# 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}.