---
tag: 034FGCEYX4
kind: statement
role: proposition
name: Formula for the modulus
msc: [30A99]
uses: [0KW2X0TFBX]
sources:
  - url: https://en.wikipedia.org/wiki/Absolute_value
    locator: Complex numbers
---

## Statement

Let $x$ and $y$ be real numbers.
Then the [[modulus|0KW2X0TFBX]] of the complex number $x + y i$ is
$$|x + y i| = \sqrt{x^2 + y^2}.$$

## Remarks

This follows from the definition of the modulus, because $\operatorname{Re}(x + y i) = x$ and $\operatorname{Im}(x + y i) = y$ for real $x$ and $y$.
The hypothesis that $x$ and $y$ are real is necessary. For example, $|i + i \cdot i| = |-1 + i| = \sqrt{2}$, but $\sqrt{i^2 + i^2} = \sqrt{-2}$ is not even real.

## Explanations

### L1 human

To find the length of a complex number, square its two real coordinates, add them, and take the square root.
For $1 - i$, this gives $\sqrt{1 + 1} = \sqrt{2}$.