Imaginary Numbers

The imaginary numbers are needed to fill in the last gap in exponentiation. The real numbers do not allow some negative numbers to be raised to fractional powers. For example they do not allow (-1)^(1/2) = √(-1).

To get around this problem we can just make up an answer. We'll call this answer i. It can't be a real number, so it is outside the real numbers.

Using i allows us to find roots for all negative numbers. For example

√(-2) = √(2 × -1) = √2 × √(-1)= (√2)i = 1.414i

Both negative and positive numbers have two square roots. Just as both 4² and (-4)² are 16, both (4i)² and (-4i)² are -16.

When we look at cube roots, thing get more complicated. Now every number has three cube roots, not just one, like it did with the real numbers. So not only does 3³ equal 27, but so does (-3/2 + (3√3/2)i)³ and (-3/2 + (3√3/2)i)³

For more on this see the complex numbers.