Elementary Number Theory – Shorter Proof of Irrationality of $\sqrt{2}$

divisibilityelementary-number-theoryprime factorizationradicalsrationality-testing

Euclid's proof of the irrationality of $\sqrt{2}$ via contradiction involves arguments about evenness or odness of $a^2 = 2 b^2$ which is then lead to contradiction in using few more steps. I wonder why one needs this line of arguments, isn't it from this expression immediately obvious that all prime factors of both $b^2$ and $a^2$ must have even exponents (and that factor $2^1$ between them makes this impossible)? Is the reason for the "more complicated" structure of the traded proof, that this argument involves implicitly more theory (like the fundamental theorem of arithmetics, maybe)?

Best Answer

Here is one of my favorite proofs for the irrationality of $\sqrt{2}$.

Suppose $\sqrt{2}\in\Bbb Q$. Then there exist an integer $n>0$ such that $n\sqrt{2}\in\Bbb Z$.

Now, let $n$ be the smallest one with this property and take $m=n(\sqrt{2}-1)$. We observe that $m$ has the following properties:

  1. $m\in\Bbb Z$
  2. $m>0$
  3. $m\sqrt{2}\in\Bbb Z$
  4. $m<n$

and so we come to a contradiction!

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