Wikipedia:Reference desk/Archives/Mathematics/2009 July 10

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July 10[edit]

Practical uses of very big numbers[edit]

Just ran across Skewes' number while deleting a nonsense page, and I'm just amazed by it. What is the practical benefit of theorising such a large number? Reading Ramsey theory, I can understand (slightly) that Graham's number, because it helps Ramsey's theory, helps us in predicting sequences of some sort of events, although I'm not sure what kind of events. I gather that the two numbers are somehow related (more than by just being very very very big numbers), but I can't see how Skewes' benefits anything "in the real world" [no slam on higher mathematics intended]. Understand, by the way, that I was a history major in college, so I'm (1) altogether unfamiliar with higher mathematics, and (2) accustomed to being asked about the utility of my field of study. Nyttend (talk) 12:20, 10 July 2009 (UTC)

As explained in the article you linked, the various things called Skewes' number were introduced because Skewes could prove that something rather interesting and unexpected happens at some point lower than that number. It is now known that this phenomenon in fact occurs at at much lower point, so Skewes' original numbers are now just historical curiosities relating to his specific proofs. Algebraist 14:04, 10 July 2009 (UTC)
Do you mean the section "Skewes' numbers"? Looking at that, I didn't realise that it was an example of the formula given in the intro; for all I knew, those were two proofs that he had done on other topics. Because there's no pi in either expression, and because the greater-than expression included e, I thought it was something different. Assuming that I understand you rightly, I can now see the point of these numbers; thanks. Nyttend (talk) 14:27, 10 July 2009 (UTC)

One practical area where theories about extremely large numbers can be useful is program verification. Say you have a computer program that defines three functions: 1) f(n) appears to compute something complicated and it's hard to tell quite what it's doing. 2) g(n) = f(n) + 1, and 3) h(n) = g(n) - f(n). You'd like to use equational reasoning to prove that h(n)=1 for all n, regardless of what f is. The problem is this reasoning can fail if f never returns. For example you could give the recursive "definition"

f(n) = f(n) + 1

and subtracting f(n) from both sides, you get 1=0, not a good basis for sound proofs of anything ;). Of course if you try to treat that definition as an executable program and actually run it, it will simply recurse forever, not giving you an opportunity to show that it's wrong. So your "proof" that h(n)=1 is only valid if you can also prove that f actually terminates and returns a value for each n (i.e. it is a total function). If f doesn't always terminate, it might also imply (if you incorrectly assume that it does terminate) that 1=0, and the implication may be much less obvious than the blatant recursive example I gave, so it could go silently screwing up the results of some fancy automated theorem prover trying to reason about the program. For example, if f(n) is defined as the smallest counterexample to Goldbach's conjecture that is greater than n and it works by searching upwards from n, then determining whether even f(0) halts is a famous unsolved math problem.

So here's where the big numbers come in. In general, deducing whether some arbitrary function terminates is called the halting problem and it is unsolvable (there is provably no algorithm that can do it, as has even been proved in verse). You are ok only if f turns out to be one of the functions whose termination you can prove. And the termination might take an extremely large number of steps: for example, f might compute the Ackermann function or even a Goodstein sequence while still being provably total. Numbers like Skewes' and Graham's are pretty big by most everyday standards, but at least you can write down formulas for computing them. The size of the Goodstein sequence grows so fast, that you can only prove nonconstructively that it does eventually finish--the number of steps in it even for fairly small n makes Graham's number look tiny.

So, you've got a situation where you can make a valid and useful deduction (e.g. h(n)=1) about a piece of software only if you can prove that for every n, there's a number t so that f(n) finishes computing in under t steps, where t might be unimaginably enormous. But you don't to compute t or care how large it is; all you have to do is prove that t exists, to ensure that your reasoning about some other part of the program is actually sound. That, then is a practical use of theories involving enormous numbers.

Harvey Friedman has written some "Lecture notes on enormous integers" [1]. The math is fairly technical but you might be able to get a sense of the topic just from the english descriptions. (talk) 05:23, 11 July 2009 (UTC)

Added: and an anecdote by Friedman about the topic of the article I linked. (talk) 21:32, 11 July 2009 (UTC)
Thanks much for a detailed explanation! I hadn't expected from the first explanation that there was a continuing modern use for these numbers. And the poem was entertaining, too :-) Nyttend (talk) 20:59, 13 July 2009 (UTC)

Complex logarithm[edit]

Hello, I am looking at a solution to the problem "Suppose f is analytic and non-vanishing on an open set U. Prove that log |f| is harmonic on U." The solution here is to show that Log(f(z)) is holomorphic on some neighborhood of each point of U, and then log |f(z)| is the real part of it so harmonic. But, I do not understand the log function very well, especially where it is holomorphic. So, the basic idea makes sense but the details of knowing that the composite is holomorphic does not make sense. I've looked through my undergrad book and it does not seem to tell where log is holomorphic (we can assume the principal branch). I also looked at the [Complex logarithm] article and it does not help me understand much either. It says under the section Logarithms of holomorphic functions that

If f is a holomorphic function on a connected open subset U of \mathbb{C}, then a branch of log f on U is a continuous function g on U such that eg(z) = f(z) for all z in U. Such a function g is necessarily holomorphic with g′(z) = f′(z)/f(z) for all z in U.

What I don't get is, f(z) could be 0 at some point and then this makes no sense for two reasons, e^z is never 0, and the derivative as shown would have a 0 in the denominator. Is the article wrong or do I not understand? Maybe a better question is, "Is the article wrong?" I think I do not understand either way. Any help would be much appreciated! StatisticsMan (talk) 15:02, 10 July 2009 (UTC)

Your remark is correct, and the article is right. The point of that definition is that it does not state that every f on a domain U admits such a g (maybe it could be good to add a small rkm there on this point). As you observe a first condition is that the image of f should be included in the image of exp (i.e. C\{0}, meaning that f does not vanish in U). You may check e.g. Rudin's Real & Complex Analysis for all the story and the connection with the Riemann mapping thm, to solve globally the problem on a domain U. As to log|f(z)|, note that the Log in the argument you quoted is a sort of auxiliary function that you just need to have locally, so there is no topology to consider: locally you have your Log for free, because exp is locally invertible. Also, note that you can prove in less elegant but more elementary way that log|f(z)| is harmonic by direct computation (you can try it if you haven't still done it). Write f(z)=u(x,y)+iv(x,y) where z=x+iy, and log|f(z)|= (1/2)log(u2+v2); then derive and use CR. --pma (talk) 15:41, 10 July 2009 (UTC)
In my study group, the guy who did the problem did it the way you mention and this way is simpler and is probably what I would come up with if I tried. But, in my complex analysis class, the professor did it the way I mentioned. I want to understand more so I am trying to understand this one as well. So, are you saying basically that as long as we have a small disc around f(p) where f is not 0 at all in the disk (would we need not 0 in the closure of the disk?), then Log of f is holomorphic on that disk? StatisticsMan (talk) 16:38, 10 July 2009 (UTC)
Exact. If a holomorphic function f:U → C has f '(p)≠0 at p in U, then it is locally invertible, meaning that there is an open nbd V subset of U such that f(V) is open and f:V → f(V) has a holomorphic inverse. It is a particular case of the inverse function theorem if you want. Here you just need a local inverse of exp(z) defined in a nbd of a given point p≠0. Such a p is therefore in the image of exp, say p=exp(a) (of course there are many such points a; we just choose one); moreover exp is locally invertible everywhere because its derivative never vanishes; so there is a local inverse of exp, call it Log, defined on a nbd of p, with values in a nbd of a. It verifies the relation exp(Log(z))=z in the domain of Log (the nbd of p), which is all you need to conclude that Re(Log(z))=log|z| (remember |exp(z)|=exp(Re(z)) ). Going back to the harmonicity of log|f|, look at any z0 in U; take a Log defined in a nbd of p:=f(z0) and write log(|f(z)|)=Re Log(f(z)), where Log(f(z)) is defined in a nbd of z0. There is possibly no Log such that Log(f(z)) is globally defined in the whole of U, but that's no problem at all.--pma (talk) 17:23, 10 July 2009 (UTC)(I've re-edited to change notations or correct, sorry)
I'm still not understanding this completely but I've thought about it a lot and I understand it better. It's starting to make sense. Thanks for the help! StatisticsMan (talk) 19:58, 10 July 2009 (UTC)
Summarizing: all you need in order to prove the harmonicity of your log|f(z)| at each z0 in U, is a holomorphic function "Log" defined in a nbd of f(z0) such that exp(Log(w))=w, hence Re Log(w)=log|w|, so log(|f|)=Re Log f(z) in a nbd of z0.--pma (talk) 20:13, 10 July 2009 (UTC)

This is probably not a good answer (since it uses ideas beyond undergraduate curriculums), but this is my solution: if f is analytic on some open set U, then \log|f| is subharmonic there (f doesn't have to vanish). If, in addition, f is nonvanishing, then 1/f is subharmonic and so -\log|f| is subharmonic. Hence, \log|f| is harmonic. I guess my point is that it is possible to approach the problem from real analysis. (This is a very good problem, so I couldn't resist.) -- Taku (talk) 22:29, 10 July 2009 (UTC)

Another possibility is: u(x,y):=log|z| is harmonic (by easy direct check, or because it's real part of the complex logarithm). It's a general fact that a conformal change of variables in any harmonic function u is still harmonic , that is u(f(x,y)) is harmonic if f is holomorphic, just because any two-variable harmonic function u is locally real part of a holomorphic. --pma (talk) 07:41, 11 July 2009 (UTC)

Quaternion algebra[edit]

The article Quaternion algebra claims:

One illustration of the strength of this analogy concerns unit groups in an order of a rational quaternion algebra: it is infinite if the quaternion algebra splits at \infty and it is finite otherwise, just as the unit group of an order in a quadratic ring is infinite in the real quadratic case and finite otherwise.

Is this correct? I would expect the unit group of a rational quaternion algebra to be infinite in both cases, since a splitting quadratic field can be embedded in the quaternion algebra in infinitely many ways. --Roentgenium111 (talk) 15:59, 10 July 2009 (UTC)

Actually, looks right to me. The norm should be positive definite in the nonsplit case, and the order should form a lattice in the algebra, and the units of the order should have norm 1, as in the Hurwitz quaternion situation. (There are only a couple imaginary quadratic fields with nontrivial units, too.) No time to think more (and be more correct. :-))John Z (talk) 23:05, 16 July 2009 (UTC)

non base 10 math[edit]

What is the purpose for non base 10 math? I can understand the uses of Hex, or binary, but are there practical uses for base 5, or base 28 etc? Googlemeister (talk) 19:54, 10 July 2009 (UTC)

There's nothing particularly useful about them as far as I'm aware. Of course, there's nothing particularly useful about base 10, either; it just so happens that we're using it. Apparently some languages count in quinary. Algebraist 20:29, 10 July 2009 (UTC)
(ec) Well maybe not such a big use as base 10, 2, 16 &c. But, for instance, they can be of use in arithmetic computations by hand, for it is very easy to reduce a number mod pk when written in base p. They give representations for p-adic extensions (via unbounded sequences of digits from the left). In any case, we have them all for free, with no particular stocking problems. Even the number 12931/3 is not so used as it is 3 or 4, but we know it is there at any need. --pma (talk) 20:39, 10 July 2009 (UTC)
Base 28 may be practical for some purposes when you work with an alphabet of 28 symbols, for example 26 letters, space and period. PrimeHunter (talk) 22:04, 10 July 2009 (UTC)
A base-85 encoding is used in some file formats. -- BenRG (talk) 08:54, 11 July 2009 (UTC)
Once upon a time I encountered an algorithm for determining base-10 square roots on mechanical calculators (i.e. from the old days before electronic calculators took off). This algorithm actually used base-20 arithmetic internally because it helped to minimize the number of mechanical components involved. Dragons flight (talk) 22:13, 10 July 2009 (UTC)
I learned that technique a long time ago as the "twenty method", but that does not appear to be a common term based on my lack of search results. It is actually a method for calculating square roots in base ten using a technique similar to long division, described here.
As for other bases, base 60 has been historically popular and is still present in daily life on clocks (hours, minutes, and seconds) and in angular measurements (degrees, minutes, and seconds. Binary is the natural base for digital circuitry, and thus computers, but for convenience, binary digits are often represented in groups of three or four yielding octal and hexadecimal. As for decimal, the obvious reason that we commonly use that system has to do with the number of fingers we have. It's no coincidence that the word digit refers to both a number or a part of your anatomy. If we had evolved from three-toed sloths, we might be using base 6 or base 12 on a daily basis, but mathematics as a whole would be the same. -- Tcncv (talk) 00:35, 11 July 2009 (UTC)
There have been computers that actually used hexadecimal internally, although a hexadecimal digit was still represented as four bits. I'm thinking of the implementation of floating-point numbers on the IBM 360, which stored a mantissa M and exponent E in order to represent the value M×16E. There was also at least one computer that used base 3 internally, the Russian-built Setun. Some computers from Digital Equipment Corporation, in the days when internal and external memory were both far more expensive than today, used software that was able to store 3 characters of text in a 16-bit word by using a character set with just 40 possible characters and treating the 3 characters as a number in base 40, which was then translated to base 2 to be stored. (403 = 64,000 < 65,536 = 216.) DEC called this RADIX-50, where the "50" meant 40 but was written in octal (base 8)!! --Anonymous, 07:20 UTC, July 12, 2009.
The most useful bases from the point of view of doing common mental arithmetic are those with a lot of factors, especially low factors. Dividing a given number by some divisor is often easier if the divisor is a factor of the base in which you have the number represented - so, for instance, it's easier to divide numbers in decimal by 2 or 5 than by 3. Thus a base with lots of low factors like 2, 3, 4, 5 will make it easier to divide numbers into halves, thirds, quarters, etc. This is why base 60 is used a lot - it's a highly composite number. Maelin (Talk | Contribs) 13:18, 11 July 2009 (UTC)
IMO the greatest significance of non-decimal representations is to demonstrate that they are possible. As noted above, base 10 is completely arbitrary, but most people don't really understand this fact. They think that 9 must be followed by a two-digit number by some sort of cosmic edict. Some then go and attribute all kinds of significances to the digits constituting a number's decimal representation. Familiarity with writing numbers in different bases helps to understand why this is absurd. -- Meni Rosenfeld (talk) 18:46, 11 July 2009 (UTC)
Very true. I once heard one guy claiming that the fact that we have ten fingers is possibly due to the Lord wanting to provide men with a kind of hand calculator "not eight, nor twelve, you see". --pma (talk) 15:18, 12 July 2009 (UTC)
Base64 is widely used on the internet, and in other computer applications. Here, the base is chosen by the number of different characters (“digits”) that are safely available in the ASCII character set, and is also a power of 2, which is handy for software efficiency and simplicity. Red Act (talk) 12:19, 13 July 2009 (UTC)

question about 0.333...[edit]

That's a number right? Am I out of my mind? I was discussing 0.999... repeating and someone said that 0.333.. is a limit, and I told him that 0.333.. isn't a limit, it's a number. He then told me he has a bs in mathematics and ask for my credentials... am I going crazy? I know it can be expressed as a limit (as well as an infinite series), but is 0.333... itself a limit? Thanks-- (talk) 20:04, 10 July 2009 (UTC)

Sure, it's a limit. It's also a number. Why do you think a limit shouldn't be a number, or a number shouldn't be a limit? --Trovatore (talk) 20:14, 10 July 2009 (UTC)
(ec)Yes, 0.333.. is a number, 1/3, and a limit (of real numbers) is itself a number. In fact you may also say that 0.333.. is not a number, nor a limit, it is a representation of a number. --pma (talk) 20:21, 10 July 2009 (UTC)
No, you're not crazy. As many other people have indicated, 0.333... represents both a number and a limit. As a side note, your acquaintance ought to be less defensive and arrogant. Asking for your credentials for asserting correctly that 0.333... is a number, or even for asserting incorrectly that it is not a limit, is obnoxious. Michael Slone (talk) 03:36, 11 July 2009 (UTC)

But how can it be a limit? The mathematical limit article says that it has to be a function in order to be a limit. And if it was a limit, wouldn't it mean that 2 or pi would also be limits?-- (talk) 20:47, 10 July 2009 (UTC)

That's not what the article says at all. Functions (and sequences, and suchlike things) can have limits, but the limits themselves are simply numbers (at least in the cases we're talking about). Any given real number is the limit of any one of many real sequences or real-valued functions. The notation '0.333…' denotes the number 1/3 by giving a specific sequence (0, 0.3, 0.33, 0.333, 0.3333,…) of which 1/3 is the limit. Similarly, I could if I wanted to refer to the number 2 with the curious notation '1.999…', using the fact that 2 is the limit of the sequence 1, 1.9, 1.99, 1.999, …. Algebraist 20:55, 10 July 2009 (UTC)

Ok, but if I showed a bunch a mathematicians the symbol "2", the first thing that will come to their mind is "oh, that's a number", not "oh, that's a limit".-- (talk) 21:35, 10 July 2009 (UTC)

I think you have some fundamental misunderstanding of what a limit is, or maybe you just expect it to carry more baggage than it does. Saying that something "is a limit" conveys exactly nothing -- anything at all can be a limit. --Trovatore (talk) 21:43, 10 July 2009 (UTC)
But if you showed them 2.000... they might think of limit first, and if you showed them 2/1 they might think of a fraction. A limit can be a way to express a number. You would normally only use the word limit about a number when the number is expressed in a way referring directly or indirectly to a limit. Compare to the possibly simpler term "sum". Is 2.0 a number or is it a sum (for example 2 + 0/10)? It can be viewed as both, and as several other things. PrimeHunter (talk) 21:56, 10 July 2009 (UTC)
Let me try an analogy: Consider f(x)=x^2. Clearly f is a function, but f(3) isn't a function, it's a number, it's 9. In the same way (0.9, 0.99, 0.999...) is a sequence, but the limit of that sequence is just a number, 1. --Tango (talk) 22:22, 10 July 2009 (UTC)
I probably do this a lot, but let me make clear that which is important in this context. Mathematicians are not particularly concerned with what 2, or 1, or 0.333... actually means but rather, they treat these "numbers" as a collection of symbols which together form a set (of symbols). But do not interpret this incorrectly - it is not that mathematicians simply remove meaning from these symbols, but rather, they study the relations between them (in fields such as number theory and calculus, for instance). For instance, assuming I am a mathematician, if you told me just the number 2, I would frankly not think of anything in particular. However, if you told me every integer, I would start thinking about prime numbers and all sorts of concepts in the realm of number theory. Therefore, a number alone does not mean anything (whether a limit, or "a function"), but if you tell me every number (that is, give me a context), then I can talk about limits and and other such concepts. --PST 03:07, 11 July 2009 (UTC)
Well, the axiomatization of real numbers I was forced to not understand as an undergrad defined the reals as (the limits of) (equivalence classes of) Cauchy sequences over the rationals - see Construction_of_the_real_numbers#Construction_from_Cauchy_sequences. So in that sense, every real number very directly is a limit. --Stephan Schulz (talk) 08:04, 11 July 2009 (UTC)
That's not an axiomatization, that's a construction. Axiomatizations of the reals don't define them to be anything at all. Algebraist 12:12, 11 July 2009 (UTC)
Anyway, I think that the doubts of the OP are due to a slight ambiguity of language. Strictly speaking a number and a limit are two different concepts (if not why two different terms). But a limit of a sequence (of real numbers) is itself a number by definition, that is, a certain number with a special property wrto that sequence. And any number is, of course, a limit of some sequence, and, according to some construction of real numbers, it is a certain limit by definition. --pma (talk) 09:42, 11 July 2009 (UTC)
The expression "0.333..." is shorthand for the series \sum_{i=1}^\infty \frac{3}{10^i}, because of the way decimal numbers are defined. Since you can't really evaluate the sum directly with that infinity there, it's also taken to mean  \lim_{n \to \infty} \sum_{i=1}^n \frac{3}{10^i} (explained in more detail at infinite series). You can evaluate this limit, which turns out to be identical to  \frac{1}{3} . So 0.333... represents a limit, and also the number which is the result of evaluating that limit. (The last part is fairly standard terminology - when a mathematician refers to a "limit", she can be either referring to the limit expression itself, or to the numerical result of evaluating that expression.) -- (talk) 22:01, 11 July 2009 (UTC)

But that's when I think it gets messy. Because if you say "oh, 0.333.. is just shorthand for \sum_{i=1}^\infty \frac{3}{10^i}", then what's to stop someone from saying \pi is shorthand for \pi = \sum_{k=0}^\infty \frac{1}{16^k} \left( \frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6}\right), or that two is shorthand for \sum_{k=0}^{\infin}\frac {1}{2^k}. Does it really matter that .333... is a repeating decimal in deciding whether to call it a limit or not?-- (talk) 13:33, 12 July 2009 (UTC)

Let's back up a little. We use a positional number system with ten as the base. That is, the '2' in "20" and the '2' in "200" mean different things. It tends to be easier to discuss these things when using a different base, so let's look at the value "258hex" in hexadecimal (base 16). Each position to the left of the decimal point is worth the next power of 16. So "258hex" in hexadecimal is equivalent to 2*162 + 5*161 + 8*160. That's the way a hexadecimal number is defined. The same holds true for decimal numbers. 376dec is defined to mean 3*102 + 7*101 + 6*100. This scheme holds for numbers to the right of the decimal place too, but in that case you're reducing the exponent by one for each position to the right. "0.02" is defined to mean 2*10-2. So when you write something like "0.333...", by the definition of what that sort of decimal representation means, you're writing a shorthand for "0*100 + 3*10-1 + 3*10-2 + 3*10-3 ...", which is equivalent to the more compact \sum_{i=1}^\infty \frac{3}{10^i}. That series is the most straightforward way of transforming "0.333..." into a form which is mathematically tractable (that is, into a form which you can then use in subsequent calculations). The other examples you give do not proceed directly from definitions. Pi is defined as the ratio of the circumference to the diameter of a circle in Euclidean geometry - the series is one of the ways to calculate that ratio. The other one is just a series which evaluates to two, and has nothing to do with the definition of two. When you start writing things like "0.333...", you're playing with notation and in doing so you must be aware of what the notation means. The most straightforward, minimal translation of the concept of "0.333..." is the above infinite series - and if you want to know what the value of an infinite series is, you use a limit. So the value of "0.333..." is defined by the limit simply because it is a repeating decimal, which is equivalent - by definition - to an infinite series. -- (talk) 16:31, 12 July 2009 (UTC)

But any number can be, by definition, an infinite series. Just because you chose to define 0.333.. as an infinite series doesn't mean that you can't arbitrarily not decide to define 2 or \pi as an infinite series. That's just the number system we use. In base-10, 0.333.. is an infinite series. But in other number bases, such as base 4, 0.333.. = 1. And numbers that are infinite series in base 4 , might not be infinite series in base 10. So defining numbers by their decimal representation is flawed, especially considering that fractions came first.-- (talk) 22:28, 14 July 2009 (UTC)

1/3 represents an element of the rational numbers, and we don't need the concept of a limit to understand what element we're talking about. There are sequences that have 1/3 as a limit, just like any other number. The simplest case, for any element of a metric space x, the sequence x, x, x, x,... has a limit of x, so being a limit is not a particularly notable property. However, the decimal representation of 1/3, "0.333...", implicitly uses the concept of the limit to describe what value we're talking about. That notation can be thought of as describing the sequence 0.3, 0.33, 0.333,... which has 1/3 as its limit. Since 1/3 can't be described as a terminating decimal, we allude to it by describing a sequence of terminating decimals that approaches it. Note that limits are only involved in the notation, and have nothing to do with the number itself. So it could be said that the expression "0.333..." refers to a limit, but you're also right that the number it describes is just a number.
An important side note though is that there are cases where there are "limits" that aren't "numbers". If M is a metric space that isn't complete, then there are Cauchy sequences where the "limit" isn't actually in the set M. For example, pi isn't in the rational numbers, but we can construct a sequence of rationals that has pi as a limit. So in the space of rationals, we can say that pi isn't a "number", even though it is a "limit" (quotation marks on "limit" because technically such a sequence doesn't have a limit). If we extend the rational numbers by adding elements that correspond to Cauchy sequences that didn't have limits before, then we get the real numbers, which said to be the completion of the rationals. Back to the example of 1/3, we can define a metric space M that is the set of numbers that can be expressed as terminating decimals. 1/3 is not a number in M, but it is a "limit". That is, we can identify 1/3 with certain Cauchy sequences that don't have a limit in M. In particular, the sequence 0.3, 0.33, 0.333... has the properties we're looking for. The completion of M turns out to also be the real numbers. On the other hand 1/3 is obviously an element of both the reals and the rationals. Rckrone (talk) 19:00, 15 July 2009 (UTC)
Not sure how much this will add to the above responses, but let's back up a little more. I'll assume we have already constructed the real numbers somehow (say, using Cauchy sequences), and start with defining a decimal expansion (other people might give a different, but ultimately equivalent definition):
A decimal expansion is a function f:\mathbb{Z} \to \{0,1,2,3,4,5,6,7,8,9\} with the property that there is some N \in \mathbb{Z} such that for every i<N we have f(i)=0.
Next, I will define the real number represented by a decimal expansion:
Let f be a decimal expansion. The real number represented by f is defined to be the sum of the infinite series \sum_{i=-\infty}^{\infty}f(i)10^{-i}.
That this is well-defined is left as an exercise. It can also be proven that every positive real number is represented by at least one, and at most two, decimal expansions.
Next we have a convention to use a string of ASCII characters to denote both any terminating decimal expansion (having only finitely many nonzero entries) and the number it represents. You start with writing the first nonzero digit, write all subsequent ones up to f(0) (and write it even if the number has no integer part), write a dot, and then write all subsequent digits in order until the last nonzero one. So the decimal expansion
f(i) = \begin{cases}0&i<-2\\6&-2\le i\le 4\\0&i>4\end{cases}
Is denoted by the string 666.6666. This string also denotes the number represented by the decimal expansion.
Next we have a convention to denote repeating decimal expansions. The formal convention uses overbars, dots or parentheses, while the informal convention says "write repeating part enough times to make it clear what it is, and then write an ellipsis". According to this convention, the string 0.333... denotes the decimal expansion
f(i) = \begin{cases}0&i\le0\\3&i>0\end{cases}.
The string also denotes the number represented by this expansion, which is \sum_{i=1}^{\infty}3\cdot10^{-i}, which is 1/3.
So, it is in this sense that 0.333... is "defined" to be \sum_{i=1}^{\infty}3\cdot10^{-i}. It's not that 1/3 somehow has the property of being an infinite series (as you have correctly noted, all real numbers are the sum of some infinite series). It's just that the decryption of what the notation "0.333..." means involves the use of an infinite series. The decryption of the notation "\pi", however, does not involve infinite series at all - it only involves our definition of what \pi is (which is usually the ratio of the circumference and diameter of a circle). -- Meni Rosenfeld (talk) 22:46, 15 July 2009 (UTC)

Distance between points -- using lat/lon[edit]

OK, I've got two points on the surface of the earth, identified only by their lat/long coordinates in DMS. I'm wanting to know the "straight-line" distance between them, in miles or km.

First, I have to convert everything to one unit, logically degrees, but seconds might actually make the endgame easier since I recall 1 second of lat is close to 1 mile. Further, I recall that longtitudinal distance has to be reduced by the cosine of the latitude ... but I can't work out the rest of it.

BUT, maybe I don't have to think much harder than that, i.e. Pythagorus is Close Enough. For "small" distances, the corrections for a spherical or even ellipsoidal surface won't make a noticeable difference from a plane. For example, if the planar distance is 1000 km and the spherical distance is 999 or 1001, that's 1/10 of 1%. At what point does "small" become significant? --DaHorsesMouth (talk) 22:20, 10 July 2009 (UTC)

1. 1 nautical mile is about 1 minute of arc at the equator, not 1 second.

2. The simplest approach to finding spherical distance is convert the lat/lon to rectangular coordinates where the center of the earth is at the origin. You get two 3-dimensional vectors from which you can easily compute the dot product which gives you the angle between the vectors. Since you know the earth's radius, the angle lets you figure out the distance. (talk) 03:17, 11 July 2009 (UTC)

Great-circle distance should be relevant here. Michael Hardy (talk) 05:19, 11 July 2009 (UTC)
Great circle is usually sufficient for most cases, but as the earth isn't an exact sphere (it is more an oblate spheroid), a more accurate method is Vincenty's formulae3mta3 (talk) 07:52, 11 July 2009 (UTC)
Actually the article you want is Geographical distance. It is all covered there. —3mta3 (talk) 11:21, 11 July 2009 (UTC)

Proves once again that knowing what something is properly called is the single best starting point for learning about it. Thanks to all! Issue is