# Constructing a larger rational whose square is also less than two [duplicate]

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Rational Numbers

Baby Rudin has a very nice construction showing that, given a positive rational number whose square is less than (greater than) two, one can always find a larger (smaller) rational whose square is also less than (greater than) two. Basically, if $p$ is a rational whose square is less (greater than) than 2, then let $q$ be $$q=p-\frac{p^2-2}{p+2}=\frac{2p+2}{p+2}$$ so that $$q^2 - 2 = \frac{2(p^2-2)}{(p+2)^2}$$

It's easy to see that $q$ satisfies the requirements.

My question: how would one come up with this on one's own? What approach could I have taken to derive this gem by myself? What magic happened behind the scenes to get here?

I tackled it from scratch before going back to Rudin, and came up with my own solution (applying Newton's method to find an approximation for the positive root of $(x^2-2)^2$, giving me $q=\frac{3p}{4}+\frac{1}{2p}$), but the resulting proof is uglier and requires restricting p to certain intervals and handling the rest as a trivial special case.