On 12/12/2010 09:59 PM, Orchid XP v8 wrote:

> A proof is usually a highly abstract sequence of logical steps which
> demonstrates that something /is/ true. Myself, I'm usually more
> interested in trying to get an intuitive feeling for /why/ it is true.

For example... There is a rather famous proof that asserts that Sqrt(2) 
is irrational. It proves this by demonstrating that any ratio equal to 
Sqrt(2) can be cancelled down an infinite number of times, which is 
absurd. Hence, such a ratio cannot exist.

Logically, this is a perfectly valid argument. But intuitively, it 
doesn't make much sense. "You can't write this ratio because if you did, 
it would be *evil*!" Um... OK? So how come you can't do it then?

On the other hand, any positive integer can be represented as the 
product of unique prime numbers, each raised to a positive integral 
power. For example, 99 = 3^2 * 11^1. And when you square a number, you 
simply double all the exponents. Hence, 99^2 = 9801 = 3^4 * 11^2.

Now A/B = Sqrt(2) is equivalent to A^2/B^2 = 2. And in order for the 
ratio to be 2, A^2 and B^2 must have identical factors, except for A^2 
having an exponent for 2 which is exactly one higher than the exponent 
for 2 in B^2. But, as just established, the exponents of all prime 
factors of a square number are always even, so this condition is 
completely impossible to satisfy.

This doesn't *prove* anything, but it does explain why you can't solve 
the equation, in a way which intuitively makes sense.

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