# What are imaginary numbers?

We’ve always been told we can’t take the square root of a negative number, since the square of a negative number is a positive number. But remember, in maths nothing is impossible, not even dividing by 0!

Imagine you want to take the square root of -1;

or the square root of -4;

or any other negative number. These are made up of -1 and their absolute value:

This means that we can consider -1 the negative unit. So, the square root of -4 is

And, more in general:

with n < 0.

But what is the square root of -1? It is called i, known as the imaginary unit. Therefore:

And in the case of -4 we have

Another example:

This is cool, isn’t it?

2i, 3i, -8i and so on are imaginary numbers, made up of the imaginary unit and the real numbers.

If the square root of -1 is i, i² = -1, and so i times i is equal to -1 as well. But what is i³?

And you can try with other different powers. When it comes to addition and subtraction, you can think of i as a constant a:

the same way 2a + a = 3a. But

the same way 2a + 1 is not 3a.

You can factor out the i as well: