---
tag: 07KBQCG9KV
kind: statement
role: proposition
name: i squared is minus one
msc: [12D99]
uses: [0ESCX36MMC, 09FTAET5Z4]
sources:
  - url: https://en.wikipedia.org/wiki/Imaginary_unit
    locator: Definition
---

## Statement

Let $i$ be the [[imaginary unit|0ESCX36MMC]] of $\mathbb{C}$.
Then $i^2 = -1$.

## Remarks

With the product of [[complex numbers|09FTAET5Z4]], $(0 + 1 i)(0 + 1 i) = (0 \cdot 0 - 1 \cdot 1) + (0 \cdot 1 + 1 \cdot 0) i = -1$.
Also $(-i)^2 = -1$. Thus $-1$ has the two square roots $i$ and $-i$ in $\mathbb{C}$.

## Explanations

### L1 human

This is the property that defines $i$.
All the rules for computation with complex numbers come from the usual rules of algebra together with $i^2 = -1$.