Math CentralQuandaries & Queries


Question from Haya:

how can i prove the the square root of 2 plus the square root of 2 is an irrational number?


Have you seen a proof that $\sqrt{2}$ is irrational? There is one in our response to an earlier question. I would use the same technique here.

Suppose that $\sqrt{2}+ \sqrt{2} = 2 \sqrt{2}$ is rational then there are integers $p$ and $q$ so that

\[2 \sqrt{2} = \frac{p}{q}.\]

Square both sides to get

\[8 = \frac{p^2}{q^2}\]


\[8 q^2 = p^{2}.\]

Can you see how to complete the proof? Write back if you need more assistance.


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