---
tag: 09285XX7AE
kind: statement
role: corollary
name: Euler's identity
msc: [33B10]
uses: [04R3CY41VJ]
sources:
  - url: https://en.wikipedia.org/wiki/Euler%27s_identity
    locator: lead section
same_as:
  wikidata: Q204819
  wikipedia: Euler's identity
---

## Statement

$$e^{i \pi} + 1 = 0.$$

## Remarks

Proof: [[Euler's formula|04R3CY41VJ]] with $z = \pi$ gives $e^{i \pi} = \cos \pi + i \sin \pi = -1$.
The identity links five constants: $0$, $1$, $\pi$, $e$ and $i$.

## Explanations

### L1 human

Walk around the unit circle from $1$ for a distance of $\pi$, which is half of the circle.
You arrive at $-1$, so $e^{i \pi} = -1$.