The Pythagorean theorem is one of the most ancient theorems of mathematics. The Pythagorean theorem states that in a right triangle (a triangle with a 90 degree angle), the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse; the legs of a right triangle are the two sides adjoining the 90 degree angle, and the hypotenuse is the opposite side. In terms of symbols, if the lengths of the legs are labeled a and b, and the hypotenuse is labeled c, the Pythagorean theorem states the famous relationship that a^2+b^2=c^2, the form in which the theorem is most commonly remembered. The Pythagorean theorem is one of the most powerful of the fundamental theorems of geometry. It is the basis for the definition of the distance between two points in the rectangular coordinate system, the key definition on which all of coordinate (or Cartesian) geometry rests.
Although the theorem is named after the Greek mathematician Pythagoras (circa 560-480 b.c.), it was in fact known much earlier: a statement of the theorem was discovered on a Babylonian tablet that may date as far back as 1900 b.c.. It is not believed, however, that the Babylonians gave a formal mathematical proof of the theorem. The Babylonians generally discovered mathematical relationships by experiment, and historians suspect that the Babylonians discovered the theorem simply by measuring the sides of a large number of right triangles, and noticing that for all of them, the relationship a^2+b^2=c^2 was true. From their observations they were willing to conclude that the relationship was true for every right triangle.
It was not until the time of the Greeks, more than a thousand years later, that the idea had evolved that mathematical truths could be proven by the rigors of logic. Historians believe that the first careful proof of the Pythagorean theorem, using deductive reasoning, was given either by Pythagoras himself or by one of his disciples at his school in Cortona, a Greek seaport in Southern Italy. Pythagoras was quick to recognize the significance of the theorem, and according to legend, he was so overjoyed that he offered a sacrifice of oxen to the gods in thanks for the discovery.
A natural question asked both by the Pythagoreans and by the mathematicians that came after them was, When will the lengths of all three sides of a right triangle be whole numbers? In other words, which whole numbers satisfy the relationship a^2+b^2=c^2? The numbers 3, 4 and 5 do, for example, since 3^2+4^2=5^2. Likewise, the numbers 5, 12 and 13 satisfy the relationship, as do 7, 24 and 25. Triples of whole numbers that satisfy the relationship are known as Pythagorean triples. Over the years, mathematicians have developed many ways to generate Pythagorean triples, and in fact have proven that there is an infinite collection of Pythagorean triples. A related, more complicated question, is, For what exponents n will the equation a^n+b^n=c^n have non-zero whole number solutions? Long before this question was finally set to rest, it was widely believed that n=2 is the only exponent for which this equation will have solutions. This statement is the famous Fermat's Theorem, which stumped mathematicians for more than two hundred years before finally giving way to proof in 199?, by the mathematician Andrew Wiles.
Ever since the school of Pythagoras produced the first proof of the Pythagorean theorem, mathematicians and amateurs have found amusement in trying to produce alternate proofs, and through the centuries dozens more proofs have been found. Leonardo da Vinci came up with a one, as did James Garfield in 1876, a few years before becoming President of the United States. The great Hungarian mathematician Paul Erdös, when a seventeen-year-old mathematical prodigy, boasted that he knew 37 different proofs. Many of these proofs are very simple, but perhaps the simplest and most elegant of all is the following proof, the one that historians ascribe to Pythagoras himself.
Start with a right triangle whose legs have lengths a and b, and whose hypotenuse has length c, and build a large square as in Figure X. There are two different ways to calculate the area of the large square. Each side of the square has length a+b, so the area of the square is (a+b)^2. On the other hand, the large square is made up of one smaller square and four copies of the right triangle, so its area is also equal to the area of the smaller square plus four times the area of the right triangle. The area of the smaller square is c^2, and the area of the right triangle is ab/2. So we have the equation
(a+b)^2=c^2+4(ab/2)
Now (a+b)^2=a^2+b^2+2ab, and 4(ab/2)=2ab, so we can rewrite our equation as
a^2+b^2+2ab=c^2+2ab
If we cancel the 2ab that appears on both sides of the equation then we are left with the relationship a^2+b^2=c^2, which is exactly what we wanted to prove.
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