Wikipedia:Reference desk archive/Mathematics/2006 October 16

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What calculations were aided by John Napier's tables of logarithms?[edit]

From what I've been able to gather, these tables were very incomplete, offering only a small percentage of what would have made up a complete and adequate set (given the base, etc.). Yet histories state that they were instantly very useful for all kinds of important calculations. So, what is an actual example of calculations for which Napier's tables were complete and adequate and used? I'm guessing that either

  • I've misunderstood what I've read, either mathematically or historically, and in fact Napier's tables were complete enough to do the whole range of calculations for which people used Briggsian tables before calculators.
  • There was a specialized kind of navigational (etc.) problem that determined the range of numbers Napier offered.
  • In fact the histories are wrong to say they were useful, and Napier produced his tables merely as a demonstration of what logarithms could accomplish, should others continue with much more industry to make much more complete tables.

Wareh 02:43, 16 October 2006 (UTC)

Napier's table is organized in such a way that it specifically aided in the calculations required for celestial navigation using the formulas of spherical trigonometry. For more details, see our article Prosthaphaeresis. His 1616 Admirable table presents in columnar form, for α going from 0° by increments of 1' to 45°, the values of α, sin α, Log(sin α), Log(tan α), Log(cos α), cos α, 90°–α (so effectively his table goes to 90°). Here Log(_) stands for Napier's Logarithme, which is to the base of exp(–10–7). At the time, computations for celestial navigation presented the most pressing need for easier multiplication, and Napier's table fulfilled this need. His particular organization was definitely easier for this specific application, although less convenient for general, arbitrary multiplication. It is clear from the accompanying text that Napier was aware of this.  --LambiamTalk 11:00, 16 October 2006 (UTC)
Oh, how cool it would have been if celestial navigation was actually to navigate from star to star! :-) —Bromskloss 12:42, 16 October 2006 (UTC)
You want astrogation. But, then again, who doesn't? --Tardis 15:22, 16 October 2006 (UTC)
Thanks for another fine answer, Lambiam. I would love to see someone as informed as you add a couple of sentences at Logarithm#History pointing to Prosthaphaeresis, and an example at Prosthaphaeresis of a calculation using the combined resources of that method and Napier's tables. But of course I am content just to have my curiosity satisfied on this point. Wareh 15:33, 16 October 2006 (UTC)

a?[edit]

So in my math class, we're doing factorials. One of the homework problems was "a?" My friend says it means a wants to be n. Nobody knows what it actually means. Therefore, I ask you; WTF does that mean? I'm not asking for the problem to be solved, just for someone to tell me what the question mark is supposed to be.

Thank you. 64.198.112.210 15:34, 16 October 2006 (UTC)

The Minkowski's question mark function uses the question mark symbol, but not like that. I doubt that subject would show up in your math class, anyway. My guess is that it's just a typo. ☢ Ҡiff 15:52, 16 October 2006 (UTC)
Perhaps if you can provide more context as to where and how it appears, we might figure out the intention. -- Meni Rosenfeld (talk) 21:18, 16 October 2006 (UTC)
If it was supposed to be a factorial, that is, a!, that can be expressed as a! = 1×2×...×(a-2)×(a-1)×(a), which is the definition of the factorial. --ĶĩřβȳŤįɱéØ 07:23, 17 October 2006 (UTC)
My guess is that it was typed by someone not so knowledgeable or not paying much attention. For example, the phrase "how much is a!?" may sound inappropriate for a test if the person is ignoring the fact the "!" is part of the notation of the exercise. So the typist could have just removed the "!" thinking it would make the text more "formal" (since "!?" makes it look like an overly enthusiastic question) ☢ Ҡiff 07:38, 17 October 2006 (UTC)

Combinatorics -- the "obnoxious geek" problem[edit]

This was motivated by an annoying guy in my class I hope I never have to work with again. Here's the situation, and it could be generalized, but I'll just wonder about my specific case:

In my class of 16, we've been put into 4 groups of 4 for team projects. We'll be put into different groups of 4 a few more times throughout the semester. My question is, how many partitions of the class into groups of 4 can be made such that any 2 individuals are in the same group at most once?

The best case scenario would be 5 partitions, in which case everyone would have been in a foursome once with every other classmate. It wasn't too hard to sketch out a way to get 3.

Part of my question is even how to understand this problem or put it into rigorous terms. Does it have another name? My initial naive approach is to phrase is something like how many copies of "4 x the complete graph on 4 vertices" can the complete graph on 16 vertices be broken into, but it seems like there must be something easier.

Most importantly, can I be assured I won't have to work with annoying guy for at least 3 more iterations of these classroom teams?

Thanks! - 142.110.227.162 16:05, 16 October 2006 (UTC)

Which of these do you mean to ask ?
  • How many groups of 4, out of 16, can be formed without two SPECIFIC members being in the same group more than once ?
  • How many groups of 4, out of 16, can be formed without ANY two members being in the same group more than once ?
In the first case, you would also have to define how "different" each group needs to be. That is, can they have 3 out of 4 members the same as from the last time ? Or 2 out of 4 ? StuRat 16:42, 16 October 2006 (UTC)

I guess I wasn't clear. I want to look at "partitions" (the act of dividing the whole class into 4 groups of 4) as a whole, and consider the largest set of partitions where any group of four contains any two individuals at most once.

i.e. what's the most number of times I can partition the class before somebody, anybody, is in the same group with a person from a previous partition? (as you can see part of my problem is with notation) 142.110.227.162 17:05, 16 October 2006 (UTC)

I think three is probably it. It is easy to see that by relabelling the students, the first two partitions can always be written (1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16) and (1 5 9 13)(2 6 10 14)(3 7 11 15)(4 8 12 16). There are 16×3×2 labellings that will give these for the first two partitions (16 ways to choose 1, 3 ways to choose 2, 2 ways to choose 3, 1 way to choose 4 through 16). In particular, you can permute your labelling of 1 2 3 4. I think you can show that these 4×3×2 relabellings are sufficient to put the third partition into the form (1 6 11 16)(2 7 12 13)(3 8 9 14)(4 5 10 15). It is easy to then prove that no additional partition is possible. Probably a cuter proof is, though. –Joke 20:40, 16 October 2006 (UTC)

For dividing N people into n m-groups (so ), you certainly can't do better than since you drain the pool of possible partners by each time. (In your case, , so that maximum is , as you said.) I think Joke is right that you can't get this maximum in your problem, but if are prime, I'm pretty sure you can. Start by writing the objects out in a square matrix: . Then form one partition by rows and n more by considering diagonals with various "slopes": 0 is {{Andy, Mark, ...}, {Sue, Louis, ...}, {Roger, Alice, ...}, ...}, 1 is {{Andy, Louis, ...}, {Sue, Alice, ...}, ...}, and 2 is {{Andy, Alice, ...}, ...}. You won't ever get any repeats until you have a slope of n, since n is prime. So you get total partitions, as required. This isn't relevant to your original question, of course, and I don't have a formula in general (it seems to be a harder version of the derangements problem), but I thought it was interesting that the maximum is sometimes obtained and (evidently) sometimes not. As an example, the partitions formed for from the set {0...9,'A'...'O'} is
{01234, 56789, ABCDE, FGHIJ, KLMNO},
{05AFK, 16BGL, 27CHM, 38DIN, 49EJO},
{06CIO, 17DJK, 28EFL, 39AGM, 45BHN},
{07EGN, 18AHO, 29BIK, 35CJL, 46DFM},
{08BJM, 19CFN, 25DGO, 36EHK, 47AIL},
{09DHL, 15EIM, 26AJN, 37BFO, 48CGK}
Hope this helps. --Tardis 23:25, 16 October 2006 (UTC)

Thanks for playing around with this and for your observations! (the obvious questions start popping up ... for a maximal n by n partition system, is n prime a necessary condition? for n m-groups, is the case n > m too easy and n < m too hard as far as achieving the maximum? I guess for n < m that's obvious by the pigeonhole principle.... are all maximal partitionings isomorphic? for that matter, are all partitionings isomorphic?) 68.144.123.4 01:09, 17 October 2006 (UTC)