## Incommensurability

### Arithmetic proofs

You can see the arithmetic proof for the shortest diagonal and the side in the case of the following polygons:

Square

Regular pentagon

Regular hexagon

Note that to say that two magnitudes $$d$$ and $$l$$ are commensurable, that is, there exists a measure common to both, is the same as saying that their ratio is a rational number. Indeed, if $$x$$ is a measure common to $$d$$ and $$l$$, then $$d$$ and $$l$$ are both integer multiples of $$x$$, that is, there exist two integer numbers $$m$$ and $$n$$ such that $$d=mx$$ and $$l=nx$$, so $$\frac{d}{l}=\frac{mx}{nx}=\frac{m}{n}$$ and $$\frac{d}{l}$$ is a rational number. Conversely, if $$\frac{d}{l}=\frac{m}{n}$$, with $$m$$ and $$n$$ positive integers, then $$\frac{1}{m}d=\frac{1}{m}\frac{d}{l}l=\frac{1}{m}\frac{m}{n}l=\frac{1}{n}l$$ and, taking $$x=\frac{1}{m}d=\frac{1}{n}l$$, we have $$d=mx$$ and $$l=nx$$, so $$x$$ is a measure common to $$d$$ and $$l$$.