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Irrationality of the square root of 2[edit]

A classic proof by contradiction from mathematics is the proof that the square root of 2 is irrational.^[2] If it were rational, it could be expressed as a fraction a/b in lowest terms, where a and b are integers, at least one of which is odd. But if a/b = √2, then a² = 2b². Therefore a² must be even. Because the square of an odd number is odd, that in turn implies that a is even. This means that b must be odd because a/b is in lowest terms.

On the other hand, if a is even, then a² is a multiple of 4. If a² is a multiple of 4 and a² = 2b², then 2b² is a multiple of 4, and therefore b² is even, and so is b.

So b is odd and even, a contradiction. Therefore the initial assumption—that √2 can be expressed as a fraction—must be false.

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• that's the answer my smart-ass classmate showed me after the exa -vest2005- ♀ (0 bytes) () 11/21/2013 postreply 08:39:53

• 我特向往有个上学的孩子，陪着把所有的书本知识用英语学一遍。懒猫那样。 -笨企鹅- ♀ (0 bytes) () 11/21/2013 postreply 08:48:25

• 拍爪 -mj2001- ♀ (0 bytes) () 11/21/2013 postreply 08:45:22