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The Pythagoreans proved that the length of the diagonal of a square with side length 1 is not a rational number. Prove that the length of the diagonal of a rectangle with sides length 1 and 2 is not a rational number. When trying this problem, I did the pythagorean theorem. I think that is too simple.

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marked as duplicate by Claude Leibovici, JonMark Perry, user91500, José Carlos Santos, TomGrubb Jun 17 '17 at 23:33

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  • $\begingroup$ well, after the theorem you should prove that what you obtain is not rational. How do you do that? $\endgroup$ – Exodd Jun 17 '17 at 1:53
  • $\begingroup$ I would prove it by contradiction just like the square root of 3 is done all the time. $\endgroup$ – Megan F Jun 17 '17 at 1:54
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What you did is correct. For a rectangle with sides of length $1$ and $2$, the Pythagorean theorem states that the diagonal of that rectangle is $\sqrt {1^2+2^2}=\sqrt 5$. Since $\sqrt 5$ is a real number that cannot be expressed as a ratio of two integers where the dominator is nonzero, the diagonal of this rectangle is an irrational number.

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