In algebra, **imaginary numbers** are part of the complex number system in which real numbers cannot measure.

The symbol for imaginary numbers is *i*. You can define what *i* is equivalent to, but you cannot measure it in real terms.

*i* is equal to the square root of -1.

This means that *i*^{2} is equal to -1.

Therefore, *i*^{3} is equal to -*i*, or the negative square root of -1.

Hence, *i*^{4} is equal to 1.

This pattern of *i*, -1, -*i*, and 1 continues ad infinitum in the same exact manner shown.

## Simplyfing In Terms of *i*[]

With *i*, you can simplify the square roots of negative numbers.

## Defining Complex Numbers[]

All complex numbers have a real part and an imaginary part. This can be shown as follows:

, where *a* is real and *b* is imaginary.

Below are several examples of complex numbers.

, ,

, ,

, and even are all complex numbers.

## Multiplying Complex Numbers[]

Multiplying complex numbers is not that much different than multiplying algebraic expressions. I'll show you how so below.

1. Copy the problem...

2. LOIF it (Last - Outside - Inside - First), which is like FOIL in reverse...

3. Combine like terms...

**WATCH OUT!** You're not done! Remember, *i*^{2} = -1.

4. Simplify the exponents...

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -16 - 6i + 14}**

5. And combine the like terms again for the final answer.