# 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*, *i*² = -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:

What about multiplication and division?

When we divide two imaginary numbers, the *i* goes away as if it was an x.

When we add or subtract an imaginary number and a real number, we get a complex number. Examples of complex numbers are 4i + 3, -2 + 9i, -5i – 8, etc…, but I will talk about them more specifically in another post. In the meantime, subscribe and stay tuned for more! And if you have any doubt of questions don’t hesitate to leave a comment; I will be happy to help you!

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