8

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.

Community
  • 1
David
  • 604
  • 2
    http://math.stackexchange.com/questions/14970/rational-numbers – Andrés E. Caicedo Feb 08 '11 at 23:15
  • 1
    It turns out that this is actually a special case of the secant method - the associativity of which is closely related to the group law on conics (folklore). This is actually a very beautiful topic which I would have said something about this had the question not been rudely closed so quickly. Perhaps some people will reopen it, but probably not, since we all know how hard that is give the seriously broken close-reopen process. Alas, complaints about such fall on deaf ears. – Bill Dubuque Feb 08 '11 at 23:24
  • 4
    @Bill: Why don't you add your answer to the previous question? – Andrés E. Caicedo Feb 08 '11 at 23:25
  • 1
    Because no one will see it. – Bill Dubuque Feb 08 '11 at 23:27
  • 8
    @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? – Qiaochu Yuan Feb 08 '11 at 23:36
  • 2
    @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. – Bill Dubuque Feb 08 '11 at 23:44
  • My apologies for asking an identical question. I did search for 'Rudin' but failed to find the previous question. Had I succeeded, I wouldn't have asked it again. – David Feb 09 '11 at 00:32
  • 6
    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. – Jonas Meyer Feb 09 '11 at 01:02
  • 1
    @Andres, what is interesting here: Bill had already answered that duplicate question. (No offense meant, Bill, it just leaves me wondering ...) – Hendrik Vogt Apr 24 '11 at 13:09

0 Answers0