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Commensurability (mathematics)

In mathematics, two non-zero real numbers ''a'' and ''b'' are said to be '''''commensurable''''' if and only if|iff ''a''/''b'' is a rational number.

History of the concept

Euclid's notion of commensurability is anticipated in passing in the discussion between Socrates and the slave boy in Plato's dialogue entitled Meno, in which Socrates uses the boy's own inherent capabilities to solve a complex geometric problem through the Socratic Method. He develops a proof which is, for all intents and purposes, very Euclidean in nature and speaks to the concept of incommensurability.Plato's ''Meno''. Translated with annotations by George Anastaplo and Laurence Berns. Focus Publishing: Newburyport, MA. 2004. The usage primarily comes to us from translations of Euclid's Euclid's Elements|''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 negative and non-negative numbers|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'''.

Commensurability in group theory

In group theory, a generalisation to pairs of subgroups is obtained, by noticing that in the case given, the subgroups of the integers as an additive group, generated respectively by ''a'' and by ''b'', intersect in the subgroup generated by ''d'', where ''d'' is the greatest common divisor|GCD of ''a'' and ''b''. This intersection has finite set|finite Index of a subgroup|index in the integers, and therefore in each of the subgroups. This gives rise to a general notion of '''commensurable subgroups''': two subgroups ''A'' and ''B'' of a group are ''commensurable'' when their Intersection (set theory)|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 projection (linear algebra)|projections that have finite-dimensional kernel and cokernel. In contrast, two subspaces

\mathrm{A}

and

\mathrm{B}

that are given by some moduli space algebraic stack|stacks over a Lie algebra

\mathcal{O},

are not necessarily commensurable if they are described by infinite dimensional representations. In addition, if the Complete space|completions of

\mathcal{O}

-type Module (mathematics)|modules corresponding to

\mathfrak{H}

and

\mathfrak{G}

are not well-defined, then

\mathfrak{G}

and

\mathfrak{H}

are also ''not commensurable.''

In physics

In physics, the terms ''commensurable'' and ''incommensurable'' are used in the same way as in mathematics. The two rational numbers ''a'' and ''b'' usually refer to periods of two distinct, but connected physical properties of the considered material, such as the crystal structure and the ANNNI model|magnetic superstructure. The potential richness of physical phenomena related to this concept is exemplified in the devil's staircase.

References

Category:Real numbers Category:Infinite group theory

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