---
tag: 0BJV24C4SY
kind: definition
name: Complex conjugate
msc: [30A99]
defines: 0FTH2E9TAS
uses: [09FTAET5Z4, 02ZE6JHJ7G]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_conjugate
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Complex_conjugate
    locator: Notation
---

## Definition

Let $z = a + b i$ be a [[complex number|09FTAET5Z4]], 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|0ZB7J071WQ]].

## 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 $7$.