# Complex numbers

We have learnt previously that we cannot find the root of a negative number, but that is not entirely true. If we take the root of a negative number, we have what is known as an imaginary number (unreal number).

If we solve a quadratic equation and arrive at a solution as:

$$z_{1}=2+\sqrt{-4}$$

This is known as a complex number and consists of two parts - a real part (2) and an imaginary part (root of -4). A complex number is often designated as *z*.

The definition of the imaginary part is

$$\sqrt{-1}=i$$

How do you calculate the root of a negative number?

**Example**

We have

$$\sqrt{-1}=i$$

And obtain

$$\sqrt{-4}=\sqrt{4}\cdot \sqrt{-1}=2i$$

The solution to our quadratic equation in the beginning of the chapter is thus:

$$z_{1}=2+2i$$

and

$$z_{2}=2-2i$$

The two roots are very similar except for the sign preceding the imaginary number. Such numbers are known as conjugates of each other. You designate a conjugate with a dash above the symbol:

$$z_{1}=\bar{z}_{2}$$

Calculating with complex numbers proceeds as in ordinary mathematics but you should remember that

$$i^{2}=\sqrt{-1}\cdot \sqrt{-1}=-1$$

## Video lesson

Rewrite the given expression in the a+bi form:

$$\frac{2+5i}{i}$$