Editing Commensurability (mathematics) (section)

==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'''.