Complex conjugate

Definition

Let $z = a + b i$ be a complex number, with $a$ and $b$ real. The complex conjugate of $z$ is the complex number $\bar{z} = a - b i$.

Remarks

Another common notation is $z^*$. Conjugation respects the operations: $\overline{z + w} = \bar{z} + \bar{w}$ and $\overline{z w} = \bar{z} \, \bar{w}$. Also $\bar{\bar{z}} = z$. A complex number $z$ is real exactly when $\bar{z} = z$. The product $z \bar{z}$ is the real number $|z|^2$, see a complex number times its conjugate.

Explanations

L1 human

To conjugate a complex number, change the sign of its imaginary part. For example, the conjugate of $2 + 5 i$ is $2 - 5 i$, and the conjugate of $7$ is $7$.