Wikipedia:Reference desk/Archives/Mathematics/2010 February 24

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February 24

The hardest Math question

I can not figure this out.

Ralph likes 25 but not 24; he likes 400 but not 300; he likes 144 but not 145. Which does he like:

	10,   50,   124,   200,   1600,  —Preceding unsigned comment added by Skivvyboy-5 (talk • contribs) 07:03, 24 February 2010 (UTC) 

Hint: Think about factors. Rckrone (talk) 07:08, 24 February 2010 (UTC)

<sarcasm>Isn't it obvious? Ralph like numbers for which $(x-350)(2x-289)(2x-49)>0$. So he likes 50, 124 and 1600. </sarcasm>-- Meni Rosenfeld (talk) 07:58, 24 February 2010 (UTC)

Hint: Which of the choices is a perfect square? PST 08:28, 24 February 2010 (UTC)

You should also be aware that there is no such thing as "the hardest question" and that mathematics is broader than that which you may encounter in these sorts of problems; mathematics requires a diverse range of skills (in fact, there exist skills and ideas in mathematics of which people are unaware), and is most certainly not restricted to "pattern-finding" (which is the skill this question tests).

On the same note, the problem is imprecise (to an extent) as Meni Rosenfeld points out; it is likely that the question looks for a property that the numbers 25, 400 and 144 satisfy, but that 24, 300 and 145 do not. However, there are many such properties, and that leaves the solver "guessing" what the questioner intends (put differently, this question is essentially a test of one's guessing ability rather than one's ability to think mathematically). Given that there are only certain "types of numbers" of which potential solvers are aware (such as "perfect squares", "prime numbers" etc.), a solver can be somewhat certain that his/her guess as to what the questioner intends is correct.

Succinctly, it is not necessarily a "bad question" but you should be aware that it does not really require mathematics to solve. In fact, it is helpful to be aware of this, because if you approach these questions by not "thinking too hard", you should arrive at the answer quickly, but if you do think too hard, you are likely to not be "guessing", and "guessing" is what many of these questions test. PST 08:28, 24 February 2010 (UTC)

For those who intend to criticize the above post: I included a response to the OP (a hint) before digressing on another topic, and thus the OP can decide whether or not he/she will read the post subsequent to the hint. Irrespective of whether he/she does so, his/her question will be answered upon reading the first nine words of my post (the hint). PST 08:33, 24 February 2010 (UTC)

multivariable simultaneous algebra

This one problem is stumping me. I'm supposed to solve for K and a.

$4.4781={K}\times 400^{a}$

$2.2581={K}\times 200^{a}$

I'm interested in how you would do this, not only the answer. Thank you very much! 198.188.150.134 (talk) 08:54, 24 February 2010 (UTC)

Try dividing the first equation by the second.

More generally, there are two basic ways to deal with such problems:

1. From the first equation find an expression for K in terms of a, and substitute it in the second.
2. Do some computation with the two equations which cancels one of the variables.

All this assumes you are comfortable with logarithms. -- Meni Rosenfeld (talk) 09:09, 24 February 2010 (UTC)

Still not 100% yet. Would you mind posting the steps of how you do this? 198.188.150.134 (talk) 09:13, 24 February 2010 (UTC)

Nevermind I finally got it! K=.012 a=.987 Thanks for your help Meni! 198.188.150.134 (talk) 09:20, 24 February 2010 (UTC)

Squares and cubes

If n^3 is a perfect square and n^2 is a perfect cube, does this prove that n = x^6 where x is an integer? —Preceding unsigned comment added by 59.189.218.80 (talk) 10:52, 24 February 2010 (UTC)

Suppose some prime p appears with exponent a in the prime factorisation of n (in other words, the highest power of p that divides n is p^a). Then then exponent of p in the prime factorisation of n³ is 3a. But since n³ is a square, then 3a must be even (why ?). What does this tell you about a ? In a similar way, you can conclude that 2a is a multiple of 3 (why ?). What else does this tell you about a ? Put these facts together, and you can state something about the exponent of any prime in the prime factorisation of n, which will tell you that n is a 6th power. Gandalf61 (talk) 13:23, 24 February 2010 (UTC)

"Groundfield"

the term groundfield (also seen as ground field) is used often, but nowhere defined. Now, I think I've got a pretty good idea of what it means -- an algebraic field used as a basis for a more complex mathematical object built on top of that field -- but it seems to be a piece of jargon, rather than a precise term. Can anyone with a real mathematics background confirm this, or is there a formal definition? -- The Anome (talk) 15:39, 24 February 2010 (UTC)

I only remember that term used when working with a vector space -- where the ground field of an F-vector space is the field F. More generally one uses the term ground ring to denote the ring R when working with R-modules. Of course the term might be ambiguous if a module M is a module over several rings -- for example if M is an A-module, where A is an algebra (with unit) (in this case the term ground ring usually refers to the ring that A is a module over). Aenar (talk) 16:44, 24 February 2010 (UTC)

what is with #9?

why when we add the integers that result from mutiplications of 9 we always get 9?is there other integer has the same thing? for example:(9).(3)=27,7+2=9 or , (9).(75)=675,5+7+6=18,8+1=9 and so on.respectfullyHusseinshimaljasimdini (talk) 16:54, 24 February 2010 (UTC)

It has an explanation at the number's article: 9. See also Casting out nines.--JohnBlackburne^words_deeds 17:16, 24 February 2010 (UTC)

See also divisibility rule. -- Meni Rosenfeld (talk) 19:10, 24 February 2010 (UTC)

Seperation of simultanious speech signals

I'm doing some experiments on speech separation.

My situation is: consider a conference with n number of speakers. There are n microphones to record the speech. But each of the microphone will get a weighted speech of all the users. I want separate each speakers using the outputs from all the microphones. Mathematically is it possible?? —Preceding unsigned comment added by JephyVarapuzha (talk • contribs) 17:32, 24 February 2010 (UTC)

If the weighting is linear (which I believe is true), and you know the weights, then yes, this is just solving a system of linear equations. Otherwise it depends. -- Meni Rosenfeld (talk) 19:16, 24 February 2010 (UTC)

If you have some criterion to work out when you have separated the signals, rather than simply remixed them, you can work out the weights yourself. See Independent component analysis and blind source separation for more information on techniques to do this. -- The Anome (talk) 21:40, 24 February 2010 (UTC)

As with any challenging engineering problem, the problem is under-defined. Each source will have its own 1/r^2 amplitude decay to each different microphone, as well its own time delay T to said microphone. Thus there will be 2mn unknowns for n speakers and m microphones (n*m amplitude decays and n*m time delays). There are some interesting things you can do with real human speech though...most conversations have one person speaking while the others listen, so it is not the case that all n speakers are talking at the same time. Also, each person's voice has a particular frequency spectrum. It is possible to use this information to separate out the different conversations using time and frequency domain analysis. It might be useful to have the microphones in pairs in order to do some triangulation/direction finding using phase correlation. I barely scratched the surface of this stuff for my undergrad thesis. Zunaid 21:35, 25 February 2010 (UTC)

what we call number 2?

we read all the time about certain numbers or integers and we call them with certain name,like perfect numbers for example my question here what we have to call number 2 if it is the only numbers as far as i know that satisfys a^a=a*a=a+a and a!=a,offcourse regardless number 1.Husseinshimaljasimdini (talk) 17:33, 24 February 2010 (UTC)

What's wrong with the name "two"? --Tango (talk) 17:35, 24 February 2010 (UTC)

Perhaps they are distinguishing between its name and what it is called like in Haddocks' Eyes? ;-) Dmcq (talk) 18:04, 24 February 2010 (UTC)

The same thing that's wrong with calling an infinitesimal "zero" - just because there is a unique number with a given property in some system, doesn't mean we shouldn't name the property - either because this uniqueness is nontrivial and we need the terminology before we establish it, or for use in more general settings.

In this case, the property is probably not important enough to be named. -- Meni Rosenfeld (talk) 19:13, 24 February 2010 (UTC)

You make a good point, but I don't think it applies here. The property will be limited to two in any reasonable number system. a+a=2a in any number system which includes the integers and a sensible multiplication and a*a=2a means a=2 in any number system with cancellation. Those are very weak restrictions to put on your number system. --Tango (talk) 19:51, 24 February 2010 (UTC)

In some sense, the answer is that we call 2 "binary", as in binary operator. As Tango points out, addition, multiplication, and exponentiation form a hyperoperation hierarchy in which for each successive operator O_n we have a O_n b = a O_n-1 a O_n-1 ... O_n-1 a (where a appears b times) for any positive integer b. So when you write a * a = a + a, a appears twice on the RHS (because + is a binary operator), and thus a solution is 2. Likewise with a ^ a = a * a. The next function in the hierarchy is tetration, and 2 is a solution to ^aa = a^a, as it is to the general equation $a\uparrow ^{n}a=a\uparrow ^{n-1}a$. 58.147.60.130 (talk) 03:00, 25 February 2010 (UTC)

is there a statistical difference between these data sets?

Is there a statistically significant difference between these data sets? :

first:

1:no    2:no    3:yes   4:yes   5:yes   6:yes   7:no    8:no    9:yes   10:yes
11:no   12:no   13:yes  14:yes  15:yes  16:yes  17:no   18:no   19:yes  20:yes
21:yes  22:yes  23:no   24:yes  25:no   26:yes  27:no   28:yes  29:yes  30:no
31:yes  32:no   33:yes  34:no   35:no   36:no   37:yes  38:yes  39:no   40:no
41:no   42:yes  43:no   44:no   45:yes  46:yes  47:yes  48:no   49:no   50:no
51:no   52:no   53:no   54:no   55:yes  56:yes  57:no   58:yes  59:yes  60:yes
61:yes  62:no   63:yes  64:yes  65:no   66:yes  67:no   68:no   69:yes  70:no
71:no   72:no   73:no   74:no   75:no   76:no   77:yes  78:yes  79:yes  80:no
81:no   82:yes  83:no   84:no   85:yes  86:yes  87:no   88:no   89:yes  90:no
91:no   92:no   93:yes  94:no   95:yes  96:no   97:no   98:yes  99:no   100:yes

second:

1:yes   2:no    3:yes   4:yes   5:yes   6:yes   7:yes   8:no    9:no    10:no
11:yes  12:yes  13:no   14:yes  15:no   16:no   17:yes  18:no   19:no   20:yes
21:yes  22:yes  23:no   24:yes  25:no   26:no   27:yes  28:yes  29:no   30:no
31:yes  32:no   33:no   34:yes  35:yes  36:no   37:no   38:yes  39:no   40:no
41:yes  42:no   43:no   44:yes  45:no   46:no   47:yes  48:no   49:no   50:yes
51:yes  52:no   53:no   54:no   55:yes  56:yes  57:yes  58:no   59:no   60:no
61:yes  62:no   63:yes  64:no   65:no   66:yes  67:no   68:yes  69:yes  70:yes
71:no   72:yes  73:yes  74:no   75:no   76:yes  77:yes  78:yes  79:yes  80:yes
81:no   82:yes  83:yes  84:yes  85:yes  86:yes  87:yes  88:yes  89:no   90:no
91:yes  92:no   93:yes  94:no   95:no   96:no   97:no   98:no   99:no   100:yes

Thank you. 82.113.106.89 (talk) 19:32, 24 February 2010 (UTC)

We'll need more information. What are those values? In particular, is the order significant? --Tango (talk) 19:48, 24 February 2010 (UTC)

They're independent Bernoulli trials so the order doesn't matter. I understand you probably want to know what those values are to be able to be able to assign some kind of prior probability, so suffice to say that these trials are a test to calibrate divination. You can consider the question as "Did the subject divine currectly?" The question is, is there any statistically significant difference between the first and second set of results? Finally, I must add that this is only a calibration, it is not a final result. Thank you for any analysis you may be able to provide; again, as you can also see from the posting history (82.113.106.89) that this is not homework.

This sounds like a homework problem. Are those yes/no's supposed to be the results of independent Bernoulli trialss? OK, identify the null hypothesis and use the binomial test. 75.62.109.146 (talk) 20:08, 24 February 2010 (UTC)

It's not homework. (You can look at the posting history - 82.113.106.89 - for a hint). Yes, they're independent Bernoulli trials. 84.153.235.131 (talk) 21:00, 24 February 2010 (UTC)

How many "yes"s and how many "no"s are in each list? With that I could answer the question (I don't necessarily agree that we don't have enough information). But I'm too rushed to count them right now. Michael Hardy (talk) 21:21, 24 February 2010 (UTC)

There are 47 "yes"s in the first list, and 51 in the second. Perhaps they are easier to analyze without the noise of numeric labels and spaces:

0011110011001111001111010101101010001100010011100000001101111011010010000000111001001100100010100101
1011111000110100100111010011001001100100100100100110001110001010010111011001111101111111001010000001

There are 29 matched 0s, 27 matched 1s, 24 0-1 columns, and (thus) 20 1-0 columns. --Tardis (talk) 00:24, 25 February 2010 (UTC)

You didn't say how many "no"s there are in each! Also: are you the original poster? If so, how do you know these are matched? Oh.... I see you did later say, not explicitly, how many "no"s there are. But how do you know they're matched? Michael Hardy (talk) 04:36, 25 February 2010 (UTC)

As there are 100 samples in each set, the number of "no"s is pretty obvious given the number of "yes"s, is it not? My comments about matching may be ignored, since the OP explicitly said that they were independent Bernoulli trials (and as such the only possible meaningful observations are that they are 47/100 and 51/100 "yes", specifically refusing to reduce the fractions even if we could). --Tardis (talk) 05:29, 25 February 2010 (UTC)

No, the difference is not at all statistically significant. Frequentists would say that the p-value is about 57%, much higher than the desired 5%. -- Meni Rosenfeld (talk) 08:33, 25 February 2010 (UTC)

With so little explanation provided by the OP, I thought a matching analysis seemed prudent as well, but you managed to post before me. I did verify that our numbers agreed.58.147.60.97 (talk) 08:38, 25 February 2010 (UTC)

The first case is a binomially distributed observation i=47, n=100 having an unknown probability p₁. The likelihood function for p₁ is $\scriptstyle p_{1}^{47}(1-p_{1})^{53}.$ This likelihood function is a beta distribution function having mean value $\scriptstyle {\frac {47+1}{100+2}}=0.47$ and standard deviation $\scriptstyle {\sqrt {\frac {48\cdot 54}{(102)^{2}\cdot 103}}}=0.05$. In short $\scriptstyle p_{1}\approx 0.47\pm 0.05$. Similarly the parameter for the second distribution is $\scriptstyle p_{2}\approx 0.50\pm 0.05$. The difference $\scriptstyle p_{2}-p_{1}\approx 0.03\pm 0.07$ is not significantly different from zero. Bo Jacoby (talk) 16:03, 25 February 2010 (UTC).

"Bo", you're finding confidence intervals for the two samples SEPARATELY, using variances for the two separately. I'd use the variance for the difference. (It's still not significant.)

Michael, The variance for the difference is the sum of the variances of the terms, so the standard deviation for the difference is the square root of the sum of the squares of the standard deviations of the terms. $\scriptstyle {\sqrt {0.5^{2}+0.5^{2}}}=0.7$. Bo Jacoby (talk) 14:47, 26 February 2010 (UTC).

"Tardis", you write: "As there are 100 samples in each set, the number of "no"s is pretty obvious given the number of "yes"s, is it not?" I'd say no: not unless you know that 100 is how many there are. As I'd said, I hadn't gone through and counted them.

Whether I'd change my assertion about the difference not being significant if I treated them as paired (or "matched"?) is something I haven't thought about yet. Michael Hardy (talk) 18:25, 25 February 2010 (UTC)

The OP did put a serial number next to each observation, so you can just look at the last one and see that it's 100. Also, if you look at the source code you'll see the OP arranged it in a neat 10*10 table. -- Meni Rosenfeld (talk) 19:47, 25 February 2010 (UTC)

Oh...I was too busy being annoyed by the lack of obviously needed summary statistics to notice the "serial numbers". To expect us to do that or to look at the source code seems rather demanding. Michael Hardy (talk) 22:11, 25 February 2010 (UTC)

I suppose the OP didn't expect us to look at the source code but rather that MediaWiki would format his table correctly. -- Meni Rosenfeld (talk) 06:40, 26 February 2010 (UTC)

I took the liberty of reformatting it.—Emil J. 11:33, 26 February 2010 (UTC)

Thanks for the analysis Michael Hardy and others. If you treat them as paired or matched (which is consistent with the methodology used to arrive at them), then in that case is there a statistically significant difference? 84.153.239.187 (talk) 13:06, 26 February 2010 (UTC) (OP)

Fisher's exact test on

 53 49
 47 51

gives a p-value of 0.67, ie not significant. The contingency table is

29 24
20 27

but Fisher's exact test isn't really relevant to this form of the data, if I understand the OP's question. More appropriate might be McNemar's test, which gives a p-value of 0.6511, ie non-significant. HTH, Robinh (talk) 22:11, 27 February 2010 (UTC)

If you treat them as paired or matched, there are 29, 24, 20, and 27 occurrences of the pairs (0,0), (0,1), (1,0) and (1,1) respectively. Knowing the probabilities p₀₀, p₀₁, p₁₀, and p₁₁=1− (p₀₀ + p₀₁ + p₁₀), the observations have a multinomial distribution. The corresponding likelihood functions for p_ij define mean values ± standard deviations. p₀₀≈0.29±0.04, p₀₁≈0.24±0.04, p₁₀≈0.20±0.04, and p₁₁≈0.27±0.04. These estimates are not significantly different from one another. So the answer is still: No sir, there is no statistical difference between the data sets. Bo Jacoby (talk) 20:17, 28 February 2010 (UTC).