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January 4

conversions

How many gallons is 60,000 liters? What fahrenheit is 500 °C?24.90.204.234 (talk) 01:56, 4 January 2010 (UTC)[reply]

We have a template, {{convert}}, for this kind of question. It returns results in the form Original number Original unit (Converted number target unit). For example, {{convert|60000|l|gal}} produces 60,000 litres (16,000 US gal), and {{convert|500|C|F}} produces 500 °C (932 °F). Intelligentsium 02:00, 4 January 2010 (UTC)[reply]

Thanks a lot.24.90.204.234 (talk) 02:36, 4 January 2010 (UTC)[reply]

Between gallons and liters is somewhat messy. In the USA, an inch is defined as 2.54 centimeters, and a gallon is 231 cubic inches. A liter is 1000 cubic centimeters.

{\begin{aligned}1{\text{ liter}}&=1000{\text{ cm}}^{3}=1000{\text{ cm}}^{3}\cdot \left({\frac {1{\text{ in}}}{2.54{\text{ cm}}}}\right)^{3}={\frac {1000}{2.54^{3}}}{\text{ in}}^{3}\\[8pt]&={\frac {1000}{2.54^{3}}}{\text{ in}}^{3}\cdot {\frac {1{\text{ gallon}}}{231{\text{ in}}^{3}}}={\frac {1000}{2.54^{3}\cdot 231}}{\text{ gallons}}.\end{aligned}}

Now multiply that by 60000 and you've got it. Do not round until after you've crunched all the numbers. Rounding should be the last step. I'm getting about 60000 liters = about 15850.3 gallons. A British or "imperial" gallon is different, I think, and maybe the British actually have inches of a different size. Michael Hardy (talk) 02:42, 4 January 2010 (UTC)[reply]

Well, actually I'm referring to American gallons.24.90.204.234 (talk) 03:40, 4 January 2010 (UTC)[reply]

Everything above is in American gallons and American inches. Michael Hardy (talk) 07:04, 4 January 2010 (UTC)[reply]

Google is your friend: [1], [2]. Of course, knowing how to do the conversions yourself is very important. -- Meni Rosenfeld (talk) 09:13, 4 January 2010 (UTC)[reply]

American and British inches are exactly the same (unless you want the obscure American Survey Inch, if it exists?). The British (Imperial) gallon is one of the few things that are bigger (and better) in England ;) One Imperial gallon = 4.54609188 litres. (Yes, I know we don't have an empire any more!) and one US "dry gallon" is 4.40488377086 liters.Dbfirs 10:22, 4 January 2010 (UTC)[reply]

But people are still being appointed to the Order of the British Empire, funnily enough. -- 202.142.129.66 (talk) 01:40, 11 January 2010 (UTC)[reply]

Cantor's Ordinal Numbers

Last night I was reading a book about some of Cantor's work. After reading the articles here on wikipedia, I had a question. Assuming the axiom of choice, what is the cardinality of the set of all transfinite cardinal numbers? I am thinking it must aleph-zero because we can write them in a sequence

0,1,2,3,\cdots ,n,\cdots ;\aleph _{0},\aleph _{1},\aleph _{2},\cdots ,\aleph _{\alpha },\cdots .

So there are countably infinite of these transfinite cardinal numbers, right? And also just to clarify,

\aleph _{0}=card(\mathbb {N} )

\aleph _{1}=card(P(\mathbb {N} ))

\aleph _{2}=card(P(P(\mathbb {N} )))

and so on, right?-Looking for Wisdom and Insight! (talk) 03:35, 4 January 2010 (UTC)[reply]

Not exactly, in answer to your latter question. Have you read the article on the continuum hypothesis? The article cardinal numbers helps answer your questions. You read it, but did not understand it?Julzes (talk) 04:03, 4 January 2010 (UTC)[reply]

I thought I had it but I guess not. The article says $2^{\aleph _{0}}=\aleph _{1}$ and $2^{\aleph _{\alpha }}=\aleph _{\alpha +1}$ and I thought that for a given set A, $card(P(A))=2^{card(A)}$ . For sure, I know that this is true for finite sets A but I thought that $2^{\aleph _{\alpha }}=\aleph _{\alpha +1}$ was motivated by this. So if $\aleph _{0}=card(\mathbb {N} )$ , then $card(P(\mathbb {N} ))=2^{card(\mathbb {N} )}=2^{\aleph _{0}}=\aleph _{1}$ . Is this not right?-Looking for Wisdom and Insight! (talk) 04:17, 4 January 2010 (UTC)[reply]

Sir (or madam), is this a mathematician's idea of a joke? Surely you are contradicted by the fact that the article does not say what you claim.Julzes (talk) 04:23, 4 January 2010 (UTC)[reply]

I'll assume you breezed over the article for a moment. The power set of one set is proven to have larger cardinality than the set itself, but the generalized continuum hypothesis is known to be independent of ZFC (and the truth or falsity's status is of disputable value for applied mathematics). What this means is that the equality you state depends on the assumption of an hypothesis that can neither be proved true nor false inside or outside of mathematics.Julzes (talk) 04:31, 4 January 2010 (UTC)[reply]

Well, I wasn't joking and that is why I posted this question because I didn't understand it. I posted it here to see if someone can clarify it a bit. I didn't do it to waste mine or anyone else's time. So you are saying that my second question cannot be answered? What about the first? Is that at least correct?-Looking for Wisdom and Insight! (talk) 04:40, 4 January 2010 (UTC)[reply]

Cardinals are weird. Correct, either set theory is uselessly broken or it is independent of ZFC that ℵ₁ is the cardinality of the set of subsets of the integers. Some people do prefer the cardinality of the set of subsets of the integers to be a different aleph number. For every cardinal X, there is a set of cardinals Y such that |Y| is strictly larger than X. In particular, in ZFC there is no "set of all cardinals". You can construct Y fairly easily, Y = { ℵ_k : k is an ordinal such that |k| ≤ |P(X)| }. Note that the aleph numbers do not "stop" at the positive integers, after ℵ₁, ℵ₂, ..., you get ℵ_ω where ω is the least ordinal that is infinite; ω = {0,1,2,...} is the set of non-negative integers (finite ordinals) under the obvious ordering in von Neumann's model of ordinals. The problem with your idea is that the "sequence" you wrote down is transfinite, not countable. JackSchmidt (talk) 04:55, 4 January 2010 (UTC)[reply]

It looks right to me. That appears to me to be a statement about the sizes of sets that Cantor proved prior to a lot of controversy during which the concept of a set was clarified.Julzes (talk) 04:51, 4 January 2010 (UTC)[reply]

(e/c) To be clear, the first question is not correct. There is no set of all transfinite cardinals, and there are sets of transfinite cardinals that are uncountably infinite. JackSchmidt (talk) 04:59, 4 January 2010 (UTC)[reply]

The funny thing is that you can run into Russell's paradox when you try to write a decent sentence answering your first question.Julzes (talk) 05:00, 4 January 2010 (UTC)[reply]

Thanks, JackSchmidt.Julzes (talk) 05:02, 4 January 2010 (UTC)[reply]

Thanks and sorry if I annoyed anyone with my slowness.-Looking for Wisdom and Insight! (talk) 05:35, 4 January 2010 (UTC)[reply]

No problem. To clarify one thing Jack Schmidt seems to say, there is mathematics done under assumptions of where in the aleph scheme the cardinality of the continuum fits (the cardinality of the continuum is 'c' written in Fraktur script and equals the cardinality of the set of subsets of the integers). To clarify an earlier parenthetical by me; in terms of real world applications of mathematics, it so far has not been decided if it matters what one assumes on the subject--and perhaps different assumptions will each find some use. Most mathematicians get along just fine without any of the mathematics around that particular question, with the following quote made by Gerald Folland in his Real Analysis: Modern Techniques and Their Applications probably being typical:

"My own feeling, subject to revision in the event of a major breakthrough in set theory, is that if the answer to one's question turns out to depend on the continuum hypothesis, one should give up and ask a different question."

I'm not sure one also shouldn't also relate what such a question is to those who specialize in the subject area, but you should get the idea.Julzes (talk) 07:24, 4 January 2010 (UTC)[reply]

I have to object to the subtext here, that CH or not CH is a matter of what we assume. There are very serious set theorists trying to find out whether or not it's true. The fact that it can neither be proved nor refuted from ZFC has little bearing on that, though it does certainly indicate that any successful approach to the question will have to go outside the sorts of argument that are familiar to mathematicians.

Folland's attitude is highly practical, given that as he was not doing research in set theory, he was unlikely (though you never know!) to stumble on anything shedding much light on CH's truth or falsity. But you'll notice the subject to revision clause, which is also very appropriate. Some people see the work of W. Hugh Woodin on Ω-logic as the possible start of Folland's "breakthrough", though few if any are bold enough to claim the question is settled. --Trovatore (talk) 07:45, 4 January 2010 (UTC)[reply]

Well, wouldn't you say that the question of whether it is "true" essentially depends on its usefulness? ZFC and ZFC+CH are either both internally consistent or both not. So far, we should probably say that at least they are equally useful and withhold judgement about whether the truth or falsity of CH has any meaning at all without appealing to what it means to what mathematics ultimately models. I personally find the possibility that both "choices" of the truth value have application in some way highly unlikely, but even that can't be determined at the present state of our knowledge, can it? I ask this as a relative ignoramus, so don't take offense. I'll look into the article on omega logic.Julzes (talk) 08:05, 4 January 2010 (UTC)[reply]

No, I wouldn't say that whether it's true depends on its usefulness. The truth or falsity of the statement the apple is red does not depend on whether it's useful for the apple to be red; it depends on whether the apple is red.

Similarly, from a realist perspective, the truth of the statement there exists a wellordering of the real numbers whose every proper initial segment is countable (aka CH) does not depend on whether such a wellordering is useful, but just on whether there is one. --Trovatore (talk) 08:09, 4 January 2010 (UTC)[reply]

Are you saying the real numbers cannot be formed from either ZFC+~CH or ZFC+CH? I'm misunderstanding something in this subject, perhaps. Pardon me if I seem a little tired, but I was simply here originally because a questioner hadn't done some simple basic research very carefully. Perhaps I'm doing the same thing.Julzes (talk) 08:42, 4 January 2010 (UTC)[reply]

I've answered my own question. Second order logic versus first-order logic. New fundamental concepts I hadn't been given any reason to study in the past. No, ZFC requires augmentation with some second order logic to get the reals because of the supremum axiom. Sorry to be so uninformed about your specialty, Trovatore, but it looks like the truth of the continuum hypothesis may be contained in the reals themselves regardless of the axiom scheme, as you said.Julzes (talk) 09:23, 4 January 2010 (UTC)[reply]

Trovatore, when you said "There are very serious set theorists trying to find out whether or not it's true", do you mean it in a realist's perspective? I.e is CH true for the platonic real numbers? I've read that people are trying to figure out new "intuitive" axioms that can be used to derive CH or its negation, for example Freiling's axiom of symmetry. I just got my books for logic and set theory today and hopefully I will understand the von Neumann universe soon. Wish me luck! Money is tight (talk) 17:50, 4 January 2010 (UTC)[reply]

It may be like the P versus NP problem. It may be that an actual proof is found using techniques we haven't dreamed of yet. It may be people just accept it as an axiom because the evidence becomes so great. Or it may just be left for ever as a problem. Dmcq (talk) 18:26, 4 January 2010 (UTC)[reply]

Term by Term Integration of a Fourier Series

On an different note, I understand that if you have a sum of functions converging uniformly to the limiting function, then that series can be (Riemann) integrated term by term. In context of Fourier series, how can you tell when a series can be integrated term by term? Because as I understand it, uniform convergence is just sufficient for term by term integration. It is not necessary. We have examples of series which can be integrated term by term even though they are not uniformly convergent. So IF we have uniform convergence, then we know we can integrate term by term but what if we don't have uniform convergence, then how can I tell if the Fourier series in question can be integrated term by term or not? I can always just find the Fourier series representation of the integral of the function and then see if it matches the term by term integration of the original Fourier series but is there any other (quicker perhaps) way?-Looking for Wisdom and Insight! (talk) 03:43, 4 January 2010 (UTC)s[reply]

There are several convergence theorem in integration theory to do that, of course. However note that Riemann integrability is unsatisfactory, for the reason that it is a quite unstable property with respect to convergence. A limit of a sequence of Riemann integrable functions in a convergence weaker than uniform may fail to be Riemann integrable. This is the main reason why at a certain moment a more general integration theory became needed (in fact, the main problem was exactly the convergence of Fourier series). Here's an elementary result about integration and series: if a series of integrable functions

\scriptstyle \sum _{k=0}^{\infty }f_{k}

satisfies

\scriptstyle \sum _{k=0}^{\infty }\|f_{k}\|_{1}<\infty

then it converges almost everywhere and in the

L^{1}

norm (in particular, you can integrate term by term). Related facs: you may see the exchange sum/integration as a particular case of a change of order in iterated abstract integration and apply the Fubini and Tonelli theorems. --pm a (talk) 11:44, 4 January 2010 (UTC)[reply]

Sums of squares

Hi. I found a spreadsheet on my hard drive that I don't remember making. It must have been a while ago. In it, I've got a column with the natural numbers, a second column with their squares, a third column with partial sums of squares (1, 5, 14, 30, 55, etc.), and a fourth column with the square roots of those partial sums. The only two integers in the fourth column seem to be 1 and 70, hence:

$\sum _{k=1}^{1}k^{2}=1^{2}$ and $\sum _{k=1}^{24}k^{2}=70^{2}$

I wonder: Are those the only two integers I'll ever see in column 4? I've only got ~~1024~~ 4096 rows so far... Thanks in advance for any insight. (Also, any ideas as to what I was thinking when I made these computations would be welcome. o_0) -GTBacchus^(talk) 07:56, 4 January 2010 (UTC)[reply]

There are no more integers in the first million rows. It seems likely that there are no more integers at all, ~~but I don't know how to prove that~~.

In case you didn't already know, the formula

\sum _{k=1}^{n}k^{2}={\frac {n(n+1)(2n+1)}{6}}

may be useful. -- Meni Rosenfeld (talk) 09:27, 4 January 2010 (UTC)[reply]

Actually, the proof should be easy. For large enough n,

{\frac {n(n+1)(2n+1)}{6}}\approx {\frac {2n+3}{6}}n^{2}

, and

{\sqrt {\frac {2n+3}{6}}}

is irrational. The rest is left as an exercise. -- Meni Rosenfeld (talk) 09:41, 4 January 2010 (UTC)[reply]

The only numbers that are simultaneously square and pyramidal (the cannonball problem) are $P_{1}=1$ and $P_{24}=4900$ , corresponding to $S_{1}=1$ and $S_{70}=4900$ (Ball and Coxeter 1987, p. 59; Ogilvy 1988; Dickson 2005, p. 25), as conjectured by Lucas (1875, 1876) and proved by Watson (1918).
— Weisstein, Eric W., Square Number., From MathWorld

--132.230.101.16 (talk) 09:47, 4 January 2010 (UTC)[reply]

I guess it's not as easy as I thought. -- Meni Rosenfeld (talk) 11:10, 4 January 2010 (UTC)[reply]

By the way, no need to go as far as (the obsolescent, IMO) MathWorld. The information is available locally, at Square pyramidal number. -- Meni Rosenfeld (talk) 12:28, 4 January 2010 (UTC)[reply]

Square pyramidal numbers that are also square numbers; that's it. Thanks to all. -GTBacchus^(talk) 17:31, 4 January 2010 (UTC)[reply]

Regarding the other question... Does this spreadsheet look like something that you would have written, but you just don't recall the act of having done so, or does it look completely alien? If the latter, it was probably written by someone else. -- Meni Rosenfeld (talk) 12:28, 4 January 2010 (UTC)[reply]

I don't think anybody but me has used my copy of OpenOffice Calc for number crunching. I'm sure I wrote it, and it may have been after reading something about figurate numbers. The formulas are all written they way I'd write them, but there's not a lot of creativity involved in typing "=sum(b$2..b15)" or something. Hard to say... -GTBacchus^(talk) 17:31, 4 January 2010 (UTC)[reply]

It's a bit off-topic, but that's a tricky age-dependent question involving human memory, and if the user feels certain (s)he neither allowed someone else to use the computer nor did the spreadsheet him- or herself, I would lean toward being mistaken on the latter. Perhaps when you were 13 or 14 years old you wanted to check something you read on the subject, GTBacchus? Or it could have been independent work (People are always rediscovering things). It could have been related to a school assignment also. Unless there is some jarring of your memory (or it occurs to you who else might have been working on the computer), we probably can't help with that question.Julzes (talk) 13:01, 4 January 2010 (UTC)[reply]

I wasn't 13 or 14, because I didn't own a computer then. I've had this one since '06, which is the year I turned 30. It was probably just some random late-night number theory doodling, I suppose. That sort of thing happens around here. Thanks again. :) -GTBacchus^(talk) 17:31, 4 January 2010 (UTC)[reply]

Actually this question has also been posed one year ago; check Wikipedia:Reference_desk/Archives/Mathematics/2009_February_25#Simultaneously_pyramidal_and_square and the links. --pm a (talk) 17:26, 4 January 2010 (UTC)[reply]

One of those links triggered the memory! I was thinking about Squaring the square, no question. I was wondering whether one could have square tiles with edges 1, 2,..., x, and arrange them to cover a square region. Apparently, the only set where you'd have a chance would be if x=24 and the playing board is 70X70. It appears from our Squaring the square article that there's no solution using that set, but that's what I was after. I think it came up when I was teaching high school in Seattle, and cutting out a lot of shapes with my geometry students. I'm glad I remembered that. -GTBacchus^(talk) 17:38, 4 January 2010 (UTC)[reply]

If anyone is able to edit SVG images, File:Squaring the square.svg could do with having the "112 x 112" in the bottom right brought properly into view. (BTW there are also other size PNGs derived from that - is that automatic?) -- SGBailey (talk) 08:20, 5 January 2010 (UTC)[reply]

The original SVG was correct, but it apparently hit a bug in the PNG rendering library (a weird one, I couldn't even reproduce it with a local copy of rsvg). I replaced all the css with regular attributes, which seems to have fixed the problem. — Emil J. 13:16, 5 January 2010 (UTC)[reply]

Rates of decline

Hi, This should be quite a simple problem, but I am having problems figuring out the quickest way to solve it... Basically a business made $10 million in the last per year, and this profit is declining at a rate of 10% per year. Estimate the total sum of profit made by the business over the next 5.7 years....

While it would be possible to solve this problem by working this out year by year (although the 0.7 of a year might pose problems) i was wondering if there was a quicker and easier way to solve it? I am sure there is!!

Many Thanks 80.47.236.247 (talk) 08:10, 4 January 2010 (UTC)[reply]

When you say it is declining at 10% per year, are you talking about an instantaneous rate or an "effective" rate? I.e. will the profit per year one year from now be 90% of what it is now (so it's an effective rate), or is the current instantaneous rate of decline such that that's what it would be if it continued at the same rate (so it's an instantaneous rate)? Michael Hardy (talk) 16:34, 4 January 2010 (UTC)[reply]

Do you know calculus? You could solve it as an integral, assuming the rate of decline is continuous. -GTBacchus^(talk) 08:15, 4 January 2010 (UTC)[reply]

I don't really know calculus very well, however yes the rate of decline is continuous... i.e. 10% down from $10m in the first year = $9m, and then $8.1m in the second year and so on.... could u explain with use of a formula how this therefore might be solved? It would be much appreciated! :) 80.47.137.127 (talk) 08:56, 4 January 2010 (UTC)[reply]

Actually what you've described is not what we consider continuous. In the discrete case, which is what you've described, you can use the formula for the sum of a geometric series. -- Meni Rosenfeld (talk) 09:32, 4 January 2010 (UTC)[reply]

Oh right.. so put simply, could you explain how to compute a 10% per year decline on $10m over 5.7 years? I am still having difficulties getting my head around this! thanks. 80.47.137.127 (talk) 10:02, 4 January 2010 (UTC)[reply]

Just looking at the 5.7 years, it raises the issue, do you earn in .7 years (0.7 * what you would have earned in the whole year) or is the 10% a gradual change over the course of the year? If you do, then you can use the formula half way down Geometric progression, k=0, n=4, r=0.9, a=10 million. Then, work out the sixth term, ar^5, multiply by 0.7 and add for your final answer. - Jarry1250 ^{[Humorous? Discuss.]} 10:31, 4 January 2010 (UTC)[reply]

Of course, it would be natural to look for a continuous decline that matches up with the discrete decline. For an article, you might look at compound interest, but it won't be quite right for your problem. One problem with trying to answer this honestly is that you are simply saying what the problem is and then asking people to work it out for you in detail. Let's get to the bottom of whether the decline is or is not continuous before trying to answer this further.Julzes (talk) 11:42, 4 January 2010 (UTC)[reply]

I think we need to know more about the background to the problem, then we can work out whether it should be modelled as a continuous or discrete decline and what ought to be done about the incomplete year. --Tango (talk) 17:12, 4 January 2010 (UTC)[reply]

jection

Bijection, injection, surjection. What is a jection? Wiktionary has nothing on the etymology. -- SGBailey (talk) 14:00, 4 January 2010 (UTC)[reply]

oed says both bijection and surjection arise from injection, which comes from the latin verb injicere Tinfoilcat (talk) 14:21, 4 January 2010 (UTC)[reply]

Usually spelled inicere, and itself derived from the verb iacere, which gives you the actual root the three -jections share. — Emil J. 15:15, 4 January 2010 (UTC)[reply]

Also related to trajectory, I think. Michael Hardy (talk) 16:23, 4 January 2010 (UTC)[reply]

And I'd conjecture that some other compound of iacio used in maths is here around. We may add "projection" if you have no objection (why should we reject it?) --pm a (talk) 17:35, 4 January 2010 (UTC)[reply]

So it's "two-way throwing", "Throwing in" and "Throwing on"? -- SGBailey (talk) 19:25, 4 January 2010 (UTC)[reply]

Sounds right. -GTBacchus^(talk) 03:19, 5 January 2010 (UTC)[reply]

Adjective, conjecture, ejection, subject. Bo Jacoby (talk) 09:43, 5 January 2010 (UTC).[reply]

"pma"'s puns will be lost on some people, I suspect.... Michael Hardy (talk) 19:29, 5 January 2010 (UTC)[reply]

Well, some people have trouble parsing anything more complex than a simple interjection... -GTBacchus^(talk) 03:50, 6 January 2010 (UTC)/me ducks the inevitable rotten fruit, hopes he's not ejected from the board for it all[reply]

BTW, to make the list complete, I wonder what kind of a map would be a dejection in maths. The Italian word kept a meaning which is somehow closer to the concreteness of mathematical objects, compared with the English meaning. --pm a 11:04, 6 January 2010 (UTC)[reply]

A quotient? A covering map? -GTBacchus^(talk) 18:07, 6 January 2010 (UTC)[reply]

"Covering map" sounds perfect --especially fitting with the Italian term "deiezione" ;-) pm a 14:19, 7 January 2010 (UTC)[reply]

Of course, all these -jections have got adjective forms as well... -GTBacchus^(talk) 18:15, 6 January 2010 (UTC)[reply]