Wikipedia:Reference desk/Archives/Mathematics/2007 April 1

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April 1

the number of cows in great britain²

plz

Europe has 130 million cows from the cattle article. Coolotter88 02:06, 1 April 2007 (UTC)

We can't do your homework for you, but the answer can be computed by taking the area integral of the cow density function over the territory of Great Britain. Note that the report was issued April 1st. If you find a different answer than 9,300,454 head of cattle (for 2005), show us your computation and we may be able to spot the error.  --Lambiam^Talk 13:09, 1 April 2007 (UTC)

Fraction exponents

How do you calculate numbers to fraction exponents, such as 6^1/2? —The preceding unsigned comment was added by 65.30.153.24 (talk) 01:06, 1 April 2007 (UTC).

Well, the simple way is to use your calculator. Keep in mind that raising a base to a fractional exponent, n, is the same as taking the n^th root of that number. So 9^1/2 = ${\sqrt {9}}$ = 3. or 8^1/3 = ${\sqrt[{3}]{8}}$ = 2. See Exponentiation#Rational_exponents for more. (the answer to your question is ${\sqrt {6}}$ or 2.44948974.)--Ybbor^TSurvey! 01:17, 1 April 2007 (UTC)

Ah, I didn't know that. Thanks much!—Preceding unsigned comment added by 65.30.153.24 (talk • contribs)

Also, note that raising to a negative exponent (2 raised to the -2) is the same as flipping the base (2/1^-2 = 1/2^2). ST47Talk 01:59, 1 April 2007 (UTC)

Because of awkwardness in typesetting, instead of √6 we sometimes write 6^1/2, which means the same thing. Similarly, we may write (x²+y²)^3/2, which means take the Euclidean length, (x²+y²)^1/2, and cube it (raise it to the power of three). Otherwise, we'd end up with something bloated, like

$\left({\sqrt {x^{2}+y^{2}}}\right)^{3}.\,\!$

To calculate a numeric value in more general cases, we (and our calculators) typically use logarithms. That is, when presented with b^r, we first compute the (natural) logarithm of b, multiply that by r, and feed the result into the exponential function:

$b^{r}=\exp(r\log b).\,\!$

The logarithm and exponential functions are heavily used elementary functions, and we can expect efficient implementations yielding good accuracy to be available in our hand calculator or standard library. For a provocative historical examination of the early computation of tables of logarithms by Henry Briggs, try this narrative by Erik Vestergaard. --KSmrq^T 02:33, 1 April 2007 (UTC)

toyota symbol

i just noticed that the toyota symbol is three elipses put together, what would be the equation to graph this? —The preceding unsigned comment was added by 72.146.114.13 (talk) 02:23, 1 April 2007 (UTC).

Hi. I got the implicit equation ${\frac {(x^{2}+9y^{2}-18y)(4x^{2}+y^{2}-4)(x^{2}+4y^{2}-16)}{576}}=0$ by trying. Of course it isn't perfect.

It comes from the product of three implicit equations each corresponding to an ellipse :

$\left({\frac {x}{4}}\right)^{2}+\left({\frac {y}{2}}\right)^{2}-1=0$

$\left({\frac {x}{3}}\right)^{2}+\left(y-1\right)^{2}-1=0$

$\left(x\right)^{2}+\left({\frac {y}{2}}\right)^{2}-1=0$

I'm sure someone can achieve better with exact references.

-Xedi 12:53, 1 April 2007 (UTC)

The denominator 576 in the equation is superflous; E/576 = 0 is equivalent to E = 0. The ellipses forming the picture should actually have a varying thickness, and can probably be approximated fairly well as the differences between the areas of an outer and an inner ellipse. The necessary data to get a professional quality result must be present one way or another in this encapsulated PostScript file.  --Lambiam^Talk 13:21, 1 April 2007 (UTC)

Stuck on proving an inequality

I have a homework problem for my analysis class that I have all worked out, except for one inequality that has resisted all my attempts to prove it. I would like to show that, for all $x\in [0,1]$ and all $n=1,2,3,\ldots$, the following inequality holds:

$\left(1+{x \over n}\right)^{n}\leq \left(1+{x \over n+1}\right)^{n+1}.$

I have tried to prove this inequality in several ways, but I can't seem to get anywhere useful; I just come back to where I started, or I meet some dead end that I've seen several times before. If anyone can help to point me in the right direction, I'd be very grateful. —Bkell (talk) 07:35, 1 April 2007 (UTC)

Alternatively, I could show that for all $x\in [0,1]$ and all $n=1,2,3,\ldots$ we have

$e^{x}\geq \left(1+{x \over n}\right)^{n}$,

but I don't see a way to prove this without proving the inequality above. —Bkell (talk) 07:39, 1 April 2007 (UTC)

Taking n-th roots at both sides, the latter inequality becomes

$e^{x/n}\geq 1+{x \over n}\,,$

or, substituting x := nx,

$e^{x}\geq 1+x\,.$

Using the property that the exponential function is its own derivative, this should not be hard to prove.  --Lambiam^Talk 07:52, 1 April 2007 (UTC)

Aha! Thank you. —Bkell (talk) 08:18, 1 April 2007 (UTC)

Or you could tackle the original inequality using the binomial expansion:

$\left(1+{x \over n}\right)^{n}=1+x+{\frac {n-1}{2!}}{\frac {x^{2}}{n}}+{\frac {(n-1)(n-2)}{3!}}{\frac {x^{3}}{n^{2}}}+\ldots$

$\left(1+{x \over n+1}\right)^{n+1}=1+x+{\frac {n}{2!}}{\frac {x^{2}}{(n+1)}}+{\frac {n(n-1)}{3!}}{\frac {x^{3}}{(n+1)^{2}}}+\ldots$

then show that after the first two terms, each term in the first expansion is smaller than the corresponding term in the second expansion. And in addition there is an extra (positive) term at the end of the second expansion. I think the inequality actually holds for all positive x. It does not necessarily hold for negative x, as the counter example x= -3, n=2 demonstrates. Gandalf61 09:21, 1 April 2007 (UTC)

Counting, continued

Continued from above, here's my counting problem. I'm trying to count the number of solutions of $a+2b+3c=n$ where $a,b$ and $c$ are non-negative integers. I can do it, I have this:

$\sum _{c=0}^{{\frac {n}{3}}-\left({\frac {1}{3}}\right)\mathrm {mod} (n,3)}{\frac {n-3c}{2}}-\left({\frac {1}{2}}\right)\mathrm {mod} (n-3c,2)+1$

The rationale behind this foul expression is as follows. Suppose we were looking at counting solutions to $a+2b=k$. Then this is simpler, just count the number of possible values for $b$. These range from 0 to ${\frac {j}{2}}$ where $j$ is the greatest $j\leqslant k$ divisible by 2, so the total number is

${\frac {k}{2}}-\left({\frac {1}{2}}\right)\mathrm {mod} (k,2)+1$

Keeping this in mind, and going back to the original problem, for given $n$ the possible values of $c$ range from 0 to ${\frac {m}{3}}$ where $m$ is the greatest $m\leqslant n$ divisible by 3. For each of these possible values then, get the remainder $n-3c$ and apply the above argument to count how many ways to carve it up. Add up all these bits and we have the answer.

Unless I've made typos, this works. I have my computer churning it out and it agrees with this guy. But there's supposed to be a nicer way - like a simple closed form function of $n$. How should I proceed? Thanks. --87.194.21.177 10:32, 1 April 2007 (UTC)

Unless I don't understand the problem, the sequence is 1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, 27, 30, 33, 37, 40, 44, 48, 52, 56, ... (sequence A001399 in the OEIS).  --Lambiam^Talk 14:11, 1 April 2007 (UTC)

Thanks, that's it. For some reason my computer messed up (okay, I messed up) for n=13 and n=26 so I missed it when I searched OEIS. That's it for sure though, thanks. --87.194.21.177 16:50, 1 April 2007 (UTC)

Approximating curves

If I pick up a pen and scribble any sort of curve onto a piece of paper, can it definitely be described mathematically, by a function? I would think this would be true, but I can't convince myself with any sort of rigour. If it is the case, how would one even go about approximating the equation of a curve? I've had a go with some online function graphers, but all the curves are too nice and smooth - to really approximate a nasty curve, would you need some horribly complicated piecewise function? Just curious, 81.102.34.92 16:44, 1 April 2007 (UTC)

If you scribble something on paper and I give you a mathematical description, claiming it is a description of your scribbles, how would you determine whether that claim is correct? A mathematical description is or can be expressed, presumably, as a finite sequence of symbols drawn from a finite repertoire of symbols, subject to certain rules to make it meaningful. That means the number of possible descriptions is countable, and since the total number is infinite, countably infinite, the same cardinality as the natural numbers, and even enumerable. However, it can be argued with some plausibility – but here we leave the realm of mathematics and move into physics – that the number of possible scribbles is uncountable, and that the probability that a scribble you produce fits any of this countable set of descriptions is zero. Similarly, an actual nail would never way exactly π gram; it may be close, but almost certainly at least a fraction of 10⁻⁴⁰ off (or more precisely, such precision has no physical meaning). So this is a negative result.

More positively, if the scribble can be viewed, abstractly, as an open bounded and connected subset of the plane produced by the ink of a pen with finite (meaning non-zero) thickness, then it can be proved that for any positive value ε, however small, there is a curve described by one of these mathematical descriptions such that it is wholly contained within your scribble, and comes within a distance ε of any point in the scribble. If S is the scribble and C the curve, this can be formulated mathematically as

C ⊆ S ⊆ C ⊕ D_ε(0,0),

in which the dilation operation denoted by ⊕ is Minkowski addition, and its second operand is the disk with radius ε centred on (0,0). To explain this fully would take up much space, but some mathematical background can be found in our article Approximation theory.  --Lambiam^Talk 17:24, 1 April 2007 (UTC)

Thanks for the help, that's cleared it up somewhat. I've always sort of assumed that anything I scribble has to be describable mathematically, since it, if drawn under "perfect" conditions" (ie a plane whose smallest division is zero, a pen that draws a line of zero width), physically exists - basically, I've always assumed for something to physically exist, it has to be mathematically describable. Is that a valid assumption? 81.102.34.92 18:27, 1 April 2007 (UTC)

That is a question about physics – properties of nature – and not about mathematics. I see no reason to assume this to be true, or even that it is a meaningful statement, and since "perfect" conditions can't exist, I consider the application to your scribbles highly dubious, even if we assume that physical reality admits of a mathematical description. If it is true, we have no way of knowing it. Of course, modelling physical reality by coercing it into a mathematical abstraction has been and continues to be fruitful, but on sufficiently close examination the models thus far always come apart.  --Lambiam^Talk 20:16, 1 April 2007 (UTC)

Generally when working with mathematically defined curves, a the curve is simplified as passing through a certain number of points, and a polynomial function is used to interpolate a curved line between them. Splines are a commonly used technique for making curves. Also, yes, where the curve is not smooth, you would add many more points to increase the accuracy. You don't necessarily need a piecewise function (like splines), but it is usually easier to work that way because otherwise your polynomial goes up a degree for every point you add (which could get very tedious to solve). - Rainwarrior 20:02, 1 April 2007 (UTC)

I don't see the above answers as being as strongly on topic as two other directly applicable topics: Parametric_equation and the issue of discontinuities (first order, second order, etc.) in the curve being drawn, see Classification_of_discontinuities.

Taken together, those two articles pretty much cover the theoretical issues of drawing curves on paper, and are in fact arguably the best starting point if one wanted to develop algorithms for drawing or modeling curves in real life. Dougmerritt 20:35, 1 April 2007 (UTC)

I'm getting bogged down in the physical reality of the situation - what I really want to know is, is any curve drawn in the (mathematically ideal 2D plane) describable by a function, no matter how messy? Surely it must be possible, even if it comes down to a gigantic string of piecewise, constant functions, each applicable only at a single point? 81.102.34.92 20:52, 1 April 2007 (UTC)

In short, the answer is "yes".

If one tried to do it non-parametrically, then even circles tend to end up being relations rather than functions, thus the first article I pointed to.

If one worries about whether piecewise sequences are an issue, well, they come up as a result of discontinuities between continuous sections of the curve/function, thus the second article.

But actually one can even have an infinite number of discontinuities, or even a function that consists of nothing but discontinuities; such things historically presented challenges to the theory of functions (and to the historical definitions of "function"). Poincare' called them "monsters". But such functions can exist in a suitably modern theory of functions. They don't really come up in regard to drawing on paper, so that's just to illustrate the utter lack of serious problems with drawing curves. Dougmerritt 21:27, 1 April 2007 (UTC)

(Below, Salix and Lambiam bring up some interesting mathematical issues, but it's not clear to me, as yet, that they involve issues that the original questioner would care about, even though they are important in abstract math. I guess we shall see, as the discussion proceeds. Dougmerritt 02:55, 2 April 2007 (UTC))

There a theorem of Hassler Whitney, stating that any close set of points in Rⁿ can be defined as a set $f^{-1}(0)$, for a smooth function f. In particular any curve in the plane could be defined in this way. See Algebraic curve. --Salix alba (talk) 23:17, 1 April 2007 (UTC)

Hmm. I'm not sure what you're driving at (tell me to go read the article you cited, if it indeed suffices). Surely "close set of points" ends up being the same thing as "smooth" in this context? Dougmerritt 02:55, 2 April 2007 (UTC)

The answer to your last formulation really hinges on what you mean by "describable". By definition, the mathematical abstraction we call a curve is a function, or, if you will, an equivalence class of functions, but then any member represents the curve. However, we have no way of exactly describing these functions in any finite way, however gigantic the strings. We can give finite descriptions that approximate them arbitrarily close. For simplicity, say you put just two points on paper, which can be described by giving the distance between them, a single real number. But is that number "describable"?  --Lambiam^Talk 23:33, 1 April 2007 (UTC)

Yes indeed. However, the original question is about an idealized model of drawing on paper. It seems to me that all straightforward such models would pretty much inherently involve "describable" points and paths. How would undescribable points or paths arise in this sort of context, despite the fact that such things are in fact important in general in mathematics? (E.g. the ineffability of almost all transcendental numbers/functions, versus the inherent finite-hood (arguably, to a first approximation) of real-world methods, measurements, and entities.) Dougmerritt 02:55, 2 April 2007 (UTC)

If in this idealized model you can only draw describable curves, then the answer is: Yes, indeed, any curve you can draw is describable. It is not clear to me, however, that this corresponds to the intuitive notion of the questioner. It would seem to imply (depending on what you mean by "describable") that you can only mark a countable subset of points in the plane, which to me would be counterintuitive. In an idealized setting, is any weight in kg you obtain by weighing an object necessarily describable? "Undescribable" weights would arise just because they happen to be the weight of the object under consideration, which presumably can be any positive real, of which there are more than even the mathematician's pen can describe.  --Lambiam^Talk 03:24, 2 April 2007 (UTC)

That's what I thought you meant. You and I differ in what we find intuitive, then. To me, when a question is grounded in real world considerations, I take for granted that all inputs and methods are describable. After all, undescribable entities cannot, by definition, be described in the real world. That's not to say that your intuition is wrong, it's just to give some explanation for why my intuition is different than yours.

Aside from that, I don't see how you can introduce the (mathematically quite valid) undescribable transcendental number of the mathematician's pen into what is supposed to be a model of the real world. Dougmerritt 03:36, 2 April 2007 (UTC)

(Correcting myself: I accidentally used your phrase "mathematician's pen" as if it meant undescribables, whereas you very clearly used it to mean "describables, even the ones that require sophistication to describe". Mea culpa. I don't think anything in the discussion really changes as a result, though. Dougmerritt 03:56, 2 April 2007 (UTC))

A fascinating discussion ! Breaking the original question down into sub-questions, I think we have the following:

1. Can an arbitrary "scribble" be approximated as closely as we like by the curve of a mathematical function that can be described in a finite number of steps ? Answer: yes. You just need to find a family of "describable" functions whose curves are dense in the space of "scribbles". Splines will probably work.
2. Is there a "describable" function whose curve is exactly the same as a specific (but arbitrary) "scribble" - this one here on this piece of paper, let's call it scribble 1784032 ? Answer: yes. I describe the function as follows: "the function whose curve is scribble 1784032". If you think this is "cheating", this may be because your definition of a "describable" function is more restrictive than mine.
3. Does any and every "scribble" correspond to the curve of a "describable" function ? Answer: no. As Lambian has pointed out, the set of "describable" functions can only be countable (regardless of how we define "describable"), whereas the set of scribbles is uncountable. So we can apply a version of Cantor's diagonal argument to show that there must be some scribble that is not included in the curves of any set of "describable" functions. Of course, if we find a specific scribble that is not covered by our set of "describable" functions - say scribble 1784032 - we can always add "the function whose curve is scribble 1784032" to the set. But there will always be another "undescribable" scribble - in fact, uncountably many.

The difference between cases 2 and 3 is subtle and non-intuitive. But most mathematicians find it both interesting and important (your mileage may vary !). Gandalf61 10:58, 2 April 2007 (UTC)

—Q: I have drawn this scribble on this piece of paper here. Can you give a mathematical description of it?

—A: Sure thing, no sweat. Let's call your scribble "Bobbie". Then here is its mathematical description: "Bobbie"! Easy, no?

 --Lambiam^Talk 19:44, 2 April 2007 (UTC)

Almost, but not quite. "Bobbie" is a label or identifier for the scribble. A mathematical description of the curve would be "the set of points that lie on the scribble Bobbie", or even "the co-ordinates of the points that lie on the scribble Bobbie" if you don't mind a frame dependent description. Gandalf61 20:27, 2 April 2007 (UTC)

Well this was certainly interesting! Thanks for summing everything up Gandalf. One last question - from a practical point of view, how would I actually go about trying to describe a scribble? I'm assuming some of the articles mentioned would help me (such as Interpolation, perhaps?) but they're all a bit abstract. What are the branches of maths I should be searching for? Having signed in, Icthyos 17:59, 2 April 2007 (UTC)

(Thanks for the signature.) As a practical matter, you might have a look at Potrace. :-) --KSmrq^T 18:34, 2 April 2007 (UTC)

See curve fitting. StuRat 02:06, 3 April 2007 (UTC)

Numbers

I'm looking for a number that is the sum of two consecutive positive integers, of three consecutive positive integers, and of four consecutive positive integers. Any suggestions? --Carnildo 18:57, 1 April 2007 (UTC)

Elementary observations:

${\begin{aligned}(k)+(k+1)=2k+1&\Rightarrow n\equiv 1{\pmod {2}}\\(k)+(k+1)+(k+2)=3k+3&\Rightarrow n\equiv 0{\pmod {3}}\\(k)+(k+1)+(k+2)+(k+3)=4k+6&\Rightarrow n\equiv 2{\pmod {4}}\end{aligned}}$

What does this tell you? --KSmrq^T 19:20, 1 April 2007 (UTC)

I find that there is no such number.

If n is your number, then n = a + (a+1) = b + (b+1) + (b+2) = c + (c+1) + (c+2) + (c+3)

Or n = 2a + 1 = 3b + 3 = 4c + 6

Then for example 2a+1 = 4c + 6 leads to c = (2a-5)/4

But (2a-5)/4 cannot be an integer for any integer value of a, because 2a is even, then 2a-5 is odd so 2a-5 cannot be a multiple of 4.

-Xedi 19:22, 1 April 2007 (UTC)

Oh well, KSmrq beat me to it. :)

It's April 1. I can always hope that someone provides me with a number that is both even and odd. --Carnildo 20:56, 1 April 2007 (UTC)

It's April 2 now. For penance, find a number that can be written in 23 different ways as the sum of two or more consecutive positive integers.  --Lambiam^Talk 03:38, 2 April 2007 (UTC)

Binomial expansion for complex powers

Possible?--Ķĩřβȳ♥♥♥ŤįɱéØ 21:11, 1 April 2007 (UTC)

See Exponentiation, particularly Exponentiation#Complex_powers_of_complex_numbers. --Ybbor^TSurvey! 21:16, 1 April 2007 (UTC)

Actually, I think the appropriate reference is the binomial theorem. --KSmrq^T 21:51, 1 April 2007 (UTC)

I'm sorry if I didn't ask my question correctly. I meant, what does it even mean to have a complex exponent? Applications? etc. --Ķĩřβȳ♥♥♥ŤįɱéØ 00:16, 2 April 2007 (UTC)

Here, the first response may be helpful.  --Lambiam^Talk 00:35, 2 April 2007 (UTC)