What are complex numbers ?and explain me how to do the four arithmetical operations with them.
A complex number is any number which can be expressed as:
In order to perform addition, add the real parts together, and add the imaginary parts together, like so:
Using the formula above we get:
To perform subtraction, simply subtract the real parts together and the imaginary parts together:
From this we get:
Complex numbers can be multiplied in a similar manner to expanding a pair of brackets:
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:
To divide this we must multiply the fraction by the complex conjugate of the bottom of the fraction. That is
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.