How does infinite descent prove the irrationality of \sqrt{2}?

[octahedron]

What is infinite descent? I saw a proof of the irrationality of \sqrt{2} using that principle. How is it any different than the proof that relies on the contradiction of having two even numbers in the fraction, which completely makes sense to me?

 

[Hurkyl]

Infinite descent is a particular method of applying of mathematical induction. The basic form of the argument is to show that whenever you have an example of some property, that you can find a smaller example. Under the right circumstances, this allows you to infer that no example exists.

One common example of the right circumstances is when 'size' is measured by natural numbers.


The term 'infinite descent' comes from one way of justifying the method -- if an example exists, you can recursively construct an infinite sequence of new examples, each smaller than the previous one. This is a contradiction if there can only possibly be finitely many examples smaller than the original.




You often see infinite descent arguments written in a different form. (I'll consider sizes measured by natural numbers, for simplicity)

1. Assume there is at least one example of some property.
2. Then, there must exist a smallest example.
3. Construct a new example smaller than that one.
4. By contradiction, infer that there are no examples of that property.

 

[octahedron]

Thank you Hurkyl! I think I get it now -- the sequence can't be infinite for we only have a finite quantity of natural numbers less than x, and if an infinite number of solutions is obtained, we get a contradiction.