---
tag: 00KRCVE7CK
kind: statement
role: proposition
name: Modulus of the conjugate
msc: [30A99]
uses: [0KW2X0TFBX, 0BJV24C4SY]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_conjugate
    locator: Properties
---

## Statement

For each complex number $z$, $|\bar{z}| = |z|$.
Here $\bar{z}$ is the [[complex conjugate|0BJV24C4SY]] of $z$ and $|\cdot|$ is the [[modulus|0KW2X0TFBX]].

## Remarks

Proof: for $z = a + b i$, $|\bar{z}| = \sqrt{a^2 + (-b)^2} = \sqrt{a^2 + b^2} = |z|$.
In the complex plane, this says that a reflection in the real axis does not change the distance to $0$.

## Explanations

### L1 human

A number and its conjugate are mirror images in the real axis.
A mirror keeps distances, so both numbers are at the same distance from $0$.