Commensurability (mathematics)

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

History of the concept

The Pythagoreans are credited with the proof of the existence of irrational numbers.^[1][2] When the ratio of lengths of two line segments is irrational, the line segments are also described as being incommensurable.

A separate, more general and circuitous ancient Greek doctrine of proportionality for geometric magnitude was developed in Book V of Euclid's Elements in order to allow proofs involving incommensurable lengths, thus avoiding arguments which applied only to a historically restricted definition of number.

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.^[3]

The usage primarily 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.

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 real line as additive group, generated respectively by a and by b, intersect in the subgroup generated by lc, where l is the LCM of m and n. This is of finite index in each of the original subgroups (but this is only the case for commensurable a and b). 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. In mathematical notation, two subgroups H₁ and H₂ of a group G are commensurable if

$[H_j: H_1\cap H_2]<\infty\text{ for }j=1,\,2.$

The relation of being commensurable in the wide sense is that H₁ be commensurable with a conjugate of H₂.^[4] Some authors use the terms commensurate and commensurable for commensurable and widely commensurable respectively.

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 $\mathrm{A}$ and $\mathrm{B}$ that are given by some moduli space stacks over a Lie algebra $\mathcal{O},$ are not necessarily commensurable if they are described by infinite dimensional representations. In addition, if the completions of $\mathcal{O}$-type modules corresponding to $\mathfrak{H}$ and $\mathfrak{G}$ are not well-defined, then $\mathfrak{G}$ and $\mathfrak{H}$ are also not commensurable.

In topology

Two topological spaces are commensurable if they have homeomorphic finite-sheeted covering spaces. Depending on the type of topological space under consideration one might want to use homotopy-equivalences or diffeomorphisms instead of homeomorphisms in the definition. Thus, if one uses homotopy-equivalences, commensurability of groups corresponds to commensurability of spaces provided one associates the classifying space to a discrete group. For example, the Gieseking manifold is commensurate with the complement of the figure-eight knot.

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 magnetic superstructure. The potential richness of physical phenomena related to this concept is exemplified in the devil's staircase.

See also

• Commensurator

References

1. Kurt Von Fritz (1945). "The Discovery of Incommensurability by Hippasus of Metapontum". The Annals of Mathematics.
2. James R. Choike (1980). "The Pentagram and the Discovery of an Irrational Number". The Two-Year College Mathematics Journal.
3. Plato's Meno. Translated with annotations by George Anastaplo and Laurence Berns. Focus Publishing: Newburyport, MA. 2004.
4. Maclachlan, Colin; Reid, Alan W. (2003). The Arithmetic of Hyperbolic 3-Manifolds. Graduate Texts in Mathematics 219. Springer-Verlag. p. 56. ISBN 0-387-98386-4. Zbl 1025.57001.