## square root of 2It 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 a^{2}/b^{2}
= 2, so that a^{2} = 2b^{2}. Since a^{2}
is even (because it has a factor of 2), a must be even, so that a
= 2c, say. Therefore, (2c)^{2} = 2b^{2},
or 2c^{2} = b^{2}, 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: A _{n} = 6A_{n-1}
- A_{n-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.
## Related category• NOTABLE NUMBERS | ||||||

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