Invisible wrote:
> 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?

Proof by contradiction isn't reducto ad absurdum.

You say "Assume that X exists." Then you prove that from there you can 
logically derive the fact that X doesn't exist.  That's a contradiction. 
Since you only made one assumption, the contradiction must be logically 
derived from the assumption you made, namely that X exists. (X in this case 
being the ratio that expresses sqrt(2).)

> 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.

That's a lot more to prove before you can prove the sqrt(2) is irrational.

> 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.

That's a very similar proof.

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

Actually, that *is* a valid proof. I'm not sure why you think it's not a proof.

-- 
Darren New, San Diego CA, USA (PST)
   Serving Suggestion:
     "Don't serve this any more. It's awful."

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