---
tag: 0KW2X0TFBX
kind: definition
name: Modulus of a complex number
aliases: [absolute-value-of-a-complex-number]
msc: [30A99]
defines: 0FFXKTKKQC
uses: [09FTAET5Z4, 02ZE6JHJ7G, 0VB8BFRABH]
sources:
  - url: https://en.wikipedia.org/wiki/Absolute_value
    locator: Complex numbers
  - url: https://en.wikipedia.org/wiki/Complex_number
    locator: Complex conjugate, absolute value, argument and division
---

## Definition

Let $z$ be a [[complex number|09FTAET5Z4]] with real part $a = \operatorname{Re} z$ and imaginary part $b = \operatorname{Im} z$.
The modulus, or absolute value, of $z$ is the nonnegative real number
$$|z| = \sqrt{a^2 + b^2}.$$

## Remarks

For a real number $z = a$, the modulus is $\sqrt{a^2} = |a|$, the [[absolute value|00S7C68D51]] of $a$.
In the [[complex plane|0CTE7A31XQ]], $|z|$ is the distance from $z$ to $0$, by the Pythagorean theorem. The distance between $z$ and $w$ is $|z - w|$.
For example, $|3 + 4 i| = 5$.

## Explanations

### L1 human

The modulus of a complex number is its length, that is, its distance from $0$ in the plane.
The point $3 + 4 i$ is the corner of a right triangle with sides $3$ and $4$, so its distance from $0$ is $5$.