alternating currents in electricity.

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