born c. 580 BC, Samos, Ionia died c. 500, Metapontum, Lucania
Pythagoras was the Greek philosopher and mathematician and founder of the Pythagorean movement that influenced the philosophy of Plato and Aristotle and the development of mathematics.
Little is known of the early life of Pythagoras. He left Italy about 532 BC to escape Samos' tyrannical rule. He then established a philosophical academy at Croton.
Pythagoras' writings have not survived, and Pythagoreans supported their teachings by citing Pythagoras. Therefore, it is difficult to distinguish their teachings from his. Pythagoras is generally thought to have originated the theory of the significance of numbers in the world. But the Pythagorean theorem for right triangles was probably developed later by the Pythagorean school. It is thought that most of the ideas originating with Pythagoras himself pertain to mysticism rather than science.
The Pythagoras theorem states that the square on the hypotenuse of an equilateral triangle is equal to the square on the other two sides. The formal proof of Pythagoras theorem is dependent on the axioms and theorems of Euclid. Therefore , the formal proof is quite lengthy, which is why many authors only give a semi-formal proof.Astute readers of naive proofs of Pythagoras theorem, as Penrose and others argue, are often left unhappy. For a fully formal proof, you need to answer questions like: what exactly is a square? To answer this, involves asking even more fundamental questions. What is a right angle? Can you build a square given the properties of right angles, parallel lines, and lines of similar length?
Imagine two straight lines crossing each other, if each of the four angles is the same each is a right angle. By taking sides of equal length crossing at right angles, and basic Euclidean arguments, you can construct a square.