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.

(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