---
tag: 02ZE6JHJ7G
kind: definition
name: Real and imaginary parts
msc: [30A99]
defines: 06J8TEPJPY
uses: [09FTAET5Z4]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_number
    locator: Definition and basic operations
---

## Definition

Let $z = a + b i$ be a [[complex number|09FTAET5Z4]], with $a$ and $b$ real.
The real part of $z$ is $\operatorname{Re} z = a$.
The imaginary part of $z$ is $\operatorname{Im} z = b$.

## Remarks

The imaginary part is a real number: $\operatorname{Im}(2 + 3 i) = 3$, not $3 i$.
The maps $\operatorname{Re}$ and $\operatorname{Im}$ are additive and $\mathbb{R}$-linear, but they do not respect products.
With the [[conjugate|0BJV24C4SY]], $\operatorname{Re} z = (z + \bar{z})/2$ and $\operatorname{Im} z = (z - \bar{z})/(2 i)$.
A complex number $z$ is real when $\operatorname{Im} z = 0$, and purely imaginary when $\operatorname{Re} z = 0$.

## Explanations

### L1 human

Each complex number has two real coordinates.
For $z = 4 - 7 i$, the real part is $4$ and the imaginary part is $-7$.