---
tag: 0R7SVNYTJS
kind: statement
role: proposition
name: pi is greater than 3
msc: [11Y60]
uses: [0C1VYVNV2N, 08W5RN375X, 0PK8QH1W1V]
sources:
  - url: https://en.wikipedia.org/wiki/Approximations_of_pi
    locator: Early history
---

## Statement

$$\pi > 3.$$

## Remarks

The [[cosine series|08W5RN375X]] alternates. At $x = 3/2$, its terms decrease in size from the term $x^4/4!$ on. So $\cos(3/2) \ge 1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720}$ at $x = 3/2$. This is about $0.070 > 0$.
Since $\cos 2 < 0$, the zero of $\cos$ in $[0, 2]$ lies in $(3/2, 2)$. So $\pi/2 > 3/2$.
Archimedes proved the sharper bounds $\frac{223}{71} < \pi < \frac{22}{7}$. Mathlib states the claim as `Real.pi_gt_three`.

## Explanations

### L1 human

We show that $\cos x$ is still positive at $x = 3/2$.
So its first zero $\pi/2$ comes later, and $\pi > 3$.