Introduction to Pythagorean Ideas
Pythagoras (569-500 B.C.E.) was both a person and a society (i.e., the Pythagoreans). He was also a political figure and a mystic. The Pythagorean society was intensively mathematical in nature, but it was also quasi-religious.
Pythagoreans are remembered for two monumental contributions to mathematics. The first of these was to establish the importance of, and the necessity for, proofs in mathematics: that mathematical statement, especially geometric statements, must be established by way of rigorous proof. The second great contribution was the discovery of, and proof of, the fact that not all numbers are commensurate.
Pythagoras proved the result that we now call the Pythagorean Theorem. It says that the legs a, b and hypotenuse c of a right triangle is related by the formula: a2+b2=c2
This theorem has perhaps more proofs than any other result in mathematics-over fifty altogether. And in fact it is one of the most ancient mathematical results. There is evidence that the Babylonians and the Chinese knew this theorem nearly 1000 years before Pythagoras.
Theorem: There is no rational number c such that c2=2.
Proof: Suppose that the conclusion is false. Then there is a rational number, expressed in lowest terms (i.e. and have no integer factors in common) such that c2=2. This translates to
α2 = 2β2
We conclude that the right-hand side is even, hence so is the left-hand side. Therefore α = 2m for some integer m.
(2m) 2 =2β2
Or 2m2= β2 .
So we see that the left-hand side is even, so β is even.
But now both α and β are even-the two numbers have a common factor of 2. This statement contradicts the hypothesis that α and β have no common integer factors. Thus it cannot be that c is a rational number. Instead, c must be irrational.
A trio of integers (a, b, c) that satisfy a2+b2=c2 is called a Pythagorean triple. The most famous and standard Pythagorean triple is (3, 4, 5). But there are many others, including (5, 12, 13), (7, 24, 25), (20, 21, 29), and (8, 15, 17). In fact, it has been known since the time of Euclid that there are infinitely many Pythagorean triples, and there is a formula that generates all of them.
A reduced Pythagorean triple must take the form
(2uv, u2 – v2, u2+v2),
with u, v relatively prime (i.e., having no common factors). Conversely, any triple of the form (2uv, u2 – v2, u2+v2) is most certainly a Pythagorean triple. That is to say, a = 2uv, b = u2 – v2, c = u2+v2
1). Take u = 2 and v = 1
2). Take u = 3 and v = 2
3). Take u = 5 and v = 3