Wikipedia:Reference desk/Archives/Mathematics/2006 November 20

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

November 20

WP Math desk

I am so glad that there is a math desk for you folks. At first I was resisting this specialized place and thought that it would be just fine for these questions to be deliberated at /Science. How wrong I was, this desk has been an inspiration to me by the quality and depth of the questions, responses and argument here. I don't understand a lot of it but love it all the same and you have provided links into areas that I may never have known about but for this desk. Thanks y'all for a very informative and interesting place. --hydnjo talk 03:52, 20 November 2006 (UTC)

The Science Reference Desk is busy enough, thanks. :P x42bn6 Talk 20:23, 20 November 2006 (UTC)

A lousy attempt..

i = sqrt(-1)

==> i^2 = -1

==> i^2 + 1 = 0 (quadratic, b=0, a=1,c=1)

==> i = 0 +/- sqrt(- (4 x 1 x 1))/2 (quadratic rule)

==> i = (0 +/- sqrt(-4))/2 (quadratic rule)

==> i = [+/- sqrt(-4)]/2 (quadratic rule)

.:. sqrt(-1) = [+/- sqrt(-4)]/2 (final result)

What's wrong with this proof?

--138.217.32.78 07:00, 20 November 2006 (UTC)

The i is not a variable, it is a number, like 1 is. So your proof looks a bit like this:

Let the Roman numeral I stand for 1. Then:

I = sqrt(1)

⇒  I^2 = 1

⇒  I^2 − 1 = 0 (quadratic, b=0, a=1,c=−1)

⇒  I = 0 ± sqrt(− (4 x 1 x −1))/2 (quadratic rule)

⇒  I = (0 ± sqrt(4))/2 (quadratic rule)

⇒  I = [± sqrt(4)]/2 (quadratic rule)

∴  sqrt(1) = [± sqrt(4)]/2 (final result)

If you read ± here as indicating two possible choices, not all of which need to apply, then there is not much wrong with this. In general, sqrt(x) = sqrt(4x)/2, also when x = −1. Since x² = (−x)², by squaring you introduce a second possibility to be considered in solving for x, in case x is an unknown variable. In general the solution set of a quadratic equation may have two different solutions. In your proof (and my mimic of it), however, x is not an unknown variable, and only one of the two "formal solutions" can be right. In this case, it is the one where ± is +.  --Lambiam^Talk 07:57, 20 November 2006 (UTC)

I couldn't see much wrong with it - the final result sqrt(-1)=[+/- sqrt (-4)]/2 equals +/- i as expected? I think this was supposed to be a trick questiom??? Though I don't understand your 'quadratic rule' if ax^2+bx+c=0 then x=b/2a +/- sqrt (b^2/4a^2 - c/a) in this case giving i=+/- sqrt(-1).... So where on earth does the 4 in "sqrt(- (4 x 1 x 1))/2" come from? (apologies if I am having a stupid day) Sorry ignore that . Not thinking.

If I was being critical I would say the problem with the proof is that it's not a proof - just a few lines of algebra that starts and ends in the same place - what does it prove - nothing. Therefore - not a proof. —The preceding unsigned comments were added by 83.100.250.33 (talk • contribs) 13:10 – 13:37, November 20, 2006 (UTC).

(-i)²=-1 also. *Max* 22:17, 27 November 2006 (UTC)

Direct proportionalities involving area, volume, and mass

Dear Wikipedians,

I am studying some basic physics. I am at a point in my book that deals with scaling--that is, how volume increases exponentially with length and area. This is the part I'm having trouble understanding: it says that since

$A\propto L^{2}$

and

$M\propto L^{3}$

then

$A\propto M^{2/3}$

where A is the surface area of a three dimensional object, M is its mass (assuming that density and shape remain constant), and L is the length of any linear dimension. I believe them, but I want to see why this is true, and I can't figure out how it follows. I do not know calculus--will it be required to solve this sort of problem? Thanks. 69.223.189.14 16:52, 20 November 2006 (UTC)

How general do you want the explanation to be? For a specific example, take eight cubes and assemble them into a single cube. Observe that:

• The big cube has the same number of edges as the little ones, but each edge is twice as long.
• The big cube has the same number of sides as the little ones, but each side has four (=2²) times the area.
• The big cube weighs eight (=2³) times as much as the little ones.

—Ilmari Karonen (talk) 17:08, 20 November 2006 (UTC)

Basically, I want to know how you can work these kinds of direct proportions on paper. I can't figure out how they're getting to $A\propto M^{2/3}$.69.223.189.14 17:29, 20 November 2006 (UTC)

Method one: cube both sides of the A proportionality and square both sides of the M proportionality:

$A^{3}\propto L^{6}\propto M^{2}$

then take cube roots to get to your eventual solution.

Method two: take logs of both sides

$\log A=2\log L+c_{1}$

$\log M=3\log L+c_{2}$

then solve the simultaneous linear equations and exponentiate the results.

—David Eppstein 18:39, 20 November 2006 (UTC)

The conclusion follows from the two stated proportions by simple algebra. In the proportions, let the constants of proportionality be κ and λ, so we have

$A=\kappa L^{2}\,\!$

and

$M=\lambda L^{3}.\,\!$

From the second equation we deduce

$L=(M/\lambda )^{1/3}.\,\!$

Substituting in the first equation gives

$A=\kappa ((M/\lambda )^{1/3})^{2},\,\!$

which simplifies to

$A=\mu M^{2/3},\,\!$

where μ equals κ/(λ^2/3). This is the desired proportionality. --KSmrq^T 23:07, 20 November 2006 (UTC)

All right, thank you all for your time. I understand now. Somehow that just wasn't clicking for me. 71.144.12.115 23:28, 20 November 2006 (UTC)

Resistance to probabilities

I don't know whether to ask this on the maths or humanities section but here goes... Given the amount of resistance there appears to be to the basic 0.999... concept, how much resistance do you think there is to basic probabilities concepts? For example, if you I were to say your chance of winning the lottery with the numbers 1,2,3,4,5,6 is the same as if with 3,9,19,25,31,40. I would suspect this would surprise a lot of people (sadly enough) but I wonder how many will actively resist the idea and try to disprove it (like with 0.999...). What do you think? Nil Einne 17:15, 20 November 2006 (UTC)

Have you seen Talk:Monty Hall problem? Good times. Melchoir 19:36, 20 November 2006 (UTC)

O why me.

The player has three choices (he doesn't know)

The games master removes one choice (he knows) leaving 2, the player has 2 choices.

The player still has 2 choices - the gold is behind one of those two doors.

Should he switch? it's 50/50. toss a coin.

Good luck Melchior and Nil Einne and happy xmas. 83.100.174.103 20:00, 20 November 2006 (UTC)

If the argument is dumbed down enough, you can quite easily convince people. The one above can be fairly quickly disproved by multiplying probabilities. x42bn6 Talk 20:22, 20 November 2006 (UTC)

So are you saying the probability of the gold being behind one of the two remaining doors is not 1/2 ?83.100.174.103 20:26, 20 November 2006 (UTC)

Good examples : people have every information under their nose and stick to propaganda. Stupid things repeated unrelentlessly. Good luck! -- DLL ^{.. T} 21:05, 20 November 2006 (UTC)

Thank you, you saved my life.83.100.253.24 21:08, 20 November 2006 (UTC)

What I've observed — remembering my own misconceptions as a child — is that clumping seems to be hard to understand. For instance, 20 heads in a row on a fair coin sounds a lot less likely than the "appropriately random" HTTHHHTHHTHHTTHTHTHH. This may be related to the true statement that precisely 1000 heads in 2000 tosses is quite unlikely: randomness gets confused with disorder in cases like these, as 1014/986 seems less orderly than 1000/1000 and thus more random and thus more likely/believable. Similarly, in a repeated random (fair) shell game it seems better to consistently pick one shell ("It has to come up this shell eventually!") rather than to change shells, which feels like opening oneself up to foolishly "dodging" the moving correct shell. Of course, it goes the other way, too: in paper rock scissors, a strategy of RRRRRRRR... (that is, hoping for an opposing S sometime) seems really bad. This is ironic because, in terms of randomness, it's entirely equivalent to the consistent shell game strategy; perhaps the difference arises because our opponents in PRS are typically not random and would quickly take advantage of so predictable a strategy. --Tardis 00:11, 21 November 2006 (UTC)

Clumping is hard to accept. (I've done some model simulation, both by hand and computer; and I've had to work with my own misconceptions a bit. The misconceptions are also reportedly rather common by roulette gamblers, as described by Darrell Huff.) My personal guess is that this is related to a biase in the selective forces on the human mind, as to understanding chance versus recognising patterns. Pre-humans and young humanity had great use of recognising patterna, less so of understanding that some incidents may happen almost simultaneously by pure chance. (My favourite example: Recognising that birds flying low in the evening is positively correlated to bad wheather next morning could be of great use, although the scientific explanation of the phenomenon still was rather far away.) Hence, we search a system as soon as there is clumping - and conversely, when we don't expect sytems, then we don't expect incidents of similar kinds grouping together. End of quasi-psychology:-)

The problem is truly that the concepts are confused, and the language of the question gets remodeled, and words inserted. Often when dealing with likelihoods of sequences, the word "like" gets inserted, which changes the question significantly. Patterns "like" HTTHHHTHHTHHTTHTHTHH may have a higher probability for a given probability model than patterns "like" HHHHHTHHHHHHHHHHHHHH because may by one's definition of "likeness" there are simply more of them. As in, for one definition of "likeness", the number of (20 long) sequences where the number of heads is 12 is much larger than the number of (20 long) sequences where the number of tails is 1. But of course the probability of hitting either of those two exact sequences are the same (with the normal assumptions). Another issue which convolutes the question further is that many words such as "like" can even have multiple different definitions in the same sentence. I could easily use two totally opposing definitions of likeness for the two different sequences, because people may be predisposed to categorizing seemingly "random" sequences differently from sequences where there is a long sequence of heads or tails. The two sequences are already in two different conceptual categories, often using different measurements of likeness.

So when you ask people, "Which is more likely, 1,2,3,4,5,6, or 34,2,46,9,54,23?" people may translate this to "Which sequence is more likely, a sequence where each number of the sequence appears in a consecutive numeric sequence (sequences like 1,2,3,4,5,6, of which there are very few) or some random sequence (a sequence "like" 34,2,46,9,54,23, of which there are many)?" They will correctly answer THAT question, but that's not the question asked.

What's required of the answerer in dealing with questions of probability is their focus strictly on what's being asked, and not they may haphazardly guess might be asked, and that's not a natural response. Root4(one) 00:13, 23 November 2006 (UTC)

Having any opinion on it, resisting or accepting, seems like a bad idea until you have gotten that recurring decimal thing defined to you. So, what does $0.{\bar {9}}$ really mean? Just saying that the decimal repeats endlessly sounds awfully vague. I want a proper definition. Is it the limit of $0.9,0.99,0.999,\ldots$, perhaps? —Bromskloss 10:32, 21 November 2006 (UTC)

I linked you, just in case you were really asking the question rather than merely suggesting that it should be understood. --Tardis 17:23, 21 November 2006 (UTC)

However, if you have lack of any opinion, you have no motivation to figure out the real answer. Sometimes having an opinion is good, even if it is wrong. Root4(one) 00:13, 23 November 2006 (UTC)

Correlation does not imply causation is a favorite for resistance to statistics. --Salix alba (talk) 17:15, 21 November 2006 (UTC)

So, is there an article for Broken symmetries lead to unequal chances? Melchoir 18:52, 21 November 2006 (UTC)

A graph of x!

I would like to know whether a continuous function exists which yields f(n)=n! for integer n, and which could perhaps then shed light on the possible meaning of (say) 2.5!

Thanks, Nathan Ntownshend 21:54, 20 November 2006 (UTC)

See Gamma function. Note that it's shifted by one -- Γ(n+1) = n!. --Trovatore 21:59, 20 November 2006 (UTC)

I cannot find any Applications of Gamma function in differential equations, probabilty from the above page, except notational convenience. Twma 02:33, 21 November 2006 (UTC)

The Gamma function features with a non-integer argument in, amongst others, the Chi distribution, Dirichlet distribution, Gamma distribution, Pearson distribution, Rayleigh distribution, and Weibull distribution. I don't see a claim that the Gamma function has applications in differential equations, but it is applied as such in, among others, the Airy function, Bessel functions (see also Bessel-Clifford function), the hypergeometric differential equation, Laguerre polynomials, and Meijer's G-function.  --Lambiam^Talk 06:49, 21 November 2006 (UTC)

Thanks. I have copied the above paragraph to the discussion of my page so that I can read it CAREFULLY after 19 Dec 2006. Please let me know if I have violated your copyright. I keep telling myself that I should learn some probability theory but never get started in the last 35 years. Twma 03:52, 22 November 2006 (UTC)

(If you're looking for a numerical value and not meaning there's also Stirling's approximation.)87.102.16.174 18:51, 21 November 2006 (UTC)

Taxes

If I am an American citizen who filed taxes already for 2005 and recieved a refund from my w-2s. But because of a ss # error in a general contractor job. I never reported income on a 1099 for the wages paid. But then filled out a w-9 for the employer to give them the right ss #. How do I go about paying those taxes... Plain and simple. I didnt file for a general contracting job because they had the wrong social security number, but now the IRS has the right info. I am sure it is only a matter of time before I get a letter. Will they ever find out? How can I fix this problem and get them their money? Can I still write off any of it? And no... Going to the irs website is even more confusing! THANKS!

Legal questions are best answered by a lawyer. In this case, an accountant may be of some help. --TeaDrinker 01:50, 21 November 2006 (UTC)

This question is double-posted; see the Miscellaneous RD.  --Lambiam^Talk 07:47, 21 November 2006 (UTC)

three legged table

I have a heavy (~30 lbs) three legged table with very oddly spaced legs. I want to measure the total weight of the table but I only have two scales. Otherwise I could use the relation a+b+c=d. With two scales I can get weights for only two legs at a time as follows: 13.25+b+14.875=d, 13.71875+b+15.34375=d, 13.3125+2.5+c=d, a+1.86875+15=d.

Without knowing the total weight of the table what is the best way of solving for the total weight of the table given this data? 71.100.6.152 04:10, 21 November 2006 (UTC)

Um... what kind of scale are you using that provides weights down to the hundred-thousandth of a pound? Anyway, why not just invert the table and set it on one scale? Or set it on edge on the two scales. Putting scales under the legs two at a time is problematic because the table is in a different orientation each time. You could perhaps do some sort of least squares fit to the data, or else just average your various readings for each leg in hopes that they have no systematic error (which is a big hope). --Tardis 06:05, 21 November 2006 (UTC)

Another practical solutions in addition to those given by Tardis. 1) Put some brick or a book under the third leg, so each time all legs end at the same level. This will reduce the table slope, so weighting two legs at a time would give more consistent results for each 'leg'. 2) Take some plank, and put it horizontally on the scale. Place the table so that two legs are supported by two ends of the plank, the third leg on the other scale. This will require some practice to get things stable, as the first scale needs to divide the plank's length in a ratio equal to the two legs' weights ratio, which is unknown in advance. ;) Add the results, then subtract the plank weight. --CiaPan 07:06, 21 November 2006 (UTC)

Do a, b and c represent the weights of the legs? Is the weight of the table only formed by the legs? Naming the legs (after the variables representing their weights) A, B, and C, why should it be the case – as I surmise from your equation – that putting say A and C each on a scale giving readings of respectively r_A and r_C, it should be the case that a = r_A and c = r_C? Finally, forgetting the context and just looking at your equations, substituting d := a+b+c, you have four equations with three unknowns (a,b, and c), so unless the equations are dependent there is no exact solution. In fact, the first two come down to a+c = 28.125 and a+c = 29.0625, which cannot hold (exactly) at the same time. For any valuation of a and c, the other two equations allow an exact solution in b and d, so the only issue is with this pair. Here is a shortcut, which (in this particular case) is equivalent to the use of a least-squares fit. Just replace it by the arithmetic average a+c = 28.59375, and you have a system of three linear equations in three unknowns, which I assume you know how to solve.  --Lambiam^Talk 07:25, 21 November 2006 (UTC)

I think there's something wrong with your data "13.25+b+14.875=d, 13.71875+b+15.34375=d"? Ignoring that I'd assume that your putting scales under each of the three legs in turn. This gives the force on each leg with the third leg acting as a pivot (Please correct me if I've misunderstood). I think you need to generate equations for force in terms of distance of the legs apart (a triangle) and the centre of mass (assuming this simply isn't in the middle of the triangle). But your data has confused me, surely there should only be three readings - you have 4??87.102.16.174 18:01, 21 November 2006 (UTC)

• The number of decimal places of the values is due to conversion of pounds and one decimal place ounce data to decimal pounds using a calculator without rounding the results.

• A single scale is inadequate to handle the weight of two legs so bridging the legs is a good idea but not possible.

• A different level due to unequal height of each leg for each reading is the most likely cause of the discrepancy in the data but the legs themselves are unstable and pivot due to age. Their warpage would seem to require their distances and the weights they provide be averaged for several readings at the positive and negative extremes for all. In absence of the distances I have averaged the weights only (13.25+b+14.875=d, 13.71875+b+15.34375=d) which yields 13.484375 and 15.109375 respectively. Leveling and averaging the data and measuring distances to get center of mass seems like a good solution rather than looking for a formula that will overcome these inconsistencies.

• A rope and pulley system with an off site weighable counterweight seems a possibility yielding the total weight as accurately as possible but I am still curious as to how best to solve this problem using only the two scales and a mathematical method or formula which knowing total weight would definitely help.

71.100.6.152 02:07, 22 November 2006 (UTC)

Assuming that the application of physics is correct (I don't understand the "experimental" set-up), and that the measurements are subject to about the same (normal) error distribution with μ = 0, the method of least squares is applicable and appropriate. Eliminate d from all equations, thereby transforming your collection of equations into a linear system in 3 unknowns. For example, for the two equations directly above, the equations you obtain by eliminating d are a+c = 28.125 and a+c = 29.0625, which are of the same type, namely giving an estimate for the sum a+c. The whole system may then be solved with the linear least squares method. If each type of equation occurs in about the same number, you may (almost) equivalently take averages first. In the (untransformed) equations of (e.g.) the type r_A+b+r_C = d, it is pointless to average the r_A and r_C values separately; you can just add them to give b+r_AC = d, where r_AC = r_A+r_C.  --Lambiam^Talk 02:59, 22 November 2006 (UTC)

I think I understand linear least squares method but in my understanding one requies a set of x and y data points from which the matrix and its inverse are derived as well as the vector. How then would b+r_AC serve to provide the corresponding x or y data points when b is unknown? Adaptron 07:26, 22 November 2006 (UTC)

See System of linear equations for how to represent a collection of linear equations in the matrix–vector form Ax = b that is used in the description of the linear least squares method.  --Lambiam^Talk 09:29, 22 November 2006 (UTC)

But why represent in that form Ax= b when in all of the equations at least one of the three independent variables is unknown and the dependent variable is always an unknown? Adaptron 23:47, 22 November 2006 (UTC)

Measure the table flipped upside-down. i assume the table-top is level...--69.11.175.210 18:37, 22 November 2006 (UTC)

Reading the dialog might invalidate your assumption. Adaptron 23:47, 22 November 2006 (UTC)

mathematicians

What is the reason that most of the great mathematicians were from Germany or France ? pavanto (59.144.104.243 08:12, 21 November 2006 (UTC))

Newton, Euler, Euclid, Archimedes? It would be easier to answer why you think most great mathematicians were from those places if you explain in more detail who these great mathematicians were — how notable does a mathematician have to be to be called great, and where are you getting the statistics on their home country? —David Eppstein 08:18, 21 November 2006 (UTC)

Well... Gauss, Reimann, Laplace, fourier, Leibnitz are few to name. But if you look at the list of mathematicians by nationality, you will know why I am asking "this" question. And by great means who had profound influence on mathematics which changed the course of mathematics.... I hope this helps -- pavanto(59.144.104.243 09:30, 21 November 2006 (UTC))

Perhaps part of the reason this may be is that when Western Europe was strengthening in math was also a time when math had come to formalisms and abstractions. Many of the disciplines within math were first being defined, and the fundamental theorems and early, important results were being discovered. Also the idea of math as a profession was first arriving. Certainly they were great mathematicians, but they were also in the right place at the right time for having very fundamental results and areas of math named after them. Also, in western culture, we are bound to focus a little more on western mathematicians. Don't forget the ancient Greeks' foundations for geometry, the Indian foundations for trigonometry, and the Persians who laid the foundations for Algebra. The Chinese developed much of number theory before the West. Very few of the discoveries of these people are named after those who discovered them so their names are not common-knowledge, or sometimes even forgotten to history. It doesn't mean they didn't have a profound influence on mathematics. —siroχo 10:15, 21 November 2006 (UTC)

This question confuses quantity with quality. The greatest mathematicians are few in number and have come from all parts of the globe. The greatest number of mathematicians are found around centers of learning. A large percentage of living mathematicians have studied or taught in the U.S., regardless of original nationality. In part this is because Germany and Russia became inhospitable to many whom the U.S. welcomed. (Emmy Noether is a famous example.)

Some measure of the importance of association over the last few centuries can be found by browsing through the Mathematics Genealogy Project. For example, Gottfried Leibniz, considered one of the inventors of calculus, was the doctoral advisor of Jacob Bernoulli, who advised Johann Bernoulli, who advised Leonhard Euler, who advised Joseph Louis Lagrange, who advised Siméon Denis Poisson, who advised Johann Peter Gustav Lejeune Dirichlet, who advised Rudolf Lipschitz, who advised Felix Klein, who advised Philipp Furtwängler, who advised Wolfgang Gröbner, who advised Bruno Buchberger, who created one of our more important tools in computer algebra. That's a chain of a dozen notable mathematicians! But we also have great mathematicians who are part of no chain, like Niels Henrik Abel of Norway. --KSmrq^T 12:17, 21 November 2006 (UTC)

The title is MATHEMATICIANS. Your interpretation is : who, where and why (had come to formalisms and abstractions as a profession) in the past! Isn't it more constructive to interpret the title as:

How can we collectively MAKE an outstanding mathematician in the future?

What is the general method to do research and development in mathematics?

What kind of scientific method is applicable to mathematical RD?

What is the necessary environment to produce an outstanding mathematician?

How to make an idea to be developed into a respectable theory?

Can you tell us the idea which you consider to have good potential but belittled by your empolyers, colleagues?

Why the people in the quoted family could become outstanding, any special way to advise the members of their family?

Please look forward. Who, where and how in the future. Twma 04:01, 22 November 2006 (UTC)

There's some discussion of the way that social and political forces in different countries at different times encouraged certain kinds of mathematicians (for example the legacy of the Ecole Polytechnique in France), in Galison, Peter (2003). Einstein's Clocks, Poincaré's Maps. ISBN 0-340-79448-8.. --ColinFine 12:34, 26 November 2006 (UTC)