Editing Commensurability (mathematics)

{{Short description|1=When two functions have co-rational periods, i.e. n T1 = m T2}}
{{other uses|Commensurability (disambiguation)}}

In [[mathematics]], two non-[[zero]] [[real number]]s ''a'' and ''b'' are said to be '''''commensurable''''' if their ratio ''{{sfrac|a|b}}'' is a [[rational number]]; 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 (group theory)|commensurability in group theory]].

For example, the numbers 3 and 2 are commensurable because their ratio, {{sfrac|3|2}}, is a rational number. The numbers <math>\sqrt{3}</math> and <math>2\sqrt{3}</math> are also commensurable because their ratio, <math display='inline'>\frac{\sqrt{3}}{2\sqrt{3}}=\frac{1}{2}</math>, is a rational number. However, the numbers <math display='inline'>\sqrt{3}</math> and 2 are incommensurable because their ratio, <math display='inline'>\frac{\sqrt{3}}{2}</math>, is an [[irrational number]]. 

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 [[Pythagoreanism|Pythagoreans]] are credited with the proof of the existence of [[irrational numbers]].<ref>{{cite journal|title=The Discovery of Incommensurability by Hippasus of Metapontum |author=Kurt von Fritz |journal=The Annals of Mathematics |year=1945 |volume=46 |issue=2 |pages=242–264  |doi=10.2307/1969021 |jstor=1969021 }}</ref><ref>{{cite journal|title=The Pentagram and the Discovery of an Irrational Number|journal=The Two-Year College Mathematics Journal |author=James R. Choike |year=1980 |volume=11 |issue=5 |pages=312–316 |doi=10.2307/3026893 |jstor=3026893}}</ref><!--Note: Von Fritz & Choike references were drawn from the Wikipedia "History of Mathematics" article--> 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 [[wikiquote:Doctrine of proportion (mathematics)|doctrine of proportionality]] for geometric [[Magnitude (mathematics)|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#History|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.<ref>Plato's ''Meno''. Translated with annotations by [[George Anastaplo]] and [[Laurence Berns]]. Focus Publishing: Newburyport, MA. 2004. {{ISBN|0-941051-71-4}}</ref>

The usage primarily comes 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 ''{{sfrac|a|b}}'' is rational is a [[necessary and sufficient condition]] for the existence of some real number ''c'', and [[integer]]s ''m'' and ''n'', such that

:''a'' = ''mc'' and ''b'' = ''nc''.

Assuming for simplicity that ''a'' and ''b'' are [[positive number|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==
{{main|Commensurability (group theory)}}
In [[group theory]], two [[subgroup]]s Γ<sub>1</sub> and Γ<sub>2</sub> of a group ''G'' are said to be '''commensurable''' if the [[intersection (set theory)|intersection]] Γ<sub>1</sub> ∩ Γ<sub>2</sub> is of [[finite index]] in both Γ<sub>1</sub> and Γ<sub>2</sub>. 

Example: Let ''a'' and ''b'' be nonzero real numbers. Then the subgroup of the real numbers '''R''' [[generating set of a group|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''<sub>1</sub> and ''G''<sub>2</sub> are ('''abstractly''') '''commensurable''' if there are subgroups ''H''<sub>1</sub> ⊂ ''G''<sub>1</sub> and ''H''<sub>2</sub> ⊂ ''G''<sub>2</sub> of finite index such that ''H''<sub>1</sub> is [[group isomorphism|isomorphic]] to ''H''<sub>2</sub>.

==In topology==
Two [[path-connected]] [[topological space]]s are sometimes said to be ''commensurable'' if they have [[homeomorphism|homeomorphic]] finite-sheeted [[covering space]]s. Depending on the type of space under consideration, one might want to use [[homotopy|homotopy equivalences]] or [[diffeomorphism]]s instead of homeomorphisms in the definition. If two spaces are commensurable, then their [[fundamental group]]s are commensurable.

Example: any two [[closed surface]]s of [[genus (mathematics)|genus]] at least 2 are commensurable with each other.

==References==
{{Reflist}}

{{Ancient Greek mathematics}}

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[[Category:Real numbers]]
[[Category:Infinite group theory]]
[[Category:Greek mathematics]]