Wikipedia:Reference desk/Archives/Mathematics/2008 December 2

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December 2

Integral of graph, area = infinite, volume = finite?

Wi-wi-wi-wikipedia!

When I'm integrating the graph y = 1/x from [1,infinity], the area is going to be infinity. But when I try to find the volume of revolution, the volume = pi!!!! Whoa! What's going on here? Why is the area infinite, but the volume finite? Hustle (talk) 00:36, 2 December 2008 (UTC)

Why is this a problem? For a fixed volume, the surface area can be made arbitrarily large. This is merely the limiting case. RayAYang (talk) 00:40, 2 December 2008 (UTC)

We're not dealing with surface area here — we are dealing with area under a curve. --98.114.146.51 (talk) 13:10, 2 December 2008 (UTC)

It is because the 'pieces' you get together when integrating the volume decrease as 1/x², while those for area are as 1/x — and the integal ∫(dx/x²) is convergent while ∫(dx/x) diverges. --CiaPan (talk) 07:34, 2 December 2008 (UTC)

This surface of revolution is called the horn of somebody-or-other, I think. The paradox is sometimes stated as "you can fill it with paint, but you can't paint it", which is counterintuitive because you would think that filling it with paint would automatically paint the inside.

But in fact you can't really fill it with paint. Once the radius gets below the size of a paint molecule, they can no longer pass. --Trovatore (talk) 01:00, 2 December 2008 (UTC)

It's Gabriel's Horn, aka Torricelli's trumpet. Algebraist 01:06, 2 December 2008 (UTC)

That real liquids consist of molecules is irrelevant here. You're dealing with idealized models of space — and paint — here. A finite volume of an idealized paint can indeed have infinite surface area. --98.114.146.51 (talk) 13:10, 2 December 2008 (UTC)

Yes, that's true, but the intuitive paradox relies on the reader's real-world experience that to paint a fixed area requires a fixed amount of paint, and therefore to paint an infinite area should require an infinite amount of paint. This is an accurate assessment, for real-world paint. But real-world paint can't fill the horn, and this resolves the paradox. --Trovatore (talk) 18:41, 2 December 2008 (UTC)

But that doesn't seem to be the OPs question - if I'm interpreting it correctly, the OP is asking how a shape with finite volume can have infinite cross-sectional area. That's not the same as it having infinite surface area. (It's a very similar question and the answer is probably closely related, but it is a different question.) --Tango (talk) 14:00, 2 December 2008 (UTC)

Didn't CiaPan answer it, though? The cross section decreases with 1/x (in $m^{2}$) and the volume decreases with 1/x² (in $m^{3}$). Zain Ebrahim (talk) 14:11, 2 December 2008 (UTC)

Any region of finite volume must have the property that the set of cross sections with infinite area has measure zero. This is easier to visualize in the plane; for any region A of finite area, the set of coordinates of vertical cross sections that have infinite measure (length) must be a measure zero subset of the x axis. — Carl (CBM · talk) 14:13, 2 December 2008 (UTC)

Maybe you mean, how can a plane region of infinite area create a finite volume of revolution? This is because the volume obtained by rotation in general goes to 0 faster than the radius of the object being rotated (as Zain says). — Carl (CBM · talk) 14:17, 2 December 2008 (UTC)

Indeed, that's what the OP is asking about (and that is the answer) - all this talk about painting things was going off in the wrong direction. --Tango (talk) 14:22, 2 December 2008 (UTC)

One way of reconciling this paradox is to think of the two integrals being considered - a volume integral and a surface area integral - as fundamentally different types of integrals. The volume integral proceeds, in the simplest case, by taking a supremum of the volumes of simple regions contained completely inside the region being integrated. The surface area integral, on the other hand, relies on the convergence of piecewise linear approximations that do not lie within the surface whose area is being found. The Koch curve is a simpler case to consider, since it does not have the rotational aspects of the horn problem. — Carl (CBM · talk) 14:23, 2 December 2008 (UTC)

Koch snowflake

Here is a related question that I think is more interesting. Suppose that you take a Koch curve. You know from the wikipedia article that the area inside the curve is given by a function of the form $Cs^{2}$ where s is the side length of the original triangle. Take the derivative of this with respect to s. Why is it that the length of the Koch curve is not equal to this derivative? What conditions would be required on the boundary of a region in order for the derivative of the area in this way to give the length of the boundary? After all, it works for circles and squares...

I'm sure this has been studied somewhere - does anyone have a reference? The same thing applies, of course, to the horn problem in three dimensions, where the derivative of the volume with respect to radius at x=1 is not equal to the surface area of the boundary. — Carl (CBM · talk) 14:36, 2 December 2008 (UTC)

It's not in general true that $|\partial S|={\frac {\partial |S|}{\partial t}}$ for any parameter t. It works in the obvious cases of spheres and rectangles (if you let the rectangle's parameter be something added to both width and height) because the "motion" of the boundary is almost always normal to the surface with the same magnitude. Sometimes you can make it work, but only by cleverness: try parameterizing a family of irregular pyramids to get their area. The family of Horns related by scaling the radius doesn't behave that way (with the radius or anything else as the parameter). The idea of moving the entire perimeter of the snowflake normal to itself by some distance is similarly nonsensical. --Tardis (talk) 16:13, 2 December 2008 (UTC)

Well, even for a nice set A (say a square or a disk) the derivative of the area of sA is not equal but just proportional to the length of the boundary of sA; and the proportionality constant depends on the set. But this is also true for the Koch interior, in a sense: the proportionality factor being infinite (for the length of the Koch curve is infinite). Anyway, I'd say that what you want here is to apply the coarea formula in the context of Hausdorff measure. PMajer.

The surprising part is actually that the opposite is not possible: you can't have infinite volume bounded by finite area. In fact, if you fix a volume, the surface area bounding a body of such volume is bounded from below. – b_jonas 20:37, 4 December 2008 (UTC)

calculator for e

I posted this on the Computing desk to ask for a software, but it just occurred to me that it might fit better on the Math. I'm looking for a software/program/other-something that can calculate e to a large number of digits. I already looked at all the stuff on the external links on the article, and only one thing helped me (link). If I could get a program like they used (they say it was in 1994 on a "VAX alpha class," so I'm guessing it's pretty simple by today's standards), then that would solve my problems; but anything else will do fine too. flaminglawyer^cneverforget 03:34, 2 December 2008 (UTC)

A VAX is a Digital Equipment Corporation mainframe computer. So, just how many digits do you need ? StuRat (talk) 04:58, 2 December 2008 (UTC)

Well, I would like to have 1 million, for starters; then I (maybe) could work my way up to larger digits. Yes, I realize that I could just look at one of this to get up to 10 million, but I'm not doing this for personal knowledge. flaminglawyer^cneverforget 05:34, 2 December 2008 (UTC)

Take a look at arbitrary-precision arithmetic. It contains references to many software packages that to big number arithmatic. Many are open source and free. Personally, I experimented with the MAPM libraries a couple of years back and found it efficient and easy to use. I assume you have some programming background. -- Tcncv (talk) 05:59, 2 December 2008 (UTC)

I have little/no programming background :/ but I'll give it a shot. And thanks for the link, never would have found it otherwise. flaminglawyer^cneverforget 06:10, 2 December 2008 (UTC)

I just downloaded the most recent version and although it hasn't has any significant updates in several years, I was able to get it to compile in MS Visual C++ 2005. In contains a sample calc program that supports RPN arithmetic. Using the "calc -d1000 1 e" command ("e" means e^x function) it calculated e to 1000 decimal digits in about two seconds on my laptop. "calc -d10000 1 e" game me 10,000 in less than a minute. If you have any questions feel free to post to my talk page. -- Tcncv (talk) 07:05, 2 December 2008 (UTC)

Download Maxima and/or PARI.

here is a typical output in Maxima (for one million digits, change fpprec:1000 to fpprec:1000000. Try also %pi and sqrt(2) instead of %e):

Maxima 5.13.0 http://maxima.sourceforge.net
Using Lisp GNU Common Lisp (GCL) GCL 2.6.8 (aka GCL)
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
This is a development version of Maxima. The function bug_report()
provides bug reporting information.
(%i1) block([fpprec:1000],bfloat(%e));
(%o1) 2.7182818284590452353602874713526624977572470936999595749669676277240766\
303535475945713821785251664274274663919320030599218174135966290435729003342952\
605956307381323286279434907632338298807531952510190115738341879307021540891499\
348841675092447614606680822648001684774118537423454424371075390777449920695517\
027618386062613313845830007520449338265602976067371132007093287091274437470472\
306969772093101416928368190255151086574637721112523897844250569536967707854499\
699679468644549059879316368892300987931277361782154249992295763514822082698951\
936680331825288693984964651058209392398294887933203625094431173012381970684161\
403970198376793206832823764648042953118023287825098194558153017567173613320698\
112509961818815930416903515988885193458072738667385894228792284998920868058257\
492796104841984443634632449684875602336248270419786232090021609902353043699418\
491463140934317381436405462531520961836908887070167683964243781405927145635490\
61303107208510383750510115747704171898610687396965521267154688957035035b0
(%i2)

and here is a typical output in PARI (for one million digits, change \p1000 to \p1000000. Try also Pi and sqrt(2) instead of exp(1)):

Reading GPRC: /cygdrive/c/PARI/.gprc ...Done.
   
                   GP/PARI CALCULATOR Version 2.3.4 (released)
            i686 running cygwin (ix86/GMP-4.2.1 kernel) 32-bit version
 compiled: Jul 12 2008, gcc-3.4.4 (cygming special, gdc 0.12, using dmd 0.125)
                (readline v5.2 enabled, extended help available)
 
                     Copyright (C) 2000-2006 The PARI Group
  
PARI/GP is free software, covered by the GNU General Public License, and
comes WITHOUT ANY WARRANTY WHATSOEVER.
  
Type ? for help, \q to quit.
Type ?12 for how to get moral (and possibly technical) support.
 
parisize = 4000000, primelimit = 500000
(08:16) gp > \p1000
realprecision = 1001 significant digits (1000 digits displayed)
(08:16) gp > exp(1)
%1 = 2.7182818284590452353602874713526624977572470936999595749669676277240766303
53547594571382178525166427427466391932003059921817413596629043572900334295260595
63073813232862794349076323382988075319525101901157383418793070215408914993488416
75092447614606680822648001684774118537423454424371075390777449920695517027618386
06261331384583000752044933826560297606737113200709328709127443747047230696977209
31014169283681902551510865746377211125238978442505695369677078544996996794686445
49059879316368892300987931277361782154249992295763514822082698951936680331825288
69398496465105820939239829488793320362509443117301238197068416140397019837679320
68328237646480429531180232878250981945581530175671736133206981125099618188159304
16903515988885193458072738667385894228792284998920868058257492796104841984443634
63244968487560233624827041978623209002160990235304369941849146314093431738143640
54625315209618369088870701676839642437814059271456354906130310720851038375051011
5747704171898610687396965521267154688957035035
(08:16) gp > quit

Mdob (talk) 22:27, 3 December 2008 (UTC)

Use a formula like $e=\sum _{0\leq k}{\frac {1}{k!}}$. This converges quickly. You can compute each term from the previous one by just dividing each by k. Just keep enough digits after the decimal throughout the whole computation. – b_jonas 20:30, 4 December 2008 (UTC)

There is an easy method that does not rely on high precision, based loosely on a method for converting a decimal to/from an expansion in another base. You create an array, the ith entry contains the numerator of a series whose denominator is i!. On the first step the array is filled with 1's to give the series above (combine the 1st and 0th terms to get 2 as first entry). On each succeeding step you read off a digit from the first element of the array and then multiply each element of the array by a base number, probably 10, though you could do 100, 1000 ... , depending on how much precision you have. Then, for each element in the array (starting at the end), "carry" the excess to the previous element. So, for example, if the 6th element was 40, then you'd carry 3, since 3*6 = 36, to the 5th element, and leave the remaining 4. Here's how it would work in detail with 6 terms (enough for 3 digits).

  Pos.  1  2  3  4  5  6
Start
        2  1  1  1  1  1
Read off 2
        0  1  1  1  1  1
Multiply by 10
        0 10 10 10 10 10
10 = 1*6 + 4 
        0 10 10 10 11  4
11 = 2*5 + 1 
        0 10 10 12  1  4
12 = 3*4 + 0 
        0 10 13  0  1  4
13 = 4*3 + 1 
        0 14  1  0  1  4
14 = 7*2 + 0
        7  0  1  0  1  4
Read off 7
        0  0  1  0  1  4
Multiply by 10
        0  0 10  0 10 40
(Carries)
        1  1  1  3  1  4
Read off 1
        0  1  1  3  1  4
Multiply by 10
        0 10 10 30 10 40
(Carries)
        8  0  0  1  1  4
Read off 8

So even from this small example you get 2.718. Expand the array to get more accuracy, Stirling's formula gives a way of approximating how many terms you need for a given number of digits. I don't have a specific reference for this, I saw on Usenet years ago, but it's the kind of thing Knuth might have.--RDBury (talk) 19:06, 6 December 2008 (UTC)

Hyperbola and its Tangent Line

This is a test problem that I wasn't able to solve.

Consider the hyperbola $b^{2}x^{2}-a^{2}y^{2}=a^{2}b^{2}$ and a tangent line to the hyperbola at $({\hat {x}},{\hat {y}})$. If the tangent line intersects the two asymptotes at points S and P, prove that $({\hat {x}},{\hat {y}})$ is the midpoint of line SP.

My textbook says that the tangent line to a hyperbola is ${\dfrac {{\hat {x}}x}{a^{2}}}-{\dfrac {{\hat {y}}y}{b^{2}}}=1$, where ${\hat {x}}$ and ${\hat {y}}$ are the coordinates of the point. The equations of the asymptotes would be $y={\dfrac {b}{a}}x$ and $y={\dfrac {-b}{a}}x$.

I substituted the equation of one of the asymptotes for the y term of the equation of the tangent line. When I simplified, I got ${\dfrac {{\hat {x}}x}{a^{2}}}-{\dfrac {{\hat {y}}x}{ab}}=1$.
$x\left({\dfrac {\hat {x}}{a^{2}}}+{\dfrac {\hat {y}}{ab}}\right)=1$

I have a lot more work but it's too long to type up. My friend tried this problem and he got a complicated mess of variables that did not simplify to the midpoint.

Well, if you got

$x\left({\dfrac {\hat {x}}{a^{2}}}+{\dfrac {\hat {y}}{ab}}\right)=1$,

that is

$x={\frac {1}{{\frac {\hat {x}}{a^{2}}}+{\frac {\hat {y}}{ab}}}}={\frac {a^{2}b}{{\hat {x}}b+{\hat {y}}a}}$

for the point on an asymptote, then you're halfway to the solution. The other coordinate is $y={\frac {b}{a}}x={\frac {ab^{2}}{{\hat {x}}b+{\hat {y}}a}}$. These two describe your point P.
Same way calculate point S on the other asymptote and check, whether ${\frac {x_{P}+x_{S}}{2}}={\hat {x}}$ and ${\frac {y_{P}+y_{S}}{2}}={\hat {y}}$ ---CiaPan (talk) 13:42, 2 December 2008 (UTC)

The special case a = b = 1 is sufficient. Simplify! Bo Jacoby (talk) 22:43, 2 December 2008 (UTC).

Rearranging for x

$y={\frac {r+x}{1-rx}}$, rearrange for x.

The problem being that x is in the numerator and denominator, and I don't understand how to seperate and get x on one side of the equation. For instance, if I try:

$y-rxy=r+x\,$

$y-rxy-r=x\,$

I'm still stuck with x on both sides of the equation. 86.144.61.245 (talk) 21:13, 2 December 2008 (UTC)

Notice that every term in your equation is either independent of x (i.e, y and r) or a multiple of x (i.e., ryx and x). You should put those terms which are a multiple of x together, and factor out the x. This will allow you to express x in terms of y and r. Eric. 131.215.158.213 (talk) 21:19, 2 December 2008 (UTC)

Please note that $rxy=ryx\,\!$, so if you denote $A=ry\,\!$ then your second equation will get the form $y-Ax=r+x\,\!$
May be it is easier to deal with this form...? --CiaPan (talk) 21:30, 2 December 2008 (UTC)

I don't know why the responders above are making this so complicated. You have

$y-rxy-r=x.\,$

Add rxy to both sides:

$y-r=x+rxy\,$

Then factor the right side:

$y-r=x(1+ry)\,$

Finally, divide both sides by (1 + ry):

${\frac {y-r}{1+ry}}=x.$

Michael Hardy (talk) 00:48, 4 December 2008 (UTC)

Oh, you don't know why? Probably because we want to direct the OP to the solution, so (s)he can find it, instead of doing homework for her/him. --CiaPan (talk) 10:09, 4 December 2008 (UTC)

Group of a given order

Newbie question: how do we find a finite group $G=\langle S\rangle$ of a given order? For example how do I find the generators of a group with $|G|=1323600000000000000000125=5^{3}*13^{2}*67*9029*1664909*62209267$?.

In GAP4 we need to input the generators if we want to work with a group. When |G| is big ($10^{20}<|G|<10^{130}$) I can only imagine examples of groups when the order is some special number like a factorial (e.g.58! or 58!/2) or a prime power. From Chapter 3 of Seress. Permutation Group Algorithms, Cambridge University Press, 2003, I know cyclic groups have 1 generator, abelian groups have 2 and most groups have less than 10. If I know the prime factorization of |G|, can I find an example of any group with that order? with few generators? Mdob | Talk 22:21, 2 December 2008 (UTC)

For any order n, there is a unique cyclic group of order n. If that doesn't solve your problem, you'll have to explain what you want more clearly. Algebraist 22:42, 2 December 2008 (UTC)

How do I find that group, Algebraist? I though it worked only with simple groups... Mdob | Talk 22:58, 2 December 2008 (UTC)

It's just the group generated by one generator, g, with the relation gⁿ=1 (where n is the desired order). --Tango (talk) 23:01, 2 December 2008 (UTC)

for n = 1323600000000000000000125, how do I find it in GAP4? Mdob | Talk 23:06, 2 December 2008 (UTC)

I've never used GAP4, but doesn't it let you just enter generators and relations? --Tango (talk) 23:08, 2 December 2008 (UTC)

Thanks Tango, and thanks, Algebraist. Mdob | Talk 23:16, 2 December 2008 (UTC)

For any desirable order n, the group G=Z/nZ with Z the integers, has order n (and is cyclic) Voyatzo (talk) 20:38, 6 December 2008 (UTC)

An aside - "abelian groups have 2 generators" - assuming this means "any abelian group can be presented with at most 2 generators", is this correct ? How would you present C₂xC₂xC₂ with just 2 generators ? Gandalf61

I think he means "finite simple group", not abelian group. That is both true, and mentioned on page 48 of the book. By "most" he means most groups presented practically as permutation groups in computational group theory. JackSchmidt (talk) 03:05, 4 December 2008 (UTC)

Of course you are correct, since are simple groups, not abelian, that have 2 generators. According to Seress (section 3.1, page 48):

"Given $G=\langle S\rangle \leq S_{n}$, the input is of length |S|n and a polynomial-time algorithm should run in $O((|S|n)^{c})$ for some fixed constant c. In practice, |S| is usually small: Many interesting groups, including all finite simple groups, can be generated by two elements, and it is rare that in a practical computation a permutation group is given by more than ten generators". Mdob (talk) 22:26, 4 December 2008 (UTC)