Commensurability (mathematics)

From Wikipedia, the free encyclopedia

This article is about the meaning of 'commensurable' and derived words in mathematics. For other senses, see commensurability.

In mathematics, two nonzero real numbers a and b are said to be commensurable if and only if a/b is a rational number.

The usage comes to us from translations of Euclid's Elements, in which two line segments a and b are called commensurable precisely if there is some third segment c that can be laid end-to-end a whole number of times to produce a segment congruent to a, and also, with a different whole number, a segment congruent to b. Euclid did not use any concept of real number, but he used a notion of congruence of line segments, and of one such segment being longer or shorter than another.

That a/b is rational is a necessary and sufficient condition for the existence of some real number c, and integers m and n, such that

a = mc and b = nc.

Assuming for simplicity that a and b are positive, one can say that a ruler, marked off in units of length c, could be used to measure out both a line segment of length a, and one of length b. That is, there is a common unit of length in terms of which a and b can both be measured; this is the origin of the term. Otherwise the pair a and b are incommensurable.

In group theory, a generalisation to pairs of subgroups is obtained, by noticing that in the case given, the subgroups of the real line as additive group, generated respectively by a and by b, intersect in the subgroup generated by dc, where d is the LCM of m and n. This is of finite index, therefore in each of them. This gives rise to a general notion of commensurable subgroups: two subgroups A and B of a group are commensurable when their intersection has finite index in each of them. Sometimes in fact this relation is called commensurate, and to be commensurable requires only to be conjugate to a commensurate subgroup.

A relationship can similarly be defined on subspaces of a vector space, in terms of projections that have finite-dimensional kernel and cokernel.

In contrast, two subspaces A and B that are given by some moduli space stacks over a Lie algebra are not necessarily commensurable if they are described by infinite dimensional representations. In addition, if the completions of -type modules corresponding to and are not well-defined, then and are also not commensurable.