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| {{About|the meaning of 'commensurable' and derived words in mathematics|other senses|Commensurability (disambiguation)}}
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| In [[mathematics]], two non-[[zero]] [[real number]]s ''a'' and ''b'' are said to be '''''commensurable''''' if ''a''/''b'' is a [[rational number]].
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| ==History of the concept==
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| 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|ref=harv}}</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|ref=harv}}</ref><!--Note: Von Fritz & Choike references were drawn from the Wikipedia "History of Mathematics" article--> When the ratio of lengths of two line segments is irrational, the line segments are also described as being incommensurable.
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| 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]].
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| [[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.</ref> | |
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| 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.
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| That ''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
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| :''a'' = ''mc'' and ''b'' = ''nc''.
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| 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'''.
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| ==Commensurability in group theory==
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| In [[group theory]], a generalisation to pairs of [[subgroup]]s is obtained, by noticing that in the case given, the subgroups of the [[integers]] as an [[group scheme|additive group]], generated respectively by ''a'' and by ''b'', intersect in the subgroup generated by ''d'', where ''d'' is the [[least common multiple|LCM]] 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. That is, two subgroups ''H''<sub>1</sub> and ''H''<sub>2</sub> of a group ''G'' are '''commensurable''' if
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| : <math>[H_j: H_1\cap H_2]<\infty\text{ for }j=1,\,2.</math> | |
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| The relation of being '''commensurable in the wide sense''' is that ''H''<sub>1</sub> be commensurable with a conjugate of ''H''<sub>2</sub>.<ref name=MR56>{{cite book | first1=Colin | last1=Maclachlan | first2=Alan W. | last2=Reid | title=The Arithmetic of Hyperbolic 3-Manifolds | series=[[Graduate Texts in Mathematics]] | publisher=[[Springer-Verlag]] | volume=219 | year=2003 | isbn=0-387-98386-4 | zbl=1025.57001 | page=56 }}</ref> Some authors use the terms '''commensurate''' and ''commensurable'' for commensurable and widely commensurable respectively.
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| A relationship can similarly be defined on subspaces of a [[vector space]], in terms of [[projection (linear algebra)|projections]] that have finite-[[dimension]]al kernel and [[cokernel]].
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| In contrast, two [[Linear subspace|subspaces]] <math>\mathrm{A}</math> and <math>\mathrm{B}</math> that are given by some [[moduli space]] [[algebraic stack|stacks]] over a [[Lie algebra]] <math>\mathcal{O},</math> are not necessarily commensurable if they are described by infinite dimensional representations. In addition, if the [[Complete space|completions]] of <math>\mathcal{O}</math>-type [[Module (mathematics)|modules]] corresponding to <math>\mathfrak{H}</math> and <math>\mathfrak{G}</math> are not [[well-defined]], then <math>\mathfrak{G}</math> and <math>\mathfrak{H}</math> are also ''not commensurable.''
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| ==In topology==
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| Two topological spaces are ''commensurable'' if they have [[homeomorphism|homeomorphic]] finite-sheeted [[covering space]]s. Depending on the type of topological space under consideration one might want to use [[homotopy|homotopy-equivalences]] or [[diffeomorphism]]s 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 to the complement of the [[figure eight knot (mathematics)|figure-eight knot]].
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| ==In physics==
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| 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 [[Cantor function|devil's staircase]].
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| ==See also==
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| * [[Commensurator]]
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:Commensurability (Mathematics)}}
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| [[Category:Real numbers]]
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| [[Category:Infinite group theory]]
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