Mathematics

This thread should be used to post your favourite theorems, not only their names, but also what they state. Fundamental Theorem of Arithmetic Each integer greater than 1 can be expressed as a product of primes, and, except for the order in which these primes are written, this can be done in only one way. and Central Limit Theorem If [math]X_1,...,X_n[/math] are independent identically distributed, with mean [math]\mu[/math] and finite variance [math]\sigma^{2}[/math], then [math]\bar X[/math] is approximately [math] N(\mu ,{{\sigma ^2 } \mathord{\left/ {\vphantom {{\sigma ^2 } n}} \right. \kern-\nulldelimiterspace} n}) [/math] for large n, no m…

Cantor used the expression 2^aleph0 in order to represent the magnitude of R set. Since base 2 can be represented as a tree diagram, we can use it in order to research a collection of infinitely many elements. For example, let us look at the infinitely long Top_to_Bottom blue tree, which is also represented as {1, 2, 4, 8, 16, ...}. It is obvious that we always find finitely many leafs in any arbitrary level of this tree, so this tree cannot have the magnitude of 2^aleph0. Furthermore, since in any arbitrary level we are still in N set, we can never define aleph0 as a transfinite number. Now let us say that we start by a collection of infinitely man…

I come across this when I was investigating the game Hex, I've heard of it before but can someone with a little bit more knowledge explain this to me. It would be appreciated,

can somebody help? prove or disprove, If G is a group in which every proper subgroup is cyclic, then G is cyclic. in additon, can u explain cyclic

after a little fooling around, here's a something that i found somewhat interesting: let S be the set of all "sqare" numbers {s such that s = n^2, where n in N} let T be the set of all "triangular" numbers {t such that t = (m^2 + m)/2, where m in N} let W be the intersection of S and T, whose elements w satisfy both w = n^2 and w = (m^2 + m)/2 where both n and m are in N. now take the ith element of the set W, wi which satisfies wi = ni^2 and w = (mi^2 + mi)/2 it can be shown that: [math]\lim_{i\to\infty} \frac {m_i}{n_i} = \sqrt{2}[/math] since ni and mi are integers, it is almost imaginable that sqrt(2) can be expressed as a ratio of integers…

There was a time in an alien society where the median age of an alien was thirty(in our solar years). Six years after this point, the median age for this alien society was 36. Is there anyway this information can imply the age or average age that aliens die in this society? How would that be represented mathematically? I think that birth rate for aliens and death rate are the two important things to take into consideration for this problem, but since I am only in basic algebra now, I don't see the way to figure out the answer, although I am curious. Does any one know how to create a mathematical framework taking into account the limited information given?

I downloaded a MS file about maths. I don't know why the symbols and numbers cannot be shown clearly and correctly. Anyone help me>?

r is circle’s radius. s' is a dummy variable (mathworld.wolfram.com/DummyVariable.html) a) If r=0 then s'=|{}|=0 -> (no circle can be found) = A b) If r>0 then s'=|{r}|=1 -> (a circle can be found) = B The connection between A,B states cannot be but A_XOR_B Also s' = 0 in case (a) and s' = 1 in case (b), can be described as s'=0_XOR_s'=1. You can prove that A is the limit of B only if you can show that s'=0_AND_s'=1 -> 1 A collaction of elements, wich can be found on many different scales, really approaching to some given constant, only if it has finitely many elements.

In order to define a framework that is researched and manipulated by a formal language, we first have to define the limitations of the formal language itself. Since any language cannot work without information, we first define what are the minimal and maximal conditions that cannot give us any information, when we try to research them. The minimal concept that cannot be researched is Emptiness, because no information can be found in it. The maximal concept that cannot be researched is Fullness, because it is beyond measurement or manipulation of information. So, formal language can work, if and only if we can measure or manipulate information by our rigoro…

How did mathematican define sin A ( A >90 degree) as [MATH]y/r[/MATH]? Are there any proofs to [MATH]sin 150 = sin 30 [/MATH]>?

this thought came to me while making my previous thread. what was used before the cartesian system?

Since everyone's posting questions here, I thought I'd post a nice question I got on my Analysis III assignment last week. It's a bit involved (and somewhat pointless) but I thought it was pretty nice for some reason: Let [math]f : [a,b] \to \mathbb{R}[/math] be a regulated function, and let [math](x_n)_{n=1}^{\infty}[/math] be a sequence in [math](a,b)[/math] such that [math](x_n) \to a[/math] as [math]n \to \infty[/math]. Show that [math]( f(x_n) )_{n=1}^{\infty}[/math] is a Cauchy sequence.

if anyone is on and able to check my work on these couple diff eqs i would really appreciate it. The prof doesnt assign problems with solutions so i cant ever check my work . That is frustrating....heres a couple x' = 3x - 2y + sin(t) y' = 4x - y - cos(t) i used the differential operator and the method of undetermined coeffcients to get this answer..dont think its right because its so ugly x = [e^(t)] * [Acos(2t) + Bsin(2t)] - (1/10)sin(t) + (7/10)cos(t) y = [Ce^(t)] * [Dcos(2t) + Esin(2t)] + (11/10)sin(t) + (3/10)cos(t) And, if you are up to it, heres another one...D is the differential operator (D-3)[x] + (D-1)[y] = t (D+1)[x] + (D+4)[y] = …

I have asked this in other forums but I don't get much of a response. In what dimension does a shadow exist. I thought that as everything in our universe is made of atoms you could not construct a 2d object. However obviously a shadow is not comprised of atoms but of a deprivity of photons. You can measue the length and width of a shadow so is a shadow a truely 2 dimensional object. I don't think a shadow can have a volume. However thinking back how can a shadow exist in two dimensions because a shadow will lay across rounded objects and I think planar objects can not be twisted.

Im trying to solve this equation for along time I tries to use the formulas for power 3 but it ddint help I think it shuld be factoring the left side but I couldnt make it z^9-z'^9=9|z|^4 please help

Pigeon Hole Theory. Is it still relevant? How can we apply it to our daily lives and mathematics? What exactly is it about? Is it applicable only for numbers?

Two questions. They may be very simple and easy for you, but a brain flash. 1.the square root of 20 consecutive positive intergers have the same intergral number, find that number. 2.(a-b)(c-d)/[(b-c)(d-a)=3 Find (a-c)(b-d)/[(a-b)(c-d)]

Any number can be distinguished form another number by at least two basic properties, which are: a) Structure b) Quantity Let me show you a nice thing, for example, let us take number 26. Now we can show that this quantity has more than one internal structure, which is based on the base value that we are using: Number 26 represented by base 10: ^0- 0123456789 |||||||||| |_|||||||| |__||||||| |___|||||| Base 10 = |____||||| |_____|||| |______||| |_______|| |________| ^1- | | 1 0 …

how would I go about finding sin(10^40) ? [10^40 in radians]

I'm coming up to a stage where I need to write an essay about some topic in Mathematics. Now, before you ask, the purpose of this thread is not to say "omg he's trying to get us to write his essay n00b fs". I'm trying to look at particularly interesting topics in mathematics to formulate some kind of essay title - and no, I don't want you to give me titles. Contrary to some common misconceptions about students, I want (and need) to do this by myself. I've had a fair amount of mathematical experience now, but I'm struggling to find some topic I find interesting enough to write an essay on. If anyone has any suggestions, then that would be much appreciated.

Why can we integrate ln?

Ok. Let's say that there is a given function y=f(x) And if your inverse of f(x) is exactly the same as f(x). What can you say about the symmetry of the graph of f(x)?

Its the last round of a game show. You have to choose between three doors labeled One, Two, and Three. Behind each door is a prize. Now two of the doors contain goats while one of the doors contains a brand new car. Your asked to choose a door number. Lets say you choose 1. The Host then reveals a door other than the one that you choose, lets say that it is door number 3, and it has a goat behind it. You are then asked if you would like to change your choice to number 2 or stick with number 1. Now the riddle is, does this make a difference? Should you change your answer or stick with it? Have you all heard of this wierd puzzle? I have a book about it, but t…

What are all of the mathematical constants that we know of (and what are they used for). By that I mean numbers such as pi, e, y and golden ratio. And could somebody please tell me how to create mathematical symbols and formulas on this site? I know it's possible, I've seen people do it on this site before.