commensurable numbers
Two positive real numbers $a$ and $b$ are commensurable^{}, iff there exists a positive real number $u$ such that
$a=mu,b=nu$  (1) 
with some positive integers $m$ and $n$. If the positive numbers $a$ and $b$ are not commensurable, they are incommensurable.
Theorem. The positive numbers $a$ and $b$ are commensurable if and only if their ratio is a rational number
$\frac{m}{n}$ ($m,n\in \mathbb{Z}$).
Proof. The equations (1) imply the proportion^{} (http://planetmath.org/ProportionEquation)
$\frac{a}{b}}={\displaystyle \frac{m}{n}}.$  (2) 
Conversely, if (2) is valid with $m,n\in \mathbb{Z}$, then we can write
$$a=m\cdot \frac{b}{n},b=n\cdot \frac{b}{n},$$ 
which means that $a$ and $b$ are multiples^{} of $\frac{b}{n}$ 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

•
The commensurability is an equivalence relation^{} in the set ${\mathbb{R}}_{+}$ of the positive reals: the reflexivity^{} and the symmetry^{} are trivial; if $a:b=r$ and $b:c=s$, then $a:c=(a:b)(b:c)=rs$, whence one obtains the transitivity.

•
The equivalence classes^{} of the commensurability are of the form
$$[\varrho ]:=\{r\varrho \mathrm{\vdots}r\in {\mathbb{Q}}_{+}\}.$$ 
•
One of the equivalence classes is the set $[1]={\mathbb{Q}}_{+}$ of the positive rationals, all others consist of positive irrational numbers.

•
If one sets $[\varrho ]\cdot [\sigma ]:=[\varrho \sigma ]$, the equivalence classes form with respect to this binary operation^{} an Abelian group^{}.
Title  commensurable numbers 
Canonical name  CommensurableNumbers 
Date of creation  20130322 18:11:14 
Last modified on  20130322 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 