Saturday, December 15, 2007
How do we know the square root of two cannot be represented by the ratio of two integers? (e.g. 7/5)
Of the proofs I have seen, Gregory Chaitin's is my favourite. It borrows from Hippasus's initial assumption and also ends with reductio ad absurdum. "Reductio ad absurdum" is a fancy way to say "this is crazy and can't be true". Instead of trying to prove what you think is true, try and prove the opposite is true, and prove it is wrong. If I want to prove that √2 cannot be a ratio of two integers (which is the literal meaning of irrational), try and prove the opposite condition:
1. Assume that √2 can be represented as the ratio of two integers, m/n = √2
2. Square both sides, m2/n2 = 2
3. Rearrange so that m2 = 2 * n2
4. Both m and n have unique prime decompositions (and if they didn't we could remove any common factors in the ratio),
m = 2a * 3b * 5c * ...
n = 2p * 3q * 5r * ...
5. If both m and n are squared, the prime decompositions of m2 and n2 are
m2 = 22a * 32b * 52c * ...
n2 = 22p * 32q * 52r * ...
6. Two (2) times n2 has a prime decomposition of
2 * n2 = 22p+1 * 32q * 52r * ...
7. If m2 = 2 * n2 (as in equation 3), then that means that 22a = 22p+1, which is impossible. 2a is an even number, and 2p+1 is an odd number! An odd number cannot be an even number - they cannot be equal. For example, 4 = 22 and 8 = 22+1, and 4 is not equal to 8 (even for large values of 4), thus their powers of two, 2 and 3, respectively, are also not equal.
8. Since this result is absurd (e.g. 4 is not equal to 8), assumption #1 must be false. This means that you cannot form a ratio of two integers to represent √2.
The beauty of this proof is that it is obviously extensible to the square root of any prime number - they're all irrational!
I first read this proof in Metamath, by Gregory Chaitin. I've really enjoyed the book. It's written in a prosaic rather than hard math style, so even a layperson could enjoy it. It explains a lot of mind blowing topics like the limits of what is provable in mathematics (incompleteness), the countability of numbers, randomness, compressability, and other tightly related topics. It has changed my views on the world (again), and given me a lot to chew on. Since I finished reading the book, I have constantly been re-reading sections of it. The image/link below will take you to amazon, where you can get it for yourself!
Update: Follow up post at Faith in √2
Burton MacKenZie http://www.burtonmackenzie.com/