# Definition: Def. 10.01: Magnitudes Commensurable and Incommensurable in Length

Those magnitudes measured by the same measure are said (to be) commensurable (in length), but (those) of which no (magnitude) admits to be a common measure (are said to be) incommensurable (in length).

### Modern Formulation

Two segments of lengths $$\alpha > 0$$ and $$\beta > 0$$ are called commensurable (in length), if there exists a segment of length $$\gamma$$ being an aliquot part of both, $$\alpha$$ and $$\beta$$, i.e. such that $\alpha=p\gamma,\quad\beta=q\gamma$ for some natural numbers $$p > 0$$ and $$q > 0$$.

Equivalently, the segments are commensurable, if their ratio is a rational number, formally, $\frac{\alpha}\beta=\frac{p\gamma}{q\gamma}=\frac pq.$

If $$\alpha,\beta$$ are not commensurable, we call them incommensurable (in length).

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### References

#### Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"