PYTHAGOREAN NUMEROLOGY

Early-Pythagorean philosophers such as Philolaus and Archytas held the conviction that mathematics could help in addressing important philosophical problems. In Pythagoreanism numbers became related to intangible concepts. The one was related to the intellect and being, the two to thought, the number four was related to justice because 2 * 2 = 4 and equally even. A dominant symbolism was awarded to the number three, Pythagoreans believed that the whole world and all things in it are summed up in this number, because end, middle and beginning give the number of the whole. The triad had for Pythagoreans an ethical dimension, as the goodness of each person was believed to be threefold: prudence, drive and good fortune.

HIPPASUS

Hippasus of Metapontum (c. 530 - c. 450 BC), was a Pythagorean philosopher. Little is known about his life or his beliefs, but he is sometimes credited with the discovery of the existence of irrational numbers. Pythagoreans knew about whole numbers (Natural Numbers starting with one) and fractions (Rational Numbers). The discovery of irrational numbers is said to have been shocking to the Pythagoreans, and Hippasus is supposed to have drowned at sea, apparently as a punishment from the gods for divulging this.

However, the few ancient sources which describe this story either do not mention Hippasus by name (e.g. Pappus) or alternatively tell that Hippasus drowned because he revealed how to construct a dodecahedron inside a sphere. The discovery of irrationality is not specifically ascribed to Hippasus by any ancient writer.

 PROOFS OF IRRATIONALITY OF ROOT(2)

Assume root 2 is rational, i.e. it can be expressed as a rational fraction of the form a/b, where a and b are two relatively prime integers. Now, since Root(2) = a/b, we have 4 = a²/b², or b² = 2a². Since 2a² is even, b² must be even, and since b² is even, so is b. Let b = 2c. We have 4c² = 2a² and thus a² = 2c². Since 2c² is even, a² is even, and since a² is even, so is a. However, two even numbers cannot be relatively prime, so Root(2) cannot be expressed as a rational fraction; hence Root(2) is irrational.

EXTENSIONS

Many proofs are available to show the irrationality of Root(2). Following are a few of them

1. Proof by infinite descent
2. Geometric proof
3. Constructive proof
4. Proof by Diophantine equations

QUESTION I: How to prove Root(n) is irrational when n is not a square number?
QUESTION II: How to prove the irrationality of π and e?