Wikipedia:Reference desk/Archives/Mathematics/2006 July 24

5k race
how far is a 5k race in miles?


 * Why not ask google? Google "5km in miles". –Joke 02:51, 24 July 2006 (UTC)


 * Why not use Fibonacci numbers? An amusing and occasionally convenient fact is that the ratio of miles to kilometers is approximately the golden ratio, as is the ratio of successive Fibonacci numbers. The first few Fibonacci numbers are
 * 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89
 * To convert miles to kilometers, step up; to convert kilometers to miles, step down. For example 55 miles (or miles per hour) is approximately 89 kilometers (or kilometers per hour). A race of 5.5 kilometers is approximately 3.4 miles. Since every mathematician either knows or can quickly calculate the first half-dozen or so Fibonacci numbers (by addition), a rough answer is available with no multiplication. Or simply use this factoid as a memory jog for the conversion factor(s).
 * Canadians visiting the United States may encounter a speed limit of 65 miles per hour, which can be converted as
 * 65 = 55 + 8 + 2 → 89 + 13 + 3 = 105.
 * Of course, the result number is memorable for mathematicians as the product 1×3×5×7, and for computer experts because of its special place in radix conversion.
 * 10510 = 6916
 * 6910 = 1058
 * Those who enjoy this kind of trivia may contemplate the observation by Tom Duff that π seconds is a nanocentury, to within half a percent. --KSmrqT 13:34, 24 July 2006 (UTC)


 * The RD should be referencing our encyclopedia, not reproducing it. The article you want is mathematical coincidence.  —Blotwell 00:53, 25 July 2006 (UTC)
 * Although that's a fine article, and worth citing, I'm amusing myself (and a few readers) by answering yet another lame (homework?) question in a way that I hope will inspire mathematical curiosity. As for reproduction, the coincidence article mentions π seconds without a reference to Duff, it mentions the golden ratio connection but doesn't include the "radix expansion", and it fails to mention the notable 105 at all — a shocking omission. :-) --KSmrqT 18:03, 25 July 2006 (UTC)
 * You can help Wikipedia by contributing to articles. Seriously, that's the other thing about the RD: when it identifies omissions in the 'pedia we should rectify them (or we'll get the same questions again).  And I hope I wasn't too brusque: the relation of the Reference Desk to the rest of WP is something I've been worrying about for a while.  —Blotwell 23:33, 25 July 2006 (UTC)
 * Ah, the optimism of youth. Or whatever. We can put information in bold letters at the top of this page and still get questions. (And we do.) We've had many questions about some named topic that could be answered by merely reading the article with that name. And if that doesn't work, often an "obvious" web search will suffice. It threatens to be so boring we might lose the readers who can help with the rare interesting question. Therefore, in the great tradition of the British panto, I try to entertain the adults as well as the children, with material that is often unsuitable for an article. (Strictly mathematical, of course!) I do wonder if this isn't an inferior imitation of a newsgroup, more awkward to search than Google Groups. --KSmrqT 02:46, 26 July 2006 (UTC)
 * It'll be a cold day in hell when the reference desk reproduces the encyclopedia. In the real world, this is will probably happen on a Friday night.

Residue number system
I've been working with the system described in this article for awhile, and I can't figure something out: is it possible to, without resorting to actually computing the numbers themselves and comparing them, determine which of two numbers in this system is larger? (eg 31>20, {1,1,1,3}>{0,2,0,6}) Black Carrot 22:02, 24 July 2006 (UTC)


 * It's always hard to prove a negative, but I don't think there's a good answer here. The reason is that the residue system in reality denotes elements of $$\mathbb{Z}/n\mathbb{Z}$$ and not normal integers, and the concept of "greater than" is not well-defined on such a set.  What you're looking for ($$a>b$$) would have to be stated as "in incrementing a we reach $$[0]_M$$ before we reach b".  You can implement this in $$O(MN)$$ time and without actually calculating the canonical integers for a and b by doing precisely that: increment a (which is equivalent to incrementing each of its residues, of course) and stop when you reach b's representation, or 0.  Whichever you find first determines your answer.  Of course, all you're doing there is determining the canonical value of $$0-a$$ by counting on your fingers.  It's faster (except in coding time, perhaps) to just find the numbers (the algorithm at Chinese remainder theorem is $$O(N\ln M)$$ or so when amortized over many numbers with the same moduli). --Tardis 19:17, 27 July 2006 (UTC)


 * I agree that the idea of "greater than" is a tad more difficult to work with here, but I don't think it's impossible to make something useful of it. For instance: Is A (that is to say, the lowest positive instance of A) greater or less than half M? If it's possible to determine this quickly, it's possible to rank them quickly. Say A is even, and that one of the moduli is 2. If you define B to be A with that particular modulus removed, is B even? If so, A<.5M, else, A>.5M . So, another way to state the problem is, is there a way to determine quickly whether B is even, with no even modulus as a guide? Black Carrot 17:58, 28 July 2006 (UTC)


 * I'm not sure that that's going to work either: with no even moduli, the overall modulus M is odd, and so half of any given equivalence class (even 0s) is even, and half is odd. The whole point of the residue arithmetic is that the residues completely fail to imply which of the equivalent numbers is under discussion, so I don't think they'd yield anything about its evenness either.  The only thing I can think of is to use whatever division algorithm the article mentions and see if 2 divides your number.  But then again that division algorithm may be more complicated and/or slower than all the other ideas!  I will note, in case it's helpful, that your classification scheme works for odd A too: $$A<M/2$$ iff (A is even) = (B is even) (because of course then $$A=B$$). --Tardis 20:00, 28 July 2006 (UTC)


 * Naturally it would work for both, that's why I mentioned it. I'm afraid I don't follow much of your argument, though. Black Carrot 21:18, 28 July 2006 (UTC)