Everything about Commensurability Mathematics totally explained
» This article is about the meaning of 'commensurable' and derived words in mathematics. For other senses, see commensurability.
Definition
In
mathematics, two non-
zero real numbers
a and
b are said to be
commensurable iff a/
b is a
rational number.
History of the concept
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's 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 didn't 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's 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
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
magnetic superstructure. The potential richness of physical phenomena related to this concept is exemplified in the
devil's staircase.
Further Information
Get more info on 'Commensurability Mathematics'.
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