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${\displaystyle \frac{m}{n}}$  ($m,n\in \mathbb{Z}$).

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

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

• The commensurability is an equivalence relation in the set ${ℝ}_{+}$ 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

  $$[\varrho ]:=\{r\varrho \mathrm{⋮}r\in {\mathbb{Q}}_{+}\}.$$

• 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.