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square root of 2
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It was the first number shown to be what is now known as an irrational number (a number that can't be written in the form a/b, where both a and b are integers.) This discovery was made by Pythagoras or, at any rate, by the Pythagorean group that he founded.
The square root of 2 is the length of the hypotenuse (longest side) of a right triangle whose other two sides are each one unit long. A reductio ad absurdum proof that √2 is irrational is straightforward. Suppose that √2 is rational, in other words that √2 = a/b, where a and b are coprime integers (that is, they have no common factors other than 1) and b > 0. It follows that a2/b2 = 2, so that a2 = 2b2. Since a2 is even (because it has a factor of 2), a must be even, so that a = 2c, say. Therefore, (2c)2 = 2b2, or 2c2 = b2, so b must also be even. Thus, in a/b, both a and b are even. But we started out by assuming that we'd reduced the fraction to its lowest terms. So there is a contradiction and therefore √2 must not be irrational. This type of proof can be generalized to show that any root of any natural number is either a natural number or irrational.
As a continued fraction, √2 can be written 1 + 1/(2 + 1/(2 + 1/(2 + ... ))), which yields the series of rational approximations: 1/1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, ... Multiplying each numerator (number on the top) by its denominator (number on the bottom) gives the series 1, 6, 35, 204, 1189, 6930, 40391, 235416, ... which follows the pattern: An = 6An-1 - An-2. Squaring each of these numbers gives 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, ..., each of which is also a triangular number. The numbers in this sequence are the only numbers that are both square and triangular.
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