---
tag: 05V7J4WD1C
kind: statement
role: proposition
name: Equality of complex numbers
msc: [30A99]
uses: [02ZE6JHJ7G]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_number
    locator: Definition and basic operations
---

## Statement

Let $z$ and $w$ be complex numbers.
Then $z = w$ if and only if $\operatorname{Re} z = \operatorname{Re} w$ and $\operatorname{Im} z = \operatorname{Im} w$.

## Remarks

Here $\operatorname{Re}$ and $\operatorname{Im}$ are the [[real and imaginary parts|02ZE6JHJ7G]].
Thus one equation between complex numbers gives two equations between real numbers.
For example, $x + y i = 3 - 2 i$ with $x, y$ real means $x = 3$ and $y = -2$.

## Explanations

### L1 human

Two complex numbers are equal when their real parts agree and their imaginary parts agree.
In the plane, this says that two points are equal when both of their coordinates agree.