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Pythagorean Theorem
The Pythagorean Theorem, sometimes known as Pythagoras Thereom, is one of the most widely known theorems in geometry.
The theorem states that, in a triangle where one angle is a right-angle (90°),
- the square on the hypotenuse is equal to the sum of the squares on the other two sides,
- where the hypotenuse is the longest side (that opposite the right angle).^{[1]}
Alternatively, if the hypotenuse has length c, and the lengths of the other two sides are a and b, then
- a^{2} + b^{2} = c^{2}.
The simplest such triangle whose sides are whole numbers has sides 3, 4, and 5 units long.
- 3^{2} + 4^{2} = 9 + 16 = 25 = 5^{2}
The picture illustrates this.
History
Circa 1800 BC to 1600 BC, Babylonian clay tablets included some examples of Pythagoras theorem.
Circa 1000 BC, Egyptian sources also give some examples of Pythagoras theorem.
Circa 550 BC, Pythagoras is credited with the first formal proof of Pythagoras theorem, although Indian and Chinese mathematicians also discovered the theorem independently. The Pythagorean school of thought believed that numbers were either whole numbers or ratios of whole numbers (rational numbers); there was some disquiet when the Pythagorean triangle with sides 1, 1, √2 was considered, since they managed to prove that √2 was irrational.
Circa 300 BC, Euclid's Elements contains the oldest extant proof.
Over the centuries, many different proofs of the theorem have been produced.
Generating Pythagorean triples
There are many different ways of generating all possible Pythagorean triangles whose sides are always whole numbers.
One of the simpler methods is Dickson's Method:^{[2]}
- The integers a = r+s, b = r+t, c = r+s+t, form a Pythagorean triple on condition that r^{2} = 2×s×t, where r, s, t are positive integers.
- The factor of 2 means that r must be even. To keep things simple and avoid ambiguity, assume s ≤ t.
As an example, consider the case when r = 2
- then r^{2} = 4
- hence s×t = 2
- hence s = 1, t = 2
- hence a = r+s = 3, b = r+t = 4, c = r+s+t = 5