# commensurable numbers

Two positive real numbers $a$ and $b$ are , iff there exists a positive real number $u$ such that

 $\displaystyle a\;=\;mu,\quad b\;=\;nu$ (1)

with some positive integers $m$ and $n$.  If the positive numbers $a$ and $b$ are not commensurable, they are incommensurable.

The positive numbers $a$ and $b$ are commensurable if and only if their ratio is a rational number $\displaystyle\frac{m}{n}$  ($m,\,n\in\mathbb{Z}$).

Proof.  The equations (1) imply the proportion  (http://planetmath.org/ProportionEquation)

 $\displaystyle\frac{a}{b}\;=\;\frac{m}{n}.$ (2)

Conversely, if (2) is valid with  $m,\,n\in\mathbb{Z}$,  then we can write

 $a\;=\;m\!\cdot\!\frac{b}{n},\quad b\;=\;n\!\cdot\!\frac{b}{n},$

which means that $a$ and $b$ are multiples   of $\displaystyle\frac{b}{n}$ and thus commensurable.  Q.E.D.

Example.  The lengths of the side and the diagonal of http://planetmath.org/node/1086square are always incommensurable.

## 0.1 Commensurability as relation

 Title commensurable numbers Canonical name CommensurableNumbers Date of creation 2013-03-22 18:11:14 Last modified on 2013-03-22 18:11:14 Owner pahio (2872) Last modified by pahio (2872) Numerical id 13 Author pahio (2872) Entry type Definition Classification msc 12D99 Classification msc 03E02 Related topic RationalAndIrrational Related topic CommensurableSubgroups Defines commensurable Defines incommensurable Defines commensurability