# Incommensurable Magnitudes have Irrational Ratio

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## Theorem

In the words of Euclid:

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

(*The Elements*: Book $\text{X}$: Proposition $7$)

## 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*.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 3*(2nd ed.) ... (previous) ... (next): Book $\text{X}$. Propositions