Complex Conjugate

The complex conjugate of a complex number simply has its imaginary part negated.

The following complex numbers are complex conjugates of each other:

$$\begin{array}{r}a+bi\\ a-bi\end{array}$$

Magenta colour-coding

All imaginary parts in expressions on this page will be highlighted a lovely shade of magenta, to make them easier to follow.

_{Yes, this took way too long to typeset.}

The complex conjugate of a variable is usually denoted by a horizontal line above the variable. For example, a complex number $z$ would have a complex conjugate $\bar{z}$.

A real number is equal to its complex conjugate, as adding or subtracting the complex part makes no difference.

Multiplying conjugates

If you multiply a complex number by its conjugate, following the process of [[Complex Arithmetic#Multiplication#Complex|multiplying two complex numbers together]] it simplifies as follows:

$$\begin{array}{rl}& (a+bi)(a-bi)\\ =& a^2+abi-abi-b^2{i}^2\end{array}$$

Notice how the imaginary terms cancel out, and that $i^2$ is $-1$:

$$\begin{array}{rl}=& a^2\cancel{+abi-abi}+b^2\\ =& a^2+b^2\end{array}$$

Now isn’t that pretty!

Tip

Multiplying a complex number by its conjugate is a reliable way to make it real (i.e. get rid of the imaginary part).

In fancy maths notation:

$$z\bar{z}\in \mathbb{R}$$

Adding and subtracting conjugates

Less useful than multiplying conjugates, but still good to know.

If you add a complex number with its conjugate, the $+bi$ and $-bi$ cancel out, leaving just two times the real part left behind:

$$\begin{array}{rl}& (a+bi)+(a-bi)\\ =& a+a\cancel{+bi-bi}\\ =& 2a\end{array}$$

If you subtract the conjugate instead, you get two times the imaginary part left behind instead:

$$\begin{array}{rl}& (a+bi)-(a-bi)\\ =& \cancel{a-a}+bi+bi\\ =& 2bi\end{array}$$