Commensurable and Incommensurable magnitudes
Book X of Euclid's Eldements studies "commensurable" (rational) and "incommensurable" (irrational) magnitudes. From Euclids' perspective, two lenghts x and y are said to be commensurable if there exists a lenght z and integers m and n, such that x = mz and y = nz. In other words, two lenghts are commensurable if they can be both measured precisely by the same measuring rod. Whole numbers are always commensurable with one another, because there is always the unit that measures both.
In modern times, commensurability is usually reprhased in terms or rationality. Namely, if x = mz and y = nz for two possible integers m and n, then x/y = mz/nz = m/n is a rational number. Hence commensurable lenghts are those with rational quotient. For example, the numbers 5 and 2 are commensurable because their ratio, 5/2, is a rational number. The numbers √5 and 2 √5 are also commensurable because their ratio, √5/ 2 √5 = 1/2, is a rational number. However, the numbers √5 and 2 are incommensurable because their ratio, √5/2, is an irrational number.
The most well-kmown and ancient pair of incommensurable magnitudes is that of a side of a square (l) and its diagonal (d); We know that
d2 = l2 + l2 = 2l2
then d2 is even, and therefore d is even. If the diagonal and side were commensurable, then their lenghts would be the ratio m:n with m and n relative prime integers (if they were not relative prime, we could divide both by the common factor). Then
d2 / l2 = m/n = 2
hence m must be even, say m = 2k; Then 4k2:n2 = 2 from which n2 = 2k2. Thus n as well is even, in contrast with the assumption that m and n are relatively prime.