Wikipedia:Reference desk/Archives/Mathematics/2006 September 27

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Math Question

Jody paid $44 bill at the hardware store using a combination of $10, $5, and $1 bills. If she paid with 13 bills in all, how many of each bill did she use?

Let i, j and k stand for the number of $10, $5, and $1 bills, respectively. Then we have 10i + 5j + k = 44 and i + j + k = 13. Subtracting the two (corresponding to the substitution k := 13 − i − j) gives 9i + 4j = 31. Modulo 4 this is i ≡ 3, so substitute i := 4m + 3, giving 36m + 27 + 4j = 31, or 36m + 4j = 4, or 9m + j = 1. Since m and j can't be negative, this is only solved by (m, j) = (0, 1). Backsubstitution gives i = 4·0 + 3 = 3 and k = 13 − 3 − 1 = 9, so (i, j, k) = (3, 1, 9). --Lambiam^Talk 01:23, 27 September 2006 (UTC)

Yo dude, I know that Lambian basically spoonfed you the answer, but make sure you can do this. An A on the homework doesn't make up for an F on a quiz. Representin', --AstoVidatu 02:46, 27 September 2006 (UTC)

Note that there aren't very many ways to get $44 from those 3 denominations, so you could just try all the possibilities, too. Obviously, you can exclude those possibilities with 14 or more $1 bills:

10+10+10+10+ 1+ 1+ 1+ 1
10+10+10+ 5+ 5+ 1+ 1+ 1+ 1
10+10+ 5+ 5+ 5+ 5+ 1+ 1+ 1+ 1
10+ 5+ 5+ 5+ 5+ 5+ 5+ 1+ 1+ 1+ 1
 5+ 5+ 5+ 5+ 5+ 5+ 5+ 5+ 1+ 1+ 1+ 1

10+10+10+ 5+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1
10+10+ 5+ 5+ 5+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1
10+ 5+ 5+ 5+ 5+ 5+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1
 5+ 5+ 5+ 5+ 5+ 5+ 5+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1

StuRat 09:44, 27 September 2006 (UTC)

Use a reasoning like: you need 4 $1 bills to get from $40 to $44 (why?), so the problem reduces to getting $40 from 9 bills. You can have 0 or 5 $1 bills (why?), 0 is impossible (why?), so you are left with $35 and 4 $5/$10 bills. Now it should be obvious.--gwaihir 10:00, 27 September 2006 (UTC)

Plural of 'series'

On the urging of another Wikipedia member, after reversions of my ideas regarding this topic, I have decided to post here and ask your opinions.

In his Principia Mathematica, Isaac Newton, when discussion series, used the latin plural 'seriei' instead of our ambiguous english habit. While the habit of singular/plural duality is tolerated for words of english origin such as 'deer', I do not believe this ambiguity belongs in Wikipedia mathematics articles.

Quite simply, the singular/plural duality of 'series', as we somewhat haphazardly have defined it, is confusing. It would not be as confusing if the word did not end with an 's', however it does. Anyone reading these articles has to do a double take to check the verb when reading an article on a series. I have asked several collegues about this already, and they voiced support to me for a clearer, less ambiguous, more grammatically correct spelling.

I thus propose the changing of 'series' in the plural to 'seriei'. One series, several seriei; again, a distinction made by Isaac Newton and lacking the ofttimes confusing singular/plural duality we have assigned to 'series' at present. We use 'criterion/criteria', and 'nucleus/nuclei', and 'formula/formulae', and 'curriculum/curricula', many of which are not exactly common distinctions made by your average english speaker, yet which go a long way towards the goal of precision and accuracy that we strive for here on Wikipedia. I propose simply extending this customary observance to another latin borrow-word which has its own well-established plural. Dbsanfte 05:19, 27 September 2006 (UTC)

I think there must be some mistake here. In Latin, the plural of series is series, and we preserve this in English. Seriei is the genitive singular. Latin nouns ending in -es do not form their plurals in -ei. Maid Marion 07:46, 27 September 2006 (UTC)

Now I feel absolutely silly. You're correct, I misread my dictionary. Please disregard this. Dbsanfte 13:03, 27 September 2006 (UTC)

Consider it thoroughly ignored. Black Carrot 14:37, 27 September 2006 (UTC)

But just to indulge in a little bit of hypothesis, if "seriei" were the Latin plural, do you seriesly (error intended) think that people of today would catch on to either the spelling or whatever the pronunciation of "seriei" is? I rather doubt it (but than, I'm a serial doubter). JackofOz 20:50, 27 September 2006 (UTC)

In general it is not incorrect to pluralize a word with S in English. There are words with latin roots which have alternative pluralization. You should find both "nebulae" and "nebulas" in your dictionary, for instance. - Rainwarrior 04:21, 29 September 2006 (UTC)

What is this question called?

I heard this question a long time ago, but I have forgotten what it is called. Suppose you have 2 types of stamps, one with a denomination of p dollars, and another of q dollars. What is the largest amount that cannot be made with a linear combination of them? The solution is (p-1)(q-1)-1 if I recall, but I'd like to find more background about this. Thoughts? --HappyCamper 17:12, 27 September 2006 (UTC)

Coin problem. Chuck 17:28, 27 September 2006 (UTC)

Largest Number Contest.

What is the largest number you can create using 38 characters? Use any notation.- R_Lee_E (talk, contribs) 17:56, 27 September 2006 (UTC)

9!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

(an iterated factorial, exclamation points are small) –Joke 18:00, 27 September 2006 (UTC)

I interpret this as saying that the expression must have 38 chars, so you may be cheating. I'm not sure what is the best way to use Conway chained arrow notation, but I'll place my bets on:

9→9→9→9→9→9→9→9→9→9→9→9→9→9→9→9→9→99

-- Meni Rosenfeld (talk) 18:10, 27 September 2006 (UTC)

A word of caution is in order here. This is a really, really large number. For example, the factorials above are, if I'm not mistaken, much less than 2→4→3. And this grows unimaginably large for longer chains. -- Meni Rosenfeld (talk) 18:24, 27 September 2006 (UTC)

That number gives me headaches :-) --HappyCamper 18:26, 27 September 2006 (UTC)

Better B(B(B(B(B(B(B(B(B(B(B(B(B(B(B(B(B(B(9)))))))))))))))))) where B(n) is the Busy beaver function. Gives a much larger number than that with the chained arror notation. (Even only 4 or 5 or busy beavers will almost certainly surpass the chained arrow number given above. This number should give even worse headaches. Note that it is arguably not in the same category as the above numbers since it is uncomputable where one could at least in principal compute the other numbers. JoshuaZ 18:33, 27 September 2006 (UTC)

To the contrary, B(...(B(9)...) is just one natural number and is thus trivially computable. The function B(...(B(n)...) would not be computable, but this is just one value. CMummert 02:33, 2 October 2006 (UTC)

This is a pretty big cardinal number:

$2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{\aleph _{\omega }}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}$

But probably someone can write a bigger one. –Joke 18:53, 27 September 2006 (UTC)

Interesting. The rules didn't specify the number has to be finite. Here is a large ordinal number (though if I'm not mistaken, it's less than the ordinal matching the cardinal above):

$\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\epsilon _{\omega }}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}$

Now for a question of my own. Joke's cardinal can be rewritten with Beth number notation:

$\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\beth _{\aleph _{\omega }}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}$

I've heard once of a Gimel number notation, which potentially could be used to write even larger numbers, but am unable to find any information about it. Has anyone ever heard of such a notation? -- Meni Rosenfeld (talk) 19:20, 27 September 2006 (UTC)

(Edit conflict.) According to Beth number, we can rewrite the above as

$\beth _{38}(\aleph _{\omega })$.

A much larger number would be

$\beth _{9\rightarrow 9\rightarrow 9\rightarrow 9\rightarrow 9\rightarrow 9\rightarrow 9\rightarrow 9\rightarrow 9\rightarrow 9\rightarrow 9\rightarrow 9\rightarrow 9\rightarrow 9\rightarrow 9\rightarrow 9\rightarrow 9\rightarrow 9}(\aleph _{\omega })$

But we're starting to get silly. –Joke 19:40, 27 September 2006 (UTC)

Yes. That number is no joke. - R_Lee_E (talk, contribs) 21:36, 27 September 2006 (UTC)

gimel(κ) is sometimes used to mean κ raised to the power cof(κ). I don't think it necessarily blows up hugely, but it is always larger than κ. As I recall the singular cardinal hypothesis can be stated as gimel(&kappa) = κ⁺ for every singular κ. --Trovatore 19:37, 27 September 2006 (UTC)

Unfortunately, this question gets a bit silly when we realise that there is probably nothing stopping me defining a function T(n) to be B(B(B(n))) (B = Busy beaver function) and so on ... - so why do we not make it a bit more of a challenge: what is the biggest number you can create within the 38 character limit if we restrict ourselves to the symbols: 0-9, +, -, x, /, (, ) i.e. no exponentials and no factorials. Any takers? Madmath789 20:14, 27 September 2006 (UTC)

Yes, but the idea is to only use standard notation, or at least one which is defined somewhere other than this discussion. Besides, under the rules you present the answer is trivially 99999999999999999999999999999999999999. -- Meni Rosenfeld (talk) 20:32, 27 September 2006 (UTC)

Wondered how long it would take to spot that :-) Madmath789 20:40, 27 September 2006 (UTC)

Allow me to cheat: $1/0$ —Bromskloss 09:32, 28 September 2006 (UTC)

Nope, 1/0 is, at best, an unsigned infinity, which isn't larger than any of the other numbers suggested here. -- Meni Rosenfeld (talk) 09:43, 28 September 2006 (UTC)

Now the only problem is determining a winner. 207.233.9.240 20:58, 27 September 2006 (UTC)

Am I allowed to use non-notable notation? User:Salix alba/Jonathan Bowers defines some truly hugh finite numbers. --Salix alba (talk) 22:01, 27 September 2006 (UTC)

Please! No!! we had enough of a problem a few weeks ago with a large number of articles he created ... Madmath789 22:05, 27 September 2006 (UTC)

Well, if you don't require the axiom of choice, we could always go with Reinhardt cardinal, which is 18 characters and vastly larger than anything else mentioned here (or derived from the processes mentioned here). For a boggling array of very big things, see list of large cardinal properties. -- Fuzzyeric 03:07, 28 September 2006 (UTC)

If we want to stay within finite numbers, I would use Steinhaus–Moser notation in conjunction with Conway chained arrow notation:

9→9→9→9→9→9→9→9→9→9→9→9→9→9→9→9→9→9 all inside a polygon with the same amount of sides (since that polygon could be drawn as a single symbol, technically it would only count as one ^_^)

--Ķĩřβȳ♥ŤįɱéØ 06:38, 28 September 2006 (UTC)

Bah... My best guess would've been B(B(B(B(B(B(B(B(B(B(B(B(G^G)))))))))))) ☢ Ҡi∊ff⌇↯ 23:06, 28 September 2006 (UTC)

If we aren't limited need to use mathematical notation strictly, len(π) or len(${\sqrt {2}}$) or even len(1/3) in an infintely powerful computer would return the number of digits in a recurring/irrational/transcendental number (ie infinity). smurrayinchester^{(User), (Talk)} 23:20, 29 September 2006 (UTC)

That's only $\aleph _{0}$. We've already done way better than that. -- Meni Rosenfeld (talk) 08:39, 30 September 2006 (UTC)

For any number n there is a notation in which the symbol Q stands for n and thus n is expressible in one symbol using this notation. Similar problems appear in Kolmogorov complexity - for any string, there is some compression algorithm (like PKZIP, but different) that reduces the string to a single character. CMummert 02:28, 2 October 2006 (UTC)

What about (The largest number defined here)+1? (R_Lee_E wrote "Use any notation". And yes, I've heard the name Gödel.) JoergenB 16:02, 2 October 2006 (UTC)

Actually, the largest number defined here is $\beth _{9\rightarrow 9\rightarrow 9\rightarrow 9\rightarrow 9\rightarrow 9\rightarrow 9\rightarrow 9\rightarrow 9\rightarrow 9\rightarrow 9\rightarrow 9\rightarrow 9\rightarrow 9\rightarrow 9\rightarrow 9\rightarrow 9\rightarrow 9}(\aleph _{\omega })$, an infinite cardinal, which of course absorbs addition with natural numbers. So your "new" number is, in fact, equal to it. -- Meni Rosenfeld (talk) 17:20, 2 October 2006 (UTC)

Math Composer ThIng

Does anyone here know of a Windows program that can be used to write out math problems and such. I'm speaking of an application like MS Word only for math and numbers. Hope I'm being specific enough. Deltacom1515 20:53, 27 September 2006 (UTC)

I'm not sure, but LyX is quite good. --HappyCamper 21:02, 27 September 2006 (UTC)

MS Word will actually work for many purposes. In MS Word (I'm assuming MS Word 2003 here; I don't know whether the feature existed in earlier versions, or if so, how good it was), go to Insert - Object... - Microsoft Equation 3.0. The interface takes a bit of getting used to, but you can do quite a bit as far as writing out mathematical formulae. Chuck 21:49, 27 September 2006 (UTC)

I would definitely go with a version of LaTeX, such as the freely available MikTeX. The Word equation editor is OK for fairly trivial uses, but for anything of any complexity or of any length, you really need power and flexibility of TeX. Madmath789 21:53, 27 September 2006 (UTC)

Thanks guys. Question answered. Deltacom1515 00:20, 28 September 2006 (UTC)

Late to the party... I use OpenOffice. Much like Word... Insert|Object|Formula. -- Fuzzyeric 03:10, 28 September 2006 (UTC)

Solids

Is there a name for this solid? It is a square prism with rounded corners, but the top and botton are rounded also

What about this one, an oval prism also with rounded top and bottom. Oblate? Prolate? Maby there is no name for these.

I actually need this info for an article. Herostratus 21:24, 27 September 2006 (UTC)

These are not shapes for which there are standard geometrical names. I've seen "pillow" used for shapes somewhat similar to the first.  --Lambiam^Talk 22:25, 27 September 2006 (UTC)

Rounded rectangular prism? Rounded cylinder? (Or whatever an oval cylinder is called.) - Rainwarrior 04:35, 28 September 2006 (UTC)

Rounded eliptical cylinder perhaps? - Rainwarrior 04:39, 28 September 2006 (UTC)

My best guess is the union between a superellipsoid and a cylinder ☢ Ҡi∊ff⌇↯ 04:51, 29 September 2006 (UTC)

Clouds can form in shapes like these, they are known as lenticular clouds. If you were defining a name for these, you could call these lenticular solids. Dysprosia 08:49, 29 September 2006 (UTC)

Thanks to all for your replies. Herostratus 07:22, 30 September 2006 (UTC)

Soap? – b_jonas 13:18, 1 October 2006 (UTC)