EECS20N: Signals and Systems

Imaginary numbers

Can we always find roots for a polynomial? The equation

x 2 + 1 = 0

has no solution for x in the set of real numbers. Thus, it would appear that not all polynomials have roots. However, a suprisingly simple and clever mathematical device changes the picture dramatically. With the introduction of imaginary numbers, mathematicians ensure that all polynomials have roots.

Using simple algebra, we discover that we need to find x such that

x 2 = -1.

This suggests that

x = √-1.

But, -1 does not normally have a square root.

The clever device is to define an imaginary number, usually written as i or j, that is equal to √-1. By definition

i ´ i = √-1 ´ √-1 = -1

This imaginary number, thus, is a a solution of the equation x 2 + 1 = 0.

For any real number y, iy is an imaginary number. Thus, we can define the set of imaginary numbers as

ImaginaryNumbers = { iy | y Reals, and i = √-1}.