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 i2 is equal to -1.

Therefore, i3 is equal to -i, or the negative square root of -1.

Hence, i4 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, i2 = -1.

4. Simplify the exponents...

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

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