Numbers Definition and 1000 Threads

An easy question. All "odd" number can be expressed as a sum of consecutive natural numbers. Example: 35=17+18 35=5+6+7+8+9 35=2+3+4+5+6+7+8Question: Demonstrate that prime numbers (except for the "2"), can only be expressed as the sum of two consecutive natural numbers.

$x_0,x_1,-----,x_{2004} \in Z , \, x_0=0, \mid x_n \mid =\mid x_{n-1}+1\mid $ $for, \,\, 1 \leq n \leq 2004$ (1) $find :\,\, min\mid x_1+x_2+x_3+ ------+x_{2004}\mid $ (2) get a set of numbers $ x_1,x_2,-----x_{2004} $ satisfying your answer

Hi lovely people, I recently came across a video http://www.youtube.com/watch?v=w-I6XTVZXww that said if you add all of the natural numbers from 1 to infinity, the answer is... What do you think it is? Infinity or something like that? They said it was -1/12. I watched the proof but I don't...

HI - what is the best textbook, website or another resource to learn about ordinal numbers (Alph -0, Alph-1, etc) and their properties? I don't even know if there is a specific subfield of set theory that they fall under. Thanks!

Hi, In BIDMAS (brackets, indices, division, multiplication, addition, subtraction) it shows clearly that indices should happen before subtraction, so why does -52=25? Because the real working (according to BIDMAS) should put the square first and then put the negative sign on, which would mean...

How to calculate <n_i ^2> for an ideal gas by the grand partition function (<n_i> is the occupation number)? In other words, I like to know how do we get to the formula <n_i>=-1/\beta (\frac{\partial q}{\partial\epsilon}) and <n_i ^2>=1/Z_G [-(1/\beta \frac{\partial }{\partial\epsilon})^2 Z_G]...

How would it make any sense to use dimensionless numbers to represent physical things? From wikipedia: If you are comparing the strength of forces, and you are using these numbers to do so, I would have thought that these numbers would represent units of - well, force. Clearly I must...

The set of the real numbers is closed. For me this is nearly trivial (*) but perhaps I miss something; a colleagues insists that there are some deeper considerations why this is far from trivial - but I don't get his point (*) A) A set is closed if its complement is open; the complement...

Compute $\sqrt{2000(2007)(2008)(2015)+784}$ without the help of calculator.

Homework Statement Show that (a^b)^c can have more values than a^(bc) Use [(-i)^(2+i)]^(2-i) and (-i)^5 or (i^i)^i and i^-1 to show this. Homework Equations The Attempt at a Solution I'm writing out the second one as the first one is long: i^i = e^ilni lni = i (\pi...

Hello. Not long ago, I did a study on numbers, raised to an exponent. I noticed that a "pattern" remained, and I could find a general formula. Let: a , n, k \in{N}, when "a" is odd number: I define: a _1, \ldots , a_k \in{N}, as the consecutive addends, such that: Let: a , n, k \in{N}...

I am studying complex numbers and, hard as I try, I cannot see great difference between them and the conjugate numbers known and used since 500 B.C. (http://en.wikipedia.org/wiki/Quadratic_formula#Historical_development) to solve a quadratic equation p/2 \pm \sqrt(p/2 ^2\pm q) where the sum...

Homework Statement convert 14.334 to the base 16 to a no. to the base 10 Homework Equations sum ofeach no.*its poand take position weigth The Attempt at a Solution the integer part 14 will be 4*16^0+1*16^1=20 to the base 10 the decimal part 0.334 is my problem i multiply 0.334 *16...

Homework Statement z=1-i e^{iz} = ? I have to solve this problem and than picture it. Homework Equations The Attempt at a Solution e^{iz} =e^{i(1-i)}=e^{i+1}=e^i*e I don't really understand how to picture this result. I assume their is an other way, in which the result has a...

I know I am jumping ahead, as I am still working through part 1 of Spivak's Calculus, and am absolutely not properly equipped to prove his properties (if they are indeed provable). However, as I am trying to work on my proofing skills, my interpretation thus far has been, that these 12...

http://en.wikipedia.org/wiki/GE_Wind_Energy The rotor swept area is 4657 m^2, and wind speed is 11.5 m/s. Produced power 1.5MW. I assumed that the air density is 1.2kg/m^3, and computed, that the theoretical upper limit for produced power would be 4.25MW. This was obtained by computing the...

Can someone explain in simple terms "K mixing level" and "K selectivity" in nuclear decay processes ? ...as it relates to the "K quantum number". And does it relate directly to the nuclear angular momentum?...and selection rules? How so? Dumb it down for me please. Thank you kindly.

Homework Statement If m and x are two real numbers where m ε Integers, then e2micot-1x{(xi+1)/(xi-1)}m, (where i=√(-1)) is equal to : (a) cos(x) + isin(x) (b) m/2 (c) 1 (d) (m+1)/2 Homework Equations The Attempt at a Solution I seriously have no clear cut idea of how to...

Hi When we talk about a fluid moving at low Mach numbers, it is said to be incompressible. But does this mean that the flow is incompressible (i.e., material derivate is zero) or does it imply that the fluid itself is incompressible (constant density)? If anybody has a reference (book...

What are the critical numbers of f(x) = (x2+3)/(x-1) I took the derivative and found f'(x) = (x+1)(x-3)/(x-1) so I thought the critical numbers were 3, ±1 ...sadly, this is incorrect. I always thought the definition of a critical number was where f'(x) = 0 or DNE. But for this to be...

Homework Statement Hey everyone I couldn't really fit a good description into the title. Basically here's what it is. Consider an atom in which all the states with principal quantum number n = 3 and angular momentum quantum number l = 2 are occupied by electrons (this is called a closed...

Homework Statement . Prove that ##\{x_n\}_{n \in \mathbb N} \subset \mathbb R## doesn't have any convergent subsequence iff ##lim_{n \to \infty} |x_n|=+\infty##. The attempt at a solution. I think I could correctly prove the implication ##lim_{n \to \infty} |x_n|=+\infty \implies## it...

Homework Statement Given two positive numbers calculate their weighted average(Note: This is not actually a homework question, but silly me have doubts, so I put it in this category) Homework Equations Two given positive numbers a,b. No separate weights are available to weigh each...

Homework Statement use Eular's formula to find the greatest prime number under : If I wasn't forced to use this method I would set up a program to loop through checking for primes Homework Equations F(n) = n^2 + n + 41(0 to 39) or depending on your PoV f(n) = n^2 - n + 41(1 to...

Hello. I contribute a small study: For : a, b, c, d, e, n \in{N} / a^3=b^3+c^3+d^3+e^3 Family 1ª) a=3n^6+3n^3+1 b=3n^6+3n^3 c=3n^4+2n d=n e=1^3 Example: For \ n=1 \ then: 7^3=6^3+5^3+1^3+1^3 For \ n=2 \ then: 217^3=216^3+52^3+2^3+1^3 For \ n=3 \ then: 2269^3=2268^3+249^3+3^3+1^3 For...

I'm using the Modern Physics by Tipler (6th edition) book. In sec 7.1 it talks about the first excited state being either E_(112 ) E_(121 ) E_(112). My question is what is the process of finding the n_(1),n_(2),n_(3) quantum numbers ? How i understand you pick random values and from their find...

Homework Statement Consider a spinless particle in a central field in a state described by: \psi_a(r) = (x^2 - y^2) e^{-\alpha r^2} \psi_b(r) = xyz e^{-\alpha r^2} Find quantum numbers l and l_z (or their appropriate superposition) for these two cases. Homework Equations...

How is it possible that the more numbers in an equation, the worse I become at it, where the more variables there are, the easier it is and the faster I can do it? Is there possibly a universal law regarding this, or am I alone in suffering this condition? I love math, but as it turns out, I...

Question 1) Write ⊆ or ⊄: {x/(x+1) : x∈N} ________ QNOTE: ⊆ means SUBSET ⊄ means NOT A SUBSET ∈ means ELEMENT N means Natural Numbers Q means Rational Numbers Question 2) Which of the following sets are infinite and uncountable? R - Q {n∈N: gcd(n,15) = 3} (-2,2) N*N {1,2,9,16,...} i.e...

Homework Statement The statement that is purported to be true is \frac{a/b}{c/d} = \frac{ad}{bc}Homework Equations The Attempt at a Solution So, I am going along with my proof, and I believe it to be going nicely. However, there is one step that I am unsure of: \frac{\frac{a}{b} d}{c} =...

Intended for high school students: Let a positive integer be called "increasing" if every one of its digits is strictly greater than the one directly to the left of it. For example, 289 is increasing but 6788 is not. How much 3 digit increasing numbers are there? How much 4 digit increasing...

Hello All, I am having 1000 consecutive numbers always starting with 1, and there could be upto 24 missing numbers from the list. My question is, what pieces of information should I have in order to find out those missing numbers. So far I came up with, 1. Sum of missing numbers. 2...

I am confused if complex numbers really are vectors. They seem to behave as vectors in addition, but not in multiplication. So why are the complex numbers defined to be vectors although they don't follow the same principles always. Another confusing thing for me is the "complex vector"...

Hello, I have an array of length M. Some of the indices are non-negative. I need to derive a function/analytic expression (preferably linear or convex) that finds the span of indices for which the indices are non-negative. Example: Let p denote the desired vector. Let, p = [0 0 0.3 2.6...

After all, numbers are part of the universe. :biggrin:

Hello everyone. Thanks for this amazing forum. I hope I can be forgiven for asking for a software problem rather than a topic specific puzzle. I am looking and happy to pay for a downloadable program I can run from my desk top which will return all possible combinations of numbers for...

At approx. step 46000 the following code puts "Inf" out, but I can't understand why since it multiplies by itself a value smaller than 1. The sin is ok, but the loop for the power fails : #include<stdio.h> #include<math.h> double power(double x, int n) { double result=1.0; for(int...

I'm modelling the Diesel cycle using excel and I'd like to have the 4 different states numbered on my P-v diagram. I can mark these four points using shapes, but I'm not sure how to do it using numbers instead of these markers. Can this be done?

I have been reading about dimensionless numbers. My question is: are there any natural dimensionless numbers? All seem to be either equations, like the fine structure constant, or ratios, like β - mass/mass. 22 of the 26 standard model inputs are mass related and only become dimensionless...

Which is the best book on complex numbers? I'm new to this topic so I would like to begin my study with the basics. I prefer books that deal with concepts in a great detail for a better understanding. The book must also contain good problem sets(high order thinking) for practise. I'm aiming to...

Homework Statement If f'(x) = x^2(x-1)^2(x-3)^2, how many local extrema does f(x) have? Homework Equations Extrema occur at critical points. Critical points are values of x such that f'(x) = 0 or = ±∞ The Attempt at a Solution Not all values that zero the derivative are critical points...

Homework Statement I'm trying to determine the number of critical numbers for the function (3x-x^3)^(1/3). Homework Equations Critical numbers are where the derivative of the function is = 0 or does not exist. Critical numbers must also exist within the domain of the function. The...

C++-- while-loop Homework Statement calculate the 3 largest number of a sequence Homework Equations you ask the user to input a sequence of numbers, and enter the value 0 to stop entering numbers. c++ The Attempt at a Solution i managed to calculate the the maximum of the series of...

For instance, math 101 is beginning algebra, math 102 is intermediate, 103 is a reasoning course, 109 and 110 are both college algebra, but 109 comes with "modeling" and either are fine but one is preferred over the other. I just want math out of the way, and 109 is the only thing that fits my...

Homework Statement Given: sin(x) = \frac{e^{ix}-e^{-ix}}{2} Show that sin(x) can be written as: sin(x) = \sum_{n=0}^n \frac{x^{(2n+1)}}{(2n+1)!} Homework Equations e^x = \sum_{n=0}^n \frac{x^{n}}{(n)!} The Attempt at a Solution I'm unsure how to treat the imaginary number in...

Other than the fact that prime numbers are infinite?

Thank you for viewing my thread. The following is a problem from my homework that is submitted online. I have already exhausted my attempts, but I'm curious as to where I went wrong. I would appreciate if someone could lead me towards the solution. Homework Statement Find the critical...

Homework Statement Give a possible set of values of the four quantum numbers for all the electrons in boron atom and a nitrogen atom if each is in the ground state. I know the principles behind this. It comes from Zumdahl's Chemistry textbook (ninth edition, p. 345, number 95). The...

I was thinking that if you could use quaternions to rotate an object using quaternion algebra that there might be a way to rotate an object using complex numbers in some fashion. I was looking at quaternion rotation of a vector and the amount of operations seemed to be a lot. Of course it levels...

I need to prove that H_n = \ln n + \gamma + \epsilon_n Using that \lim_{n \to \infty} H_n - \ln n = \gamma we conclude that \forall \, \epsilon > 0 \,\,\,\, \exists k \,\,\,\, such that \,\,\, \forall k \geq n \,\,\, the following holds |H_n - \ln n -\gamma | < \epsilon H_n <...