# Hurwitz's Theorem (Number Theory)

## Theorem

Let $\xi$ be an irrational number.

Then there are infinitely many relatively prime integers $m, n \in \Z$ such that:

- $\left|{\xi - \dfrac p q}\right| < \dfrac 1 {\sqrt 5 \, q^2}$

## Proof

### Lemma 1

Let $\xi$ be an irrational number.

Let $A \in \R$ be a real number strictly greater than $\sqrt 5$.

Then there may exist at most a finite number of relatively prime integers $p, q \in \Z$ such that:

- $\size {\xi - \dfrac p q} < \dfrac 1 {A \, q^2}$

$\Box$

### Lemma 2

Let $\xi$ be an irrational number.

Let there be $3$ consecutive convergents of the continued fraction to $\xi$.

Then at least one of them, $\dfrac p q$, say, satisfies:

- $\left|{\xi - \dfrac p q}\right| < \dfrac 1 {A \, q^2}$

$\Box$

There are an infinite number of convergents to $\xi$.

Taking these in sets of $3$ at a time, it can be seen from Lemma 2 that there are an infinite number of approximations that satisfy the given inequality.

From Lemma 1 it is seen that this inequality is the best possible.

$\blacksquare$

## Source of Name

This entry was named for Adolf Hurwitz.

## Sources

- 1979: G.H. Hardy and E.M. Wright:
*An Introduction to the Theory of Numbers*(5th ed.): $11.8$: The measure of the closest approximation to an arbitrary irrational

- Dec. 2002: Manuel Benito and J. Javier Escribano:
*An Easy Proof of Hurwitz's Theorem*(*Amer. Math. Monthly***Vol. 109**,*no. 10*: pp. 916 – 918) www.jstor.org/stable/3072459