# The Imaginary Number i

## Complex Numbers: An Introduction

A complex number can be expressed in the form:

a + bi

where a and b are real numbers.

The number i is defined as the square root of -1. It is an imaginary number, since the square root of any negative number is not a real number.

Complex numbers are not algebraic expressions. They are numbers, containing a real part (a) and an imaginary part (bi).

## Using i

In the world of real numbers, an expression like the square root of -9 is unsolvable. (Remember, because the square root of a negative number is not real.) However, by using the imaginary number i we can break this problem down as the square root of (9)(-1).

This can be expressed as the square root of 9 (which is 3) times the square root of -1 (which is i). Thus, the answer to this otherwise unsolvable problem is 3i.

## i to a Power

As with any real number, i0 = 1 and i1 = i. We know that i2 = -1. Also, i3 = (-1)(i) or –i. These four values (1,i, -1, –i) are the only possible results of raising the imaginary number i to any power.

To figure out any exponent higher than 4, just divide the exponent by four and use the remainder (0, 1, 2, or 3) as the new exponent. Match it up with the choices here and you have your answer. (Check the tutorial below to see this in action.)

## The Imaginary Number i

What is the imaginary number i, and how is it used? This tutorial will teach you all about it.