---
tag: 0E70M92FE3
kind: statement
role: theorem
name: Convergents converge to the number
msc: [11A55]
uses: [0SRWNR42BQ, 0SJ7Z6VKWK]
sources:
  - url: https://en.wikipedia.org/wiki/Simple_continued_fraction
    locator: Infinite continued fractions and convergents
---

## Statement

Let $x$ be an irrational real number, and let $p_n / q_n$ be the [[convergents|0SRWNR42BQ]] of its [[continued fraction expansion|0SJ7Z6VKWK]].
Then
$$\left| x - \frac{p_n}{q_n} \right| < \frac{1}{q_n q_{n+1}} \le \frac{1}{q_n^2}$$
for each $n \ge 0$, and $p_n / q_n \to x$ as $n \to \infty$.

## Remarks

The convergents with even $n$ increase toward $x$, and the convergents with odd $n$ decrease toward $x$.
For a rational $x$, the last convergent is equal to $x$.
Conversely, each infinite simple continued fraction converges to an irrational number, and this number has that continued fraction as its expansion.
For example, $\frac{355}{113}$ is a convergent of $\pi$, and $\left| \pi - \frac{355}{113} \right| < 3 \cdot 10^{-7}$.

## Explanations

### L1 human

The convergents of $x$ get closer and closer to $x$, and the error is smaller than one over the square of the denominator.
This makes them very good rational approximations of $x$.