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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, two non-
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s ''a'' and ''b'' are said to be ''commensurable'' if their ratio ' is a
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rati ...
; otherwise ''a'' and ''b'' are called ''incommensurable''. (Recall that a rational number is one that is equivalent to the ratio of two integers.) There is a more general notion of commensurability in group theory. For example, the numbers 3 and 2 are commensurable because their ratio, , is a rational number. The numbers $\sqrt$ and $2\sqrt$ are also commensurable because their ratio, $\frac=\frac$, is a rational number. However, the numbers $\sqrt$ and 2 are incommensurable because their ratio, $\frac$, is an
irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
. More generally, it is immediate from the definition that if ''a'' and ''b'' are any two non-zero rational numbers, then ''a'' and ''b'' are commensurable; it is also immediate that if ''a'' is any irrational number and ''b'' is any non-zero rational number, then ''a'' and ''b'' are incommensurable. On the other hand, if both ''a'' and ''b'' are irrational numbers, then ''a'' and ''b'' may or may not be commensurable.

History of the concept

The Pythagoreans are credited with the proof of the existence of
irrational numbers In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...
. When the ratio of the ''lengths'' of two line segments is irrational, the line segments ''themselves'' (not just their lengths) 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 A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
.
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
's notion of commensurability is anticipated in passing in the discussion between
Socrates Socrates (; ; –399 BC) was a Greek philosopher from Athens who is credited as the founder of Western philosophy and among the first moral philosophers of the ethical tradition of thought. An enigmatic figure, Socrates authored no te ...
and the slave boy in Plato's dialogue entitled
Meno ''Meno'' (; grc-gre, Μένων, ''Ménōn'') is a Socratic dialogue by Plato. Meno begins the dialogue by asking Socrates whether virtue is taught, acquired by practice, or comes by nature. In order to determine whether virtue is teachabl ...
, 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 from translations of
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
'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 ' is rational is a necessary and sufficient condition for the existence of some real number ''c'', and
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
s ''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 A ruler, sometimes called a rule, line gauge, or scale, is a device used in geometry and technical drawing, as well as the engineering and construction industries, to measure distances or draw straight lines. Variants Rulers have long ...
, marked off in units of length ''c'', could be used to measure out both a
line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between i ...
of length ''a'', and one of length ''b''. That is, there is a common unit of
length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Inte ...
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

In
group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
, two
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
s Γ1 and Γ2 of a group ''G'' are said to be commensurable if the intersection Γ1 ∩ Γ2 is of finite index in both Γ1 and Γ2. Example: Let ''a'' and ''b'' be nonzero real numbers. Then the subgroup of the real numbers R generated by ''a'' is commensurable with the subgroup generated by ''b'' if and only if the real numbers ''a'' and ''b'' are commensurable, in the sense that ''a''/''b'' is rational. Thus the group-theoretic notion of commensurability generalizes the concept for real numbers. There is a similar notion for two groups which are not given as subgroups of the same group. Two groups ''G''1 and ''G''2 are (abstractly) commensurable if there are subgroups ''H''1 ⊂ ''G''1 and ''H''2 ⊂ ''G''2 of finite index such that ''H''1 is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to ''H''2.

In topology

Two path-connected
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...
s are sometimes said to be ''commensurable'' if they have homeomorphic finite-sheeted covering spaces. Depending on the type of space under consideration, one might want to use homotopy equivalences or
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two ...
s instead of homeomorphisms in the definition. If two spaces are commensurable, then their fundamental groups are commensurable. Example: any two
closed surface In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as ...
s of genus at least 2 are commensurable with each other.

References

{{DEFAULTSORT:Commensurability (Mathematics) Real numbers Infinite group theory