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{{About|the meaning of 'commensurable' and derived words in mathematics|other senses|Commensurability (disambiguation)}} | |||
In [[mathematics]], two non-[[zero]] [[real number]]s ''a'' and ''b'' are said to be '''''commensurable''''' if ''a''/''b'' is a [[rational number]]. | |||
==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|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. | |||
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.</ref> | |||
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. | |||
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 | |||
:''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'''. | |||
==Commensurability in group theory== | |||
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 | |||
: <math>[H_j: H_1\cap H_2]<\infty\text{ for }j=1,\,2.</math> | |||
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. | |||
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]]. | |||
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.'' | |||
==In topology== | |||
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]]. | |||
==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 [[ANNNI model|magnetic superstructure]]. The potential richness of physical phenomena related to this concept is exemplified in the [[Cantor function|devil's staircase]]. | |||
==See also== | |||
* [[Commensurator]] | |||
==References== | |||
{{Reflist}} | |||
{{DEFAULTSORT:Commensurability (Mathematics)}} | |||
[[Category:Real numbers]] | |||
[[Category:Infinite group theory]] | |||
Revision as of 02:46, 29 January 2014
29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church. 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 integers as an additive group, generated respectively by a and by b, intersect in the subgroup generated by d, where d is the LCM of a and b. This intersection has finite 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 has finite index in each of them. That is, two subgroups H1 and H2 of a group G are commensurable if
![{\displaystyle [H_{j}:H_{1}\cap H_{2}]<\infty {\text{ for }}j=1,\,2.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed46e020c4fdc0acf52b2de10487e1bef2209cc)
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 and that are given by some moduli space stacks over a Lie algebra are not necessarily commensurable if they are described by infinite dimensional representations. In addition, if the completions of -type modules corresponding to and are not well-defined, then and 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 to 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
References
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A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
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In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ Plato's Meno. Translated with annotations by George Anastaplo and Laurence Berns. Focus Publishing: Newburyport, MA. 2004.
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