---
tag: 0ESCX36MMC
kind: object
name: Imaginary unit
aliases: [i]
msc: [12D99]
sources:
  - url: https://en.wikipedia.org/wiki/Imaginary_unit
    locator: Definition
  - url: https://en.wikipedia.org/wiki/Imaginary_unit
    locator: Integer powers
same_as:
  wikidata: Q193796
  wikipedia: Imaginary unit
---

## Description

The imaginary unit is the [[complex number|09FTAET5Z4]] $i = 0 + 1 i$. It satisfies $i^2 = -1$, see [[i squared is minus one|07KBQCG9KV]].
It is a solution of the equation $x^2 + 1 = 0$. The other solution is $-i$.
In the [[complex plane|0CTE7A31XQ]], $i$ is the point $(0, 1)$, at distance $1$ above $0$.

## Remarks

The powers of $i$ repeat with period $4$: $i^0 = 1$, $i^1 = i$, $i^2 = -1$, $i^3 = -i$ and $i^4 = 1$.
Multiplication by $i$ turns the complex plane by a quarter turn about $0$, counterclockwise.
In electrical engineering, people often write $j$ for $i$.

## Explanations

### L1 human

The imaginary unit is a new number whose square is $-1$.
No real number has this property, so $i$ is not on the real number line.
It sits one unit above $0$ in the plane of complex numbers.