commensurable numbers

Two positive real numbers a and b are commensurableMathworldPlanetmathPlanetmath, 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 mn  (m,n).

Proof.  The equations (1) imply the proportionPlanetmathPlanetmath (

ab=mn. (2)

Conversely, if (2) is valid with  m,n,  then we can write


which means that a and b are multiplesMathworldPlanetmathPlanetmath of bn and thus commensurable.  Q.E.D.

Example.  The lengths of the side and the diagonal of 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