Proving irrationality

[AntonioLao]

One way to prove that a given real number is irrational is a proof by contradiction. Given the propositional statement that √2 is not a rational number and supposing this statement is false, that is, suppose √2 is rational then there are integer m and n such that √2=m/n. Furthermore, m/n is reduced to lowest terms, meaning m and n have no common factors except 1. Taking the square of √2=m/n gives m=2n. If m was odd then m is also odd. But m=2n is clearly even. Therefore, m is even and there exists integer p such that m=2p is always true. On the other hand, 2n=4p or n=2p is also even, therefore n is also even. Since both m and n are shown to be even numbers, the fraction m/n is not in lowest terms contrary to the supposition. This is clearly a contradiction. Therefore, √2 is not a rational number.

[mkirkpatrick]

Without irrationality this universe could not function,it is wonderful that you can prove
this Antonio!





regards michael.

[AntonioLao]

Mike,

I didn't prove it. I simply rephrased what other mathematicians have done. They also assert that there are infinitely more irrational numbers than rational numbers. There is also a proof but might be too much for typing to show it here.