Rationality Quotes September 2012

Jayson_Virissimo

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Here's the new thread for posting quotes, with the usual rules:

Please post all quotes separately, so that they can be voted up/down separately. (If they are strongly related, reply to your own comments. If strongly ordered, then go ahead and post them together.)
Do not quote yourself
Do not quote comments/posts on LW/OB
No more than 5 quotes per person per monthly thread, please.

If √2 is rational, then √2 can be written as z/y for some integers z and y, where z and y are coprime. Then, 2=z²/y².

Consider the hypothetical integer z. It is equal to √2*y. Since y and z are coprime, y cannot contain a factor of √2. Thus, z does not contain a factor of 2; the highest integer power of 2 that is a factor of z is 2^0.

On the other hand, z² does have a factor of 2; it is equal to 2*y² (since y has no factor of √2, y² therefore has no factor of 2).

Therefore, to claim that p² contains exactly twice as many powers of 2 as p is exactly equivalent to claiming that √2 is irrational.

Isn't the point of math that all mathematical truths are logically equivalent? (In beore Gödel.)

2Kindly14y

Even if √2 is rational, it is not an integer, and (especially in your second paragraph) you are trying to do things with it that only make sense with integers. I don't think that continuing this conversation can be productive, since you seem to be objecting to standard techniques. A rigorous textbook in number theory can probably explain the proof that √2 is irrational more thoroughly, and you can follow back the lemmas and see exactly where your confusion lies.