Thursday, December 1, 2016

elementary number theory - How can you prove that the square root of two is irrational?


I have read a few proofs that $\sqrt{2}$ is irrational.



I have never, however, been able to really grasp what they were talking about.


Is there a simplified proof that $\sqrt{2}$ is irrational?


Answer



You use a proof by contradiction. Basically, you suppose that $\sqrt{2}$ can be written as $p/q$. Then you know that $2q^2 = p^2$. However, both $q^2$ and $p^2$ have an even number of factors of two, so $2q^2$ has an odd number of factors of 2, which means it can't be equal to $p^2$.


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