Pythagoras of Samos, c.560 - c.480 B.C., was a Greek philosopher, early scientist and religious leader responsible for important developments in the history of mathematics, astronomy, and music theory.
Pythagoras, who died at the beginning of the fifth century B.C., was known to his
contemporaries, and later even to Aristotle, as the founder of a religious
brotherhood in southern Italy, where Pythagoras played a political role in the
sixth century B.C. (see videos below). He is famous for 'the Pythagorean Theorem'
which of course is, a2 X b2 = c2.
The computation of square roots is a related problem and has inspired mathemeticians to discover some wonderful geometric constructions. One of them allows us to construct √n for any integer n. It could be called the square root spiral, and it is a geometric feedback loop. The construction which yields the family of Pythagorean trees and their relatives is very much related to the construction of the square root spiral. The construction proceeds along the following steps and is shown here (image of a Pythagorean tree, and in one of the videos).
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PYTHAGORAS Video Series
Introduction:
PYTHAGORAS Video Series: Part 1
PYTHAGORAS Video Series: Part 2
PYTHAGORAS Video Series: Part 3
THE INCOMMENSURABILITY OF SIDE AND DIAGONAL OF THE SQUARE
The discovery that the ratio of diagonal and side of the square is not equal to the ratio of two integers produced the necessity to extend the number system to irrational numbers. The length of the diagonal in the unit square, √2 is irrational. Let us give the argument. Assume that p>q are positive integers with √2 = p/q. We may also assume that p and q have no common divisor. Then p2+ 2q2, i.e. p2 must be an even number. But this implies thatpitself must be even, because the square of an odd number is odd.
Thus p2 = 2r. But then p2 = 2 q2which means that q must be even as well. But this contradicts the assumption that p and q have no common divisor. So that means, √ 2 is irrational. This proof is found in the tenth book of Euclid around 300 B.C.
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