Distances between Rational and Irrational Numbers
Date: 01/13/2005 at 13:05:51 From: Herman Subject: Math, irrational and rational numbers I know that if a rational number p/q is closer to an irrational number x than it is from another rational number r/s (the fractions are reduced so that the numerator and denominator do not have common divisors) then q has to be larger than s, but I do not know how to prove it. That is, if |x - p/q| < |x - r/s| with x irrational and p/q, r/s irreducible rationals then q > s. I do not know where to start. For example in pi=3.1415..., it is clear than in the sequence 3/10, 31/100, 314/1000, the denominator is getting large. What bothers me is that the fractions are not necessarily simplified, that is 314/1000 when simplified should be 157/500 and yes 500 > 100. This involves things on number theory (relative prime) and real analysis. My problem arose when trying to find a function that is discontinuous in all rationals and continuous in all irrationals. I believe the function is f(x) = 0 if x is irrational and f(x) = 1/q if x = p/q reduced rational. When proving continuity I need to be able to show that for each epsilon > 0, there exists delta > 0 such that |1/q| < epsilon always that |p/q - x| < delta. If I show what I have in my question, then I just need to find one (the first) p/q and after that I do not have to do anything.
Date: 01/13/2005 at 14:19:26 From: Doctor Vogler Subject: Re: Math, irrational and rational numbers Hi Herman, Thanks for writing to Dr. Math. There is a great deal of theory involved in good rational approximations to irrational numbers. What you stated in that regard is not correct. But you don't need anything nearly as sophisticated to accomplish what you're trying to do. I assume that you can show that your function is discontinuous at rational points. For that, let x = p/q be rational, choose a special epsilon (anything smaller than 1/q will do) and show that any interval around x contains points whose image is 0. To show that your function is continuous at an irrational point x = r, let epsilon > 0. You want to show that 0 <= f(x) < epsilon near x = r. So find some q so that 1/q < epsilon. Now show that there are only finitely many rational points anywhere near x = r (say within 1 of r) whose denominators are no bigger than q. Show that r is not one of those points, and then let delta be small enough that the interval (r-delta, r+delta) misses all of those points. Can you fill in all of the details? If you have any questions about this or need more help, please write back and show me what you have been able to do, and I will try to offer further suggestions. - Doctor Vogler, The Math Forum http://mathforum.org/dr.math/
Date: 01/13/2005 at 14:41:46 From: Herman Subject: Thank you (Math, irrational and rational numbers) Doctor Vogler. This is the first time I use "Ask Dr. Math". I was expecting an answer for the next week and I got it minutes! Your answer is perfect, beautiful and clear! I am so happy to see how this worked out. I was thinking on this problem for the last few days. You said that what I stated in my question was not correct. Could you please provide a counter-example? Thank you very, very much! Herman
Date: 01/13/2005 at 15:02:23 From: Doctor Vogler Subject: Re: Thank you (Math, irrational and rational numbers) Hi Herman, You stated: I know that if a rational number p/q is closer to an irrational number x than it is from another rational number r/s (the fractions are reduced so that the numerator and denominator do not have common divisors) then q has to be larger than s. This can be interpreted in a couple of different ways, but you clarified when you also wrote That is, if |x - p/q|<|x - r/s| with x irrational and p/q, r/s irreducible rationals then q > s. Here is a large collection of counterexamples: Pick any x, and pick any p/q. In fact, pick any s > q as well, and I will find a number r so that the (reduced) fraction r/s has |x - p/q| < |x - r/s|. Choose an integer N > x + |x - p/q| and then let r/s = N + 1/s = (Ns + 1)/s, which is to say, let r = Ns + 1. Then you have |x - r/s| = r/s - x > N - x > |x - p/q|. Anyway, the deep research I mentioned is called "Transcendental Number Theory" and some of its first main theorems can be seen on the "more" link at Amazon: http://www.amazon.com/exec/obidos/tg/detail/-/052139791X/ which shows you a few pages of Alan Baker's book on the subject. If you have any questions about this or need more help, please write back and show me what you have been able to do, and I will try to offer further suggestions. - Doctor Vogler, The Math Forum http://mathforum.org/dr.math/
Date: 01/13/2005 at 15:22:55 From: Herman Subject: Thank you (Math, irrational and rational numbers) Great! Thank you very much! Herman
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