Commensurability (mathematics)

In mathematics, two non-zero real numbers a and b are said to be commensurable if a/b is a rational number.

The Pythagoreans are credited with the proof of the existence of irrational numbers. 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 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.

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

...
Wikipedia