# Incommensurable Magnitudes have Irrational Ratio

## Theorem

In the words of Euclid:

Incommensurable magnitudes have not to one another the ratio which a number has to a number.

## Proof

Let $A$ and $B$ be incommensurable magnitudes.

Suppose $A$ and $B$ have the ratio which a number has to a number

Then by Proposition $6$ of Book $\text{X}$: Magnitudes with Rational Ratio are Commensurable, $A$ and $B$ are commensurable.

From this contradiction follows the result.

$\blacksquare$

## Historical Note

This proof is Proposition $7$ of Book $\text{X}$ of Euclid's The Elements.