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