# Complex conjugates

From Graphic maths

**Level: A Level Maths**

Complex conjugates are pairs of numbers which have the same real part, and where the imaginary part differs only in sign. For example and are conjugates.

## Notation and terms

For any complex number :

- is the complex conjugate
- is the complex conjugate
- and are called a
*complex conjugate pair*

For a complex number , the complex conjugate is written as .

## Argand diagram

In the Argand diagram below:

All conjugate pairs are symmetrical about the x (real) axis.

## Special properties

The sum of any complex number , and its complex conjugate is always real. This is because the two imaginary parts cancel out:

The product of any complex number , and its complex conjugate is always real:

The two terms in cancel out, and since is -1, the final result is: