User:Superhiggs/sandbox

this is just an example.

On the constructive proof of the irrationality of square root of two
I don't like how this proof has been written in the original article. I will then try to produce a new version.

Here is the old proof.

Constructive proof
In a constructive approach, one distinguishes between on the one hand not being rational, and on the other hand being irrational (i.e., being quantifiably apart from every rational), the latter being a stronger property. Given integers a and b, because the valuation (i.e., highest power of 2 dividing a number) of 2b2 is odd, while the valuation of a2 is even, they must be distinct integers, so that, applying the law of trichotomy in the context of an effectively computable predicate over $$\mathbb{N}$$, we obtain $$|2 b^2 - a^2| \geq 1$$. Then


 * $$|\sqrt2 - a / b| = \frac{|2b^2-a^2|}{b^2(\sqrt{2}+a/b)} \ge \frac{1}{b^2(\sqrt2 + a / b)} \ge \frac{1}{3b^2},$$

the latter inequality being true because we assume $$\tfrac{a}{b} \le 3- \sqrt{2}$$ (otherwise the quantitative apartness can be trivially established). This gives a lower bound of $$\frac{1}{3b^2}$$ for the difference $$|\sqrt2 - a / b|$$, yielding a direct proof of irrationality not relying on the law of excluded middle; see Errett Bishop (1985, p. 18). This proof constructively exhibits a discrepancy between $$\sqrt{2}$$ and any rational.

Constructive proof
Let $$a$$ and $$b$$ be two positive non-zero integers. The constructive proof shows that there is a finite difference between $$\sqrt{2}$$ and the rational number $$a/b$$, without using reductio ad absurdum. Since $$\sqrt{2} < 2$$ we can assume $$a/b < 2$$, which implies $$\sqrt{2} +a/b< 4$$. Moreover, since 2 divides $$2 b^2$$ an odd number of times and $$a^2$$ an even number of times, $$a^2$$ and $$2 b^2$$ must be distinct, i.e. $$|2 b^2 - a^2| \geq 1$$. It clearly follows that


 * $$|\sqrt2 - a / b| = \frac{|2b^2-a^2|}{b^2(\sqrt{2}+a/b)} \ge \frac{1}{b^2(\sqrt2 + a / b)} > \frac{1}{4b^2}.$$

This proof constructively exhibits a discrepancy between $$\sqrt{2}$$ and any rational.