Commensurability (mathematics)

In mathematics, two non-zero real numbers a and b are said to be commensurable if their ratio 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 in group theory. For example, the numbers 3 and 2 are commensurable because their ratio, 3/2, is a rational number. The numbers and are also commensurable because their ratio, , is a rational number. However, the numbers and 2 are incommensurable because their ratio, , 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. The Pythagoreans are credited with the proof of the existence of irrational numbers. 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 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.