Wikipedia:Reference desk/Archives/Mathematics/2008 February 9

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February 9

Centre of gravity of a country and capital cities

Is there a standard, generally accepted, method of determining the centre of gravity of an irregular shape such as that of most countries (as defined by their internationally recognised borders)? How would this work if we have to take far-distant offshore island components (e.g. Hawaii in the case of the USA), non-contiguous components (e.g. Alaska), and exclaves, into account? Has any research been done on the capital cities that are furthest from or nearest to their country's centre of gravity (however that might be calculated)? Thanks. -- JackofOz (talk) 00:19, 9 February 2008 (UTC)

There is a standard definition of the centre of gravity of a body, yes. The problem is that the natural definition of centre of gravity will give you something inside the earth, which is presumably not what you're after (as an extreme example, if your country covered the entire earth's surface, its centre of gravity would have to be the centre of the earth). This problem shouldn't be too extreme for actually existing countries, especially reasonably small ones: you could just take the point on the earth's surface closest to the true centre of gravity. I know very little about computers, but I suspect that with one you could very easily calculate the centre of gravity of any given country, with or without exclaves and such. I have never seen this concept elsewhere (except in the motel in the centre of America featured in American Gods) and am unaware of any research into centres (centroids?) of countries, distance from capitals, etc. Algebraist 00:53, 9 February 2008 (UTC)

I have seen many claims of a particular town being the "geographical center" of a country. How they've arrived this I don't know. --Kvasir (talk) 06:43, 9 February 2008 (UTC)

From that disambig and the linked articles, it seems to be standard to use the centroid=centre of mass, but unfortunately none of these pages explain how they get around the problem that the earth isn't flat. Algebraist 10:33, 9 February 2008 (UTC)

Yes, I noted that too. Presumably what they're actually calculating is the centroid of a 2-dimensional map projection of the country, rather than its real 3-dimensional surface. And, as we know, map projections, while useful, are inherently erroneous because it's not possible to accurately map 3 dimensions onto 2. The best we can do is minimise the error. When it comes to relatively small countries, eg. Switzerland, the variation from the "true" centroid would be small and probably insignificant for most purposes; but with the Russias, Canadas, Brazils and Australias of the world, it would be a different story. That's in absolute distance terms; but when compared with the extreme internal distances in such countries, it would still be relatively insignificant. Wouldn't it? -- JackofOz (talk) 10:48, 9 February 2008 (UTC)

That'll depend on what projection you use. I expect you can get the errors small by choosing the right projection, I'm just not sure what the right projection is. Algebraist 11:59, 9 February 2008 (UTC)

Can't you just find the centre of gravity inside the Earth, draw a line through it and the centre of the Earth and take the point where it intersects the surface as the centre of the country? --Tango (talk) 13:38, 9 February 2008 (UTC)

That's what I suggested above. What I don't know is whether anyone is actually using it. Like JackofOz, I suspect not. Algebraist 14:27, 9 February 2008 (UTC)

OK. Thanks for the comments so far. Here's a slightly different question: In front of me is a map of Afghanistan. I have no issues with whatever projection was used; I'm just interested in the irregular shape I've got. How do I go about working out its centroid? Algebraist suspected above that with a computer I could very easily calculate this. How, exactly? -- JackofOz (talk) 23:28, 10 February 2008 (UTC)

Have you tried reading Centroid? There is a formula in there - for an irregular shape like a country you'll end up doing numerical integration. --Tango (talk) 23:40, 10 February 2008 (UTC)

I don't know how to do it with a computer, but I can do it with a piece of string. Take a cardboard cut-out of the country, and suspend it from a point on its edge such that it can swing freely. Attach the string to the same point, and weight down the end of the string. Then, trace the string onto the cutout - the centre of mass will be somewhere on this line. Repeat the process, suspending the cutout at a different point, and the intersection of the lines will be the centre of mass. Confusing Manifestation(Say hi!) 23:41, 10 February 2008 (UTC)

That works. Or you can just balance it in the tip of a pencil, moving it until it balances, or, even better, rest it on 3 pencils (you'll probably need a friend to help) and move the pencils towards each other without touching the cardboard. Because of the distribution of weight between the 3 pencils, it should (if memory serves) work out that you end up with the 3 pencils all at the centre of mass, since the ones nearer the centre will have more weight on them, so will move less. (I've only seen it done in the 2D case, but I think the 3D case works as well...) --Tango (talk) 18:48, 11 February 2008 (UTC)

Algebra

We have an equation. Let's say it's $x^{2}-y^{2}=1$ . If we reduce this equation, we come out with $y=x-{\sqrt {1}}$ . Now, if we plug a number in for x, let's say 2, we get $y=2-{\sqrt {1}}=1$. Now, if we substitute back into the original equation, it comes out $2^{2}-1^{2}=1$. This is obviously incorrect for the fact that $2^{2}-1^{2}=3$. Would someone please explain to me why this does't work? Thanks, Zrs 12 (talk) 02:13, 9 February 2008 (UTC)

You have rearranged $x^{2}-y^{2}=1$ to give $y^{2}=x^{2}-1\Rightarrow y={\sqrt {x^{2}-1}}$, which is not equal to $x-{\sqrt {1}}$ if $x\neq 1$. -mattbuck (Talk) 02:17, 9 February 2008 (UTC)

Oh, okay. It's because ${\sqrt {x^{2}-1}}\neq {x}-{\sqrt {1}}$ if $x<1<x$? Zrs 12 (talk) 02:32, 9 February 2008 (UTC)

If $x<1<x$, then God didn't make little green apples and it don't rain in Indianapolis in the summertime. --Trovatore (talk) 03:30, 9 February 2008 (UTC)

$x<1<x$ means $x<1$ and $1<x$, which is never true. So your statement is vacuously true, but not the same as mattbuck's. -- BenRG (talk) 05:53, 9 February 2008 (UTC)

More generally, the reason is that you can't distribute the square root operation over addition to each term underneath the square root. In other words, ${\sqrt {a+b}}\neq {\sqrt {a}}+{\sqrt {b}}$. --Kinu ^t/_c 02:56, 9 February 2008 (UTC)

You should say $x>1$ or $x<1$. Visit me at Ftbhrygvn(Talk|Contribs|Log) 07:36, 9 February 2008 (UTC)

Yes, that was one thing POUNDED into me in Algebra II. YOU CANNOT SQUARE ROOT TERM BY TERM. Gah, I'll never forget that. Just kidding, Thanks Mrs. Harvey. Hpfreak26 (talk) 22:59, 12 February 2008 (UTC)

All maths should be done in rings of characteristic 2! --Tango (talk) 23:42, 12 February 2008 (UTC)

Yep, that's what I always say. Students everywhere will no longer live in fear of a mistake involving a stray minus sign. -- Meni Rosenfeld (talk) 17:53, 13 February 2008 (UTC)

Do some countries have different order of operations

or is it the same in all parts of the world.

also, do some parts of the world use math that is 'false' im a different part of the world and vice versa, like one ton plus one ton doesn't equal two tons? —Preceding unsigned comment added by 212.51.122.26 (talk) 11:31, 9 February 2008 (UTC)

While different countries could conceivably have different conventions, basic arithmetic is pretty absolute. 1+1=2, wherever you go. You just have to take one bean and another bean, put them in a pot together and count them and you'll find you have 2 beans - there's no convention involved in that. The order of precedence of operations is purely convention, and different countries could teach it differently, I don't know of any that do, though. --Tango (talk) 13:44, 9 February 2008 (UTC)

Thank you. But your bean example implies that 'reality' makes sense everywhere, not only in Western countries. But isn't the absolute, objective reality you describe -- wherein you can be sure there will be two beans in a pot after adding one to an empty one followed by another one without taking out the first -- limited to the Western tradition???? Surely it's not the same in a Buddhist temple...? —Preceding unsigned comment added by 212.51.122.26 (talk) 13:54, 9 February 2008 (UTC)

Not really. Reality is reality. 1+1=2 in your bedroom, on Mars and indeed in a Buddhist Temple. -mattbuck (Talk) 14:00, 9 February 2008 (UTC)

There is perhaps a possibility for cultural differences. The statement "1+1=2" requires a concept of "oneness" and "twoness" as abstract entities separate from any physical meaning. For example, I might have two books on a shelf, but that's two books; and I might have two beans in a pot, but that's two beans. In our culture we have developed an abstract idea of "twoness" which is shared by the two books on the shelf and the two beans in the pot, but which does not require beans or books or in fact any physical manifestation. We are able to use this concept to express general statements such as "1+1=2" which can be interpreted for books or beans or whatever, rather than having to make individual statements such as "one book and one book make two books" or "one bean and one bean make two beans".

This abstraction is so fundamental to us that most of us don't even recognize it. But there are some languages in the world (I think Japanese and some Polynesian languages, for example) in which the way you say "two books" is quite different from the way you say "two beans". This is because these languages developed from roots that came before people learned to abstract the concept of "twoness" away from real physical objects. —Bkell (talk) 15:15, 9 February 2008 (UTC)

Japanese has a system of counters which I think was borrowed from Chinese. English also has nouns that require counters, like "clothing". You can't have two clothings, but you can have two articles of clothing; "article" is the counter. I suppose this distinction (between mass nouns and count nouns) reflects some aspect of human psychology, but I'm darned if I can imagine what it is. -- BenRG (talk) 15:39, 9 February 2008 (UTC)

We also have several words for "twoness" in English, with rules for which word is correct depending on context. You can have a brace of pheasants or a brace of ducks, but not a "brace of pigs". You can have a couple of minutes or a pair of shoes, but not a "pair of minutes" or a "couple of shoes" (unless you mean two shoes from different pairs). The word "pair" can also refer to just one object, as in a pair of trousers, a pair of scissors, a pair of tweezers or a pair of glasses. However, I don't think these peculiarities of language cause English speakers to have any confusion about the concept of "twoness". Gandalf61 (talk) 15:42, 9 February 2008 (UTC)

If you don't have a concept of "twoness" then you can't have a concept of arithmetic, so the question is pretty redundant. I know there are cultures which only have names (and even concepts) for small numbers ("1,2,3,4,many") - I would assume they don't really have a well developed idea of addition. --Tango (talk) 18:38, 9 February 2008 (UTC)

Order of operations is a concept from grade-school arithmetic that doesn't have any exact counterpart in (higher) mathematics. I wouldn't be surprised at all if the rules are different in different places. Where I come from they teach that A × B ÷ C × D means ((A × B) ÷ C) × D, but a mathematician is more likely to interpret $AB/CD\,\!$ as ${\frac {AB}{CD}}$. In grade-school arithmetic A ÷ B ÷ C means (A ÷ B) ÷ C, but a mathematician would never write $A/B/C\,\!$; it's a meaningless string of symbols unless the / is meant to stand for something other than division. To a mathematician A + B − C + D doesn't mean ((A + B) − C) + D, it means A + B + (−C) + D, and you can do the operations in any order because addition is associative. In fact the whole concept of "doing operations from left to right" is almost never found in mathematics. Lambda calculus is the only clear exception that comes to mind. The set difference operator might count, but I don't know how often it's chained that way.

Regarding whether 1+1=2, it's important to understand that concepts like 1 and addition began as abstractions of certain things in the real world. They aren't the right concepts for everything. If you combine "one" of something with "one" of something and don't get "two" of something, that doesn't mean there's something wrong with the idea 1+1=2, it means you tried to apply it where it (evidently) doesn't apply. But these concepts are useful in surprisingly many situations. They've been independently rediscovered many times by different civilizations. They're useful all over the world because the laws of physics are the same all over the world. In fact the laws are the same not just all over the world, but for billions of light years in every direction and billions of years into the past, for reasons no one understands. -- BenRG (talk) 15:24, 9 February 2008 (UTC)

On the subject of 'doing operations from left to right', you might find reverse Polish notation interesting. Algebraist 15:59, 9 February 2008 (UTC)

Many older, and some modern texts indeed use $AB/CD\,\!$ as ${\frac {AB}{CD}}$. However, with the advent of computer algebra systems which rigorously follow a set order of operations, it seems to me that a lot of people try to avoid use of things such as $AB/CD\,\!$ because it’s ambiguous in that it actually means ${\frac {AB}{C}}\cdot D$, but many times that’s not the intention. GromXXVII (talk) 16:10, 9 February 2008 (UTC)

I know Mathematica parses A B / C D as (A B / C) D, but I don't think that's what A B / C D means. I think A B / C D means "A times B over C times D", and has the same ambiguity as the English phrase (and for the same reason). I'm not even sure mathematicians want this kind of thing to be unambiguous. Formal precision distracts from the interesting aspects of a problem. If they wanted to be exact in every detail they'd be computer programmers. -- BenRG (talk) 14:39, 10 February 2008 (UTC)

Those are bad examples - multiplication and division commute with each other and are associative, as are addition and subtraction, as they're just inverses (addition and subtraction are the same operation, just one of them uses the negative of the number). It's when you mix them that it becomes ambiguous. "1 + 2 * 3" could be any number of things, by convention, it's taken to mean "1 + (2 * 3)". That applies as much in school as it does in higher level maths - the only difference is mathematicians usually just use juxtaposition to denote multiplication, so it becomes much clearer which operations take precedence, "x+yz" clearly means "x + (y * x)", you wouldn't read it as "(x + y) * z", even if you hadn't learnt BODMAS. --Tango (talk) 18:38, 9 February 2008 (UTC)

I don't know what it means to say that two binary operations commute with each other. The unary operations (x ↦ x × a) and (x ↦ x / b) commute, but (x ↦ a × x) and (x ↦ b / x) do not. Division and subtraction are not associative: (3 − 2) − 1 ≠ 3 − (2 − 1).  --Lambiam 00:51, 10 February 2008 (UTC)

The key point is that division and subtraction aren't really binary operations in their own right, they are simply inverses of multiplication and addition. "subtracting x" just means "adding -x", viewed that way, it clearly commutes with addition since it is addition and addition is commutative. --Tango (talk) 13:25, 10 February 2008 (UTC)

Also, division isn't a binary operator at all outside of arithmetic (and programming languages). In typeset formulas the division slash is only a vertical-space-saving variant of the horizontal bar. There's no general rule governing how much of the formula on each side is gobbled up by the slash. Ambiguous cases like "1/2 (x + y)" are pretty common in practice. I'd probably guess that that means "(1/2) (x + y)", and I'd probably guess that "x/2y" means "x/(2y)". Google Calculator parses "1 km / 2 km" as "(1 km) / (2 km)" but "1 / 2 km" as "(1 / 2) km". -- BenRG (talk) 14:39, 10 February 2008 (UTC)

What is division outside arithmetic? It's an arithmetic operation. Don't get hung up on the notation. ${\frac {ab}{cd}}$ means (ab)/(cd) - a horizontal bar has implied brackets. --Tango (talk) 20:27, 10 February 2008 (UTC)

Well, my examples were meant to illustrate something different, but I think even in yours there's a big difference between BOMDAS and precedence. Given a + b + c · d, I think BOMDAS would say to multiply c and d first, but precedence just says that it means a + b + (c · d), and you can perfectly well start by adding a and b. -- BenRG (talk) 14:39, 10 February 2008 (UTC)

BODMAS is the rules of precedence of arithmetic operations. What order you do parts of the sum that are independent of each other obviously doesn't matter, it's when they depend on each other that it matters. If your sum is "(a+b)+(c.d)" then obviously it makes no difference if you do the (a+b) or the (c.d) bit first, as long as you do them both before the addition in the middle. If you did the middle addition before the multiplication, you would get the wrong answer. (The fact that (a+b)+(c.d)=a+(b+(c.d)) is simply associativity of addition, which had nothing to do with precedence since only one operation is involved.) --Tango (talk) 15:26, 10 February 2008 (UTC)

My pocket is a set.

My pocket is a set.

I add the first ten fibonacci numbers one at a time. [1]

How many elements does my pocket have?

[1] If this formulation is nonsensical after all, then "it is equal to the union of ten single-element sets, each of whose sole element is a successive number of the fibonacci sequence". or if I'm still not phrasing it right, could I get a little help? —Preceding unsigned comment added by 79.122.91.85 (talk) 23:46, 9 February 2008 (UTC)

Are you asking how many elements are in $\bigcup _{n=1}^{10}\{F_{n}\}$? If so, the answer is 9 (since $F_{1}=F_{2}$, using the convention in Fibonacci number). -- Meni Rosenfeld (talk) 23:59, 9 February 2008 (UTC)

Are you quite sure that your pocket is a set ? Perhaps it is more like a multiset (my pockets certainly are). This may be the source of your confusion. Gandalf61 (talk) 15:07, 10 February 2008 (UTC)

I'd say my pockets are like sets. They often contain very similar objects, but never perfectly identical ones (other than things like air molecules, which I don't count among the contents). —Keenan Pepper 17:19, 10 February 2008 (UTC)

But if hypothetically you did find two identical objects and put them in your pocket, and then reached into your pocket a short time later, you would be surprised if you only found one object in there (unless there is perhaps a hole in your pocket ?).

My semi-serious point is that the analogy set up by the initial statement "my pocket is a set" is subtly misleading. Gandalf61 (talk) 18:33, 10 February 2008 (UTC)

Good point. —Keenan Pepper 23:14, 11 February 2008 (UTC)