Wikipedia:Reference desk/Archives/Mathematics/2010 March 12

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March 12

little project

I've been studding acase related to prime numbers and how to generate them and iam very convinced it is not possible to follow acertain pattern to generate infinte set of aprime numbers, so i came up with this postulate.Lets, p0, be aprime number≥7 and c(n) are constants generated by acertain pattern or afunction where; {P0+c(n)}; is aset of prime numbers,Now,the maximum prime number, [p0+c(max)]<[P0×P0].It means that any pattern will fail to exceed[p0×p0].I did lots of experiments about this.I want to know if my project is rite and if it can be proven?Husseinshimaljasimdini (talk) 00:05, 12 March 2010 (UTC)

You're going to have to be more precise about what you mean by "a certain pattern or a function." For example, I could define a function c(n) to be

c(n) = [the (n + 5)th prime number] − 7.

Then, for example, c(0) = 11 − 7 = 4, c(1) = 13 − 7 = 6, and so on. This is a perfectly reasonable definition of a function. If I do this, and let p0 = 7, then of course p0 + c(n) is a prime number for every nonnegative integer n. —Bkell (talk) 00:59, 12 March 2010 (UTC).

MY point here is that every pattern no matter what it was; or how it defined would fail under this condition, —Preceding unsigned comment added by Husseinshimaljasimdini (talk • contribs) 10:19, 12 March 2010 (UTC) As amatter of fact to be more specific ,let the set p0+c(0)=p1,.....p0+c(n)=pn,are positve prime numbers < [P0×P0].and my example is 41+2=43+4=47+6=53+.......this pattern fails until 1601+80=41×41.Husseinshimaljasimdini (talk) 10:09, 12 March 2010 (UTC)

The problem is still that you haven't clarified what you consider to be a "pattern". Your example is apparently Euler's trinomial n² - n + 41 which gives distinct primes for n = 1 to 40: 41, 43, 47, 53, ..., 1601. It's mentioned at Formula for primes#Prime formulas and polynomial functions. Most primes from 41 to 1601 are not included in the sequence so it seems you don't require consecutive primes. A sequence with larger jumps can easily end at a larger prime. 41 + 420n is prime for n = 0 to 5: 41, 461, 881, 1301, 1721, 2141. For an example where the number of primes is as large as the initial prime, 5 + 6n is prime for n = 0 to 4: 5, 11, 17, 23, 29. These were simple linear functions. If we allow arbitrary functions then there is no limit as Bkell showed. PrimeHunter (talk) 13:37, 12 March 2010 (UTC).

O!i see. i like these answers i appreciate you. I have also noticed something i`d like to include here,can we say or state that prime numbers can be devided generally into tow types, the first one is the prime numbers that satisfy the following way;for istance; 43.17=731, now if we reverse this number we will get 137 which is aprime;or 17.2=34,which leades to 43 and so on.The second type is the primes that stay primes when we reverse the digits such as 113,101..and so on.Husseinshimaljasimdini (talk) 14:58, 12 March 2010 (UTC)

That is one way to pick a subset of the primes. See List of prime numbers for many others. Palindromic primes are those which read the same backwards (usually with base-10 implied) such as your 101. Emirps (also called reversible primes) are primes which become another prime when read backwards (again, base-10 usually implied) such as your 113. I (Jens Kruse Andersen) happen to be the discoverer of the largest known emirps [1] and several other computational emirp results.[2][3][4][5][6][7] Primality of a number is not a base-dependent property. Studying special properties of the decimal digits of numbers is usually considered recreational mathematics. PrimeHunter (talk) 16:25, 12 March 2010 (UTC)

How many types of numbers are there?

I've known about complex numbers for a while, but I recently discovered the article on dual numbers. I was wondering, are there any other types of numbers? If so, what are they and how many are there? Furthermore, is it possible to have a "complex dual" number (i.e. a number in the form a + bi + cε?--203.22.23.9 (talk) 02:57, 12 March 2010 (UTC)

Well, the thing is, there's no agreed general definition of what it means to be a "number". So you have things like Cayley numbers that may not really strike you as all that "numerical". That makes this really a language question — how many classes of mathematical objects are called such-and-such numbers?

Ones you might be interested in looking at: hyperreal numbers, surreal numbers, transfinite ordinal numbers and cardinal numbers. --Trovatore (talk) 03:09, 12 March 2010 (UTC)

(edit conflict)The concept of number is a moving target and so it makes no sense to count how many types of number there are. See natural number, positive number, rational number, real number, prime number, square number. Lots of mathematical constructions generalize numbers, and some of them are even called numbers. Elements of groups, rings, and fields generalize numbers but are often not called numbers. Bo Jacoby (talk) 03:20, 12 March 2010 (UTC).

Another thought - Is there a set of numbers which can be used to solve equations such as |x| = -1 , in the same sense that the complex numbers are used to solve equations such as x² = -1?--203.22.23.9 (talk) 03:33, 12 March 2010 (UTC)

I can't make any sense of it, but that doesn't prove you shouldn't think about it. (Actually I did think about it, when I was a kid -- a point closer to a given point than that point is! But absolute values are not really analogous to quadratics -- they don't have algebraic properties that obviously generalize. Again, this is not a proof. Feel free to work on it; let us know if you get anywhere!) --Trovatore (talk) 03:41, 12 March 2010 (UTC)

Try Split-complex number. Staecker (talk) 12:23, 12 March 2010 (UTC)

(First off, no one has mentioned the quaternions yet.) If you're going to allow |x| = −1, then your "absolute value" operation is no longer a norm. Norms have very useful properties—you can think of a norm as a generalized absolute value that can be applied to objects other than standard numbers. But I suppose there could be such a thing as a "signed norm" that is allowed to be negative; there are already concepts like signed measures and complex measures, which extend the idea of a measure (essentially a length, area, or volume) by allowing it to be negative or complex, respectively. —Bkell (talk) 03:55, 12 March 2010 (UTC)

Something similar is considering spaces that have "norms" that aren't positive-definite, like Minkowski space. In that context some vectors will satisfy |x|² = -1. But that also breaks from the standard definition of what a norm is. Rckrone (talk) 06:13, 12 March 2010 (UTC)

If |x|² = -1 then |x| = i (or -i), what value for x leads to that? --203.22.23.9 (talk)

Minkowski space describes events in space and time. We can choose some coordinates and describe a point in the space as w = (t,x,y,z) where t describes the point in time and x, y, z describe the position in space. The Minkowski norm of this point is |w|² = -t² + x² + y² + z² (compare this to the usual Euclidean norm which would be |w|² = t² + x² + y² + z²). So for example the point w = (1,0,0,0) has |w|² = -1. The Minkowski norm is useful in special relativity because the value remains unchanged when we change our reference frame with a Lorentz transformation (similar to how the Euclidean norm doesn't change when we rotate the coordinate system), even though it doesn't meet all the criteria of the definition of a norm since |w|² can be negative. Rckrone (talk) 07:16, 12 March 2010 (UTC)

Absolute value of a quaternion

Resolved

What's the absolute value of a quaternion?
Absolute value of a complex number:
$|a+bi|={\sqrt {a^{2}+b^{2}}}$
Absolute value of a quaternion:
$|a+bi+cj+dk|=??$ --220.253.247.165 (talk) 06:55, 12 March 2010 (UTC)

${\sqrt {a^{2}+b^{2}+c^{2}+d^{2}}}$ In general the abs of a hypercomplex number is the sqrt of the number multiplied by its conjugate. Zunaid 07:16, 12 March 2010 (UTC)

Domain of a function

f(x)=√(x-4)÷x-2 then find the domain of a given function —Preceding unsigned comment added by 121.52.145.148 (talk) 07:26, 12 March 2010 (UTC)

I think it's {do, your, own, homework}. --Trovatore (talk) 08:17, 12 March 2010 (UTC)

Also, please be more clear when defining your function. It is difficult to determine whether the function f is defined by $f(x)={\sqrt {\frac {x-4}{x-2}}}$, $f(x)={\frac {\sqrt {x-4}}{x-2}}$, $f(x)={\sqrt {\frac {x-4}{x}}}-2$, or $f(x)={\frac {\sqrt {x-4}}{x}}-2$. If you use brackets to clarify the order of operation, we may be able to guide you towards the solution. We cannot give you the solution, of course, as this will not help you to solve any other problems of a similar nature. PST 08:36, 12 March 2010 (UTC)

How about assuming there is already a proper number of parens in appropriate places? May be there is no need to complicate things, if they are not obviously simplified too much. --CiaPan (talk) 12:12, 12 March 2010 (UTC)

Statistical significance when discussing a population

I understand the concept of statistical significance when dealing with sampling. E.g., if you draw a sample of 1000 American household's incomes for two years you can get a probability that the population income differs between the two years, and hence determine whether any statistically significant change in household income can be found.

But often social scientists use the concept of statistical significance when discussing a population, not a sample drawn from a population. For example, it can be claimed that there is a significant downward trend in GDP growth, or that there is a significant correlation between health and GDP per capita. But how can you talk of significance when you already know the population? I understand that you can use R2 to determine the fit of a regression, but is statistical significance a meaningful concept in this case? What's the intuition? Jacob Lundberg (talk) 17:51, 12 March 2010 (UTC)

Actually, you don't know the population. For example, I may have data on GDP from the beginning of time to the present, but that is not the whole population because it does not include the future. Wikiant (talk) 17:57, 12 March 2010 (UTC)

Jacob Lundberg, you have to postulate that the population is a draw from an infinite set of unrealized populations. Deming has a paper on this topic from his time at the Census. 0¹⁸ (talk) 18:34, 12 March 2010 (UTC)

Here is a link to the article at jstor [8]. 0¹⁸ (talk) 19:06, 12 March 2010 (UTC)

feedback rating

Feedback ratings tend to show how many buyers were satisfied with their purchase and not only do buyers rely on this statistic but auction houses rely on it too in deciding whether an item was defective that was returned. While 4,999 buyers that were satisfied with a food product versus only one that died will still have a 99.98% (4,999/5,000) rating while the risk of death may be ignored. What type of feedback rating calculation regardless of complexity can overcome this dilemma and give a much lower feedback rating to reflect the significance of a defective product so that potential buyers will be appropriately warned? 71.100.11.118 (talk) 22:59, 12 March 2010 (UTC)

Feedback systems tend to be game-able toys (and it's in the venue's interest for them to be game-able) but the article trust metric and its references might give you some ideas. It's silly in general to try to compress complex multi-dimensional data into a single numerical score, although that doesn't stop many from trying. 66.127.52.47 (talk) 23:54, 12 March 2010 (UTC)

Any rating system is going to have a lot of subjectivity from the raters. But let's ignore some of that; there is still a lot of subjectivity from whoever designed the rating system. From a purely mathematical point of view, you're probably going to want some kind of weighted average, but what weights do you assign? If "satisfied" is given a positive weight of 10, say, what weight should "death" be given? If it's given a weight of, say, −1000, then mathematically you're saying that 100 satisfied customers balance out one dead customer. Is that the correct value to put on death? I don't know; it doesn't seem right. On the other hand, if death is given a weight of −∞, then (under typical rules for arithmetic with ∞) your weighted average will be −∞ even if you have six billion satisfied customers and just one customer who died in a freak accident. That doesn't seem right either. So, if you're going to use a weighted-average approach, you're going to have to place numerical values on the relative worths of satisfaction and death, which is certainly going to involve a subjective judgement. —Bkell (talk) 00:03, 13 March 2010 (UTC)

Feedback ratings aren't usually used to assess safety. There are usually government standards to satisfy for that sort of thing, eg. the kitemark system in the UK. You are right that giving one number like that doesn't account for different levels of satisfaction. Often the survey will let people choose from "Very poor, poor, satisfactory, good, very good" or similar and then the number you see is everyone that said "satisfactory" or better and they don't give a breakdown. This kind of loss of information is inevitable if you want to reduce things to a single number. --Tango (talk) 00:22, 13 March 2010 (UTC)

So you want to depend upon the lightning speed government to tell you it might be best if you did not stick that food in your mouth instead of having a rating system that would alert you to the danger if the person who eat the food before your just dropped dead? 71.100.11.118 (talk) 13:04, 14 March 2010 (UTC)

I've discovered an interesting problem with weighted feedback. Many people will give the maximum rating or the minimum rating, but very few choose any value in between. I suspect that people who would otherwise give a low rating actually give the minimum rating, to maximize their impact on the overall rating, and others do something similar at the high end. To deal with this, I would look at previous ratings by each responder, and "normalize" them. Do they always give a 0 or 100 ? Then maybe those ratings should actually count as a 30 or 70. StuRat (talk) 02:30, 13 March 2010 (UTC)

Actually lots of people give responses in the middle. It is (on a scale of 1-9) numbers around 3 and 7 that tend to be ignored. --Tango (talk) 02:58, 13 March 2010 (UTC)

You might be interested in the idea of 'quality adjusted life year' by NICE which has to deal with the problem of putting a price onto people's lives. If one in 5000 people dies who would otherwise live a healthy life your talking about a cost of more than a million pounds, so you'd only need a gain of a few hundred pounds for everyone else to easily offset it according to that. However in fact a death rate of 1 in 5000 would get people in a tizzy if you announced a new product a free HD television and cable for life and said there was a downside that unfortunately one in 5000 of those given the free HD would die of an electric shock. Dmcq (talk) 08:41, 13 March 2010 (UTC)