# Magnitudes with Irrational Ratio are Incommensurable

## Theorem

In the words of Euclid:

If two magnitudes have not to one another the ratio which a number has to a number, the magnitudes will be incommensurable.

## Proof

Let $A$ and $B$ be magnitudes which do not have to one another the ratio which a number has to a number.

Suppose $A$ and $B$ are commensurable.

Then from Ratio of Commensurable Magnitudes it follows that $A$ and $B$ have to one another the ratio which a number has to a number.

From this contradiction follows the result.

$\blacksquare$

## Historical Note

This proof is Proposition $8$ of Book $\text{X}$ of Euclid's The Elements.