# Magnitudes with Irrational Ratio are Incommensurable

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## 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.*

(*The Elements*: Book $\text{X}$: Proposition $8$)

## 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*.

## Sources

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