# What are complex numbers ?and explain me how to do the four arithmetical operations with them.

##### 1 Answer

See below.

#### Explanation:

A complex number is any number which can be expressed as:

**Addition**

In order to perform addition, add the real parts together, and add the imaginary parts together, like so:

Example, consider

Using the formula above we get:

**Subtraction**

To perform subtraction, simply subtract the real parts together and the imaginary parts together:

Example, consider:

From this we get:

**Multiplication**

Complex numbers can be multiplied in a similar manner to expanding a pair of brackets:

Remember

For example:

**Division**

This is a bit more complicated.

Remember above we stated that the complex conjugate of a complex number is:

Using multiplication from above: a complex number multiplied by it's own conjugate results in the following:

So, given:

To divide this we must multiply the fraction by the complex conjugate of the **bottom** of the fraction. That is

Now,

Remember:

All is left is to split up the fraction and we get:

Let's try the example:

First multiplying the top and the bottom by the conjugate of the bottom we get:

Division is a bit intimidating at first glance but should come with practice.