# Imaginary numbers

An imaginary number is a number that when squared results in a negative value. Imaginary numbers are indicated using an "i." For example, 3i is the imaginary analogue of the real number 3. Imaginary numbers are used as part of complex numbers to perform various types of calculations, such as Fourier transforms.

Typically, the square of any number is positive because a positive number multiplied by a positive number is positive, while a negative number multiplied by a negative number is also positive. This is the case for real numbers. Thus, in order for the square of a number to be negative, it must be "imaginary."

While the number 1 is the unit value for real numbers, the imaginary unit is i. In other words:

i^{2} = -1

or

i =

The use of imaginary numbers allows us to solve problems that would otherwise have no solution.

Example

Solve: x^{2} + 9 = 0

x^{2} + 9 = 0

x^{2} = -9

x =

x = ± 3i

## Powers of i

The powers of i (i^{0}, i^{1}, etc.) follow a pattern. Using i^{0} through i^{3} as a reference:

i^{0} = 1

i^{1} = i

i^{2} = -1

i^{3} = -i

This pattern continues in both directions, so i^{4} and i^{0} have the same value: 1. Similarly, i^{-1} has the same value as i^{3}.