Irrationality of Square Roots

Abstract

Indeed, if or were a fraction with denominator q, then m +na would also be fraction with denominator q. Such a fraction is either zero or at least \/q in magnitude. Let us first note some previous proofs of irrationality based on this criterion. Arbi trarily small numbers m + na have been constructed using Euclid's Algorithm, starting with a and 1. To prove that one gets arbitrarily small nonzero numbers, one must show that the sequence of numbers produced by Euclid's algorithm does not terminate. For certain numbers a, this can be done by finding a pair of consecutive numbers whose ratio is the same as the ratio of a previous pair of consecutive numbers. The sequence of ratios of consecutive numbers is periodic from then on. Kaiman, Mena, and Shariari [1] give a geometric proof that this sequence is periodic for a = \?2. Geometric proofs must be tailored to each specific number and they are bound to get very complicated. For instance, one can show by computation that for a = \/43, the ratios repeat only after 10 steps; a geometric proof would therefore have to contain dozens of points and line segments. Using algebra, Joseph Louis Lagrange proved that Euclid's algorithm is periodic for all square roots. This result can be found in books discussing continued fractions. For other kinds of irrationals such as cube roots Euclid's algorithm is not periodic and the author does not know of a direct way of showing it will not terminate.