Possible Duplicate:
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.


marked as duplicate by Andrés E. Caicedo, user3302, Qiaochu Yuan Feb 8 '11 at 23:19

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ math.stackexchange.com/questions/14970/rational-numbers $\endgroup$ – Andrés E. Caicedo Feb 8 '11 at 23:15
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    $\begingroup$ @Bill: Why don't you add your answer to the previous question? $\endgroup$ – Andrés E. Caicedo Feb 8 '11 at 23:25
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    $\begingroup$ @Bill: I really do not understand your objection. This question is a model example of an exact duplicate, and two other users had already indicated agreement. If you edit your answer to the previous question, it will be bumped and newly visible, and the OP will certainly be interested in clicking the link. Why discuss the same question in a new thread when all of the answers could be usefully accumulated in the previous one? Do you seriously think three other users would not have closed as an exact duplicate? $\endgroup$ – Qiaochu Yuan Feb 8 '11 at 23:36
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    $\begingroup$ @Qiaochu: Look at all the low-voted answers at the tail end of my 500+ answers. Most of those were all answers to old questions that I browsed when I first joined. Apparently the software does a poor job of exposing new answers to old questions. So I will not waste my time composing a long answer that will get denigrated by mindless software. I've lost count of the number of times that questions have been quickly slammed closed in my face while I was composing answers that would have been very informative. I've had enough. I quit. You were warned. Goodbye and good luck. You will need it. $\endgroup$ – Bill Dubuque Feb 8 '11 at 23:44
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    $\begingroup$ David, no need to apologize. Duplicates are just closed because it is easier to have the answers collected in one thread. You asked a fine question, deserving the upvotes it has received. $\endgroup$ – Jonas Meyer Feb 9 '11 at 1:02

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