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.

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