Numbers Definition and 1000 Threads

It all starts with a shuffling. We have a sorted list, we take out groups of numbers and we insert them in random positions: Then, it is about sorting the numbers back using the same approach, in a minimum numbers of steps (which doesn't have to be identical to a number of steps taken during...

Say I draw a set of numbers {x1,x2,x3,...} from a normal distribution and apply a function f(x) to these. Will the new set of numbers {f(x1),f(x2),f(x3),...} be gaussianly distributed? I guess it depends on f(x), since for example f(x)=x would certainly mean that the new set is gaussianly...

Homework Statement In the exercise, we solved the 2D Harmonic Oscillator in kartesian (x,y) and polar (r,φ) coordinates. We found out that both have the same energy levels, but they look very different, when I plot them. What am I missing? The polar solution seems more like it. Homework...

Hi, I’m having a problem with a particular case of binomial proportion. I want calculate a confidence Intervals for a binomial proportion for an efficiency. This kind of intervals are usually defined for ratios between integers numbers but in my case I had to subtract from both numerators and...

How many ways can the numbers 1, 2, 5, 10, 20, 50, and 100 be combined to form the number 200? This is a good example of the sort of problem that gives me trouble. Usually, as here, I have a few ideas but find it difficult to proceed because I get hung up on organizing the calculations to be...

micromass submitted a new PF Insights post Things Which Can Go Wrong with Complex Numbers Continue reading the Original PF Insights Post.

For positive real numbers $a,\,b,\,c$, prove the inequality: a + b + c ≥ \frac{a(b + 1)}{a + 1} + \frac{b(c + 1)}{b + 1}+ \frac{c(a + 1)}{c + 1}

If $a$ and $b$ are two positive real, and that $a^3+b^3=a-b$, prove that $2\left(\sqrt{2}-1\right)a^2-b^2\lt 1$.

If $z = e^{(2-\frac{i \pi}{4})}$ what's $z^5$? The only way I can think of doing this is expanding $(2-\frac{i \pi}{4})^5$, but I think I'm supposed to use a simpler method (not sure what it's).

What's the ratio $\displaystyle \frac{e^{i\sqrt{x}}-1}{e^{i\sqrt{x}}+1}$ equal to? I can't work it out to anything I recognize. :confused: The answer is $\displaystyle i\tan(\frac{1}{2}\sqrt{x})$. I suppose I could work backwards from the answer, but I won't have the answer in the exam.

I am bit confused with the Eueler representation of Complex Numbers. For instance, we say that e^{i\pi}=cos(\pi)+isin(\pi)=-1+i0=-1. The derivation of e^{i\theta}=cos(\theta)+isin(\theta) is carried out using the Taylor series. I quite understand how ##e^{i\pi}## turns out to be ##-1## using...

Is it possible to divide and multiply complex numbers and real numbers and if so, how does one do that? If not, why so? Cheers for your help!

I need to calculate some quantiles for a sample of 108 real numbers with unknown mean and unknown variance. I currently store and sort those numbers, but I would try a streaming method where the numbers are not stored. In a paper is written: "If the size of the input stream, N is known, then the...

I know that 1. To show the relation is reflexive, we need to show that for any x, using the definition of R, we have xRx. The definition of R means that we must have |x - x| is even.2. To show that R is symmetric, we would have to show that if xRy then yRx. In the context of the definition...

I've had discussions with laypeople (of which, I am one) about real-world manifestations of imaginary numbers. We can never seem to find a satisfactory, concise example. I know they are used in real-world calculations for things like EM wavelengths in electronics, but if you aren't into...

Homework Statement Arg z≤ -π /4 Homework EquationsThe Attempt at a Solution I'm confused whether the answer to that would be more than -45° or less. Should the approach to arguments be the same as in negative numbers?

I would appreciate any clarification (or opinion) on the relationship between Quantum Numbers and Quantum Information. My question is related to the puzzle wether matter merely represents information or - at the basement level of reality - actually can be said to be information. Just to give...

in every radixial representation, except of course for those cases in which the numerator is a factor of some natural-number power of the radix? For the radixial system we know (i.e., because we are bilateral and have arms that have 5 fingers), this would mean that any possible natural number...

Why some row numbers like 13 ( know the reason of unlucky number), 18, 19, 20, 21, 32, 33 and 42 are missing on flights?

Does anyone know a good resource for exercises on these topics?

Homework Statement Suppose that the characteristic equation to a second order, linear, homogeneous differential equation with constant coefficients yielded two complex roots: \begin{array}{l} {\lambda _1} = a + bi\\ {\lambda _2} = a - bi \end{array} This would yield a general solution of: y =...

Cardinality of the set of binary-expressed real numbers This article gives the cardinal number of the set of all binary numbers by counting its elements, analyses the consequences of the found value and discusses Cantor's diagonal argument, power set and the continuum hypothesis. 1. Counting...

Does the set of random integers with random lengths (the number of digits), which hypothetically would generate random numbers with random lengths for eternity, produce all possible integers? It seems to me that this is a natural conclusion but I've never seen a proof of this. A more incredible...

So what am I talking about? An extension of the complex numbers between ℂ1 and the Quaternions. A tricomplex number can be written as τ = {a + bi + cj | ∀(a,b,c)∈ℝ } where: i2 = j2 = i×j = -1 = -(j×i) Thus of course, j×i = +1 What is remarkable is that such objects are closed under...

For prime numbers, $a$, $b$, $c$, $a^2 + b^2 \ne c^2$. Prove this by contradiction. So, I get that $a^2 = c^2 - b^2 = (c - b)(c +b)$ And I get that prime numbers are the product of 2 numbers that are either greater than one, or less than the prime numbers. But I'm unsure how to go from here.

Homework Statement If a force F = F_0 cos (\omega t) = \Re{\{F_0 e^{i \omega t}\}} is applied to a body of mass m attached to a spring of constant k, and x = \Re\{z\} . Show that the following equation holds: m \ddot{z} = - k z + Fe^{i \omega t} . Homework Equations Newton's second law. The...

this is just an arithmetic series but with a small difference. i will show that below The attempt at a solution the general arithmetic formula ## S_N=\sum_{n=1}^\infty n## for my problem ## S_N=\sum_{n=1000}^{2000} n ## i have to rewrite it so i will just add the even numbers ##...

In general, one thinks of complex numbers as being absolutely required in Quantum Physics but as being optional in Classical Physics. But what about modern classical electromagnetic field theory (gauge theory) in which the electromagnetic field is coupled to the field of charged particles by...

Homework Statement My Program is not showing the sum value or not returning it. A blank space is coming.Why that is so? Homework Equations Showing the attempt below in form of code. The Attempt at a Solution #include<iostream.h> #include<conio.h> Prime_Sum(int arr[30][30],int m, int n); void...

I have a simple algorithm that appears to generate many primes (or semi-primes with relatively large factors). By 'relatively large', I mean large in relation to inputs. I have tested this algorithm for small values, and of the forty (six-digit) numbers produced, 22 are prime, 16 are...

So, I'm to understand that "0 - -a = a" and I can understand this, in the context of say, being at an arbitrary reference point called "0" and then on a 2 dimension Cartesian graph of + shape and the operator "-", which can mean to reverse (so turn around 180 degrees) and then the second minus...

I have to solve the following equation: z4=i*(z-2i)4 Now, i tried to move everything but i (imaginary number) to the left side and then find the 4-th root of i, when i did that, i had four solutions, with one of them being eiπ/8. But i don't know what to do with the left side, since i get way...

I am seeing in "slow motion" the development of vectorial system. I am reading the book "A History of Vector Analysis" (by Michael J.Crowe); it seems from my acquaintance that the vector concept came from the quaternions concept; and the quaternions concept came from the act of search for...

Hello, At my exam I had to proof the title of this topic. I now know that it can easily be done by making a bijection between the two, but I still want to know why I didn't receive any points for my answer, or better stated, if there is still a way to proof the statement from my work. My work...

Would it be possible to write an equation utilizing a summation of a modular function of a Cartesian function, whose degree is dependent upon the index of the root, in that it predicts the digits less than 1 of the root, that when summed equals the computed value sqrt( n )? I already have what...

Would it be a useful result to know there is at least one prime between 16x^2+4x-1 and 16x^2+8x-5 for any odd natural number x?

A number can be random even if limitations are applied to the outcome - e.g. selecting a random integer between 1 and 5 restricts the outcome to one of 5 numbers, but the outcome is still random. The same would be true of between 1 and 2; although there are heavy restrictions, an unbiased...

Hi, How and why set of natural numbers is closed?

Homework Statement I have two complex numbers that are non real, k and z. K and z are going to be complex conjugates if and only if the product (x-k)(x-z) is a polynomial with real coefficients. Here is my answer : k=a+bi z=c+di (x-k)(x-z) = x^2 -(k+z)x+kz Homework EquationsThe Attempt at...

Hello, My task is to write a program that after reading a positive integer i (from 0 to 1000) will write how many "nice numbers" are in the interval [0..i]. "Nice numbers" are numbers that are divisible by their every digit. For example 612 is a nice number becasuse it is divisible by 6,1 and 2...

In Dirac's "The Principles of Quantum Mechanics," ket vectors are multipled by complex numbers (c1 |A> + c2 |A> = c1 + c2 |A>) and I was curious what affect this has a) on the ket vector and b) on the entire system? Also is (c1 |A> + c2 |A> = c1 + c2 |A>) equal to (|A> + |A> = |A>)? Thank you...

Homework Statement I'm reading Zee's book Einstein Gravity, I'm in the section where he said that given an array of two numbers p=(ap1, bp2), it is not a vector unless a=b. He just stated it without really showing how it must be like that. I know that a vector should satisfy a transformation...

I have to say that I am a bit confused with the use of complex numbers. I know that: 1. They have been created by mathematicians to solve the "real"ly unsolved equation of x^2=-1. 2. They are used in many aspects of physics, like waves and quantum theory, with terrific correspondence to the...

If i understand correctly, the discovery of complex numbers was linked to solving real number problems, s.a. finding square roots of negative numbers. In other words, at first there was a problem that was formulated using real numbers only that had no real number solutions, which lead to...

I have figured out a formula that generates prime numbers along with the proof that all such generated numbers are primes. The way it works is that you have to input consecutive prime numbers staring from 2 and ending at some Pn. And no it's not primorial minus or plus 1. Is this of any value...

I was wondering, why is the set of complex numbers needed to solve problems that the set of reals doesn't permit to ? I mean, in relation to the fundamental theorem of algebra, that is.

Homework Statement Hi,I have a problem regarding to one of the questions in my homework.Actually I am not trying to ask for the solution.I am just not sure what the question is asking for.Please see the attachedHomework EquationsThe Attempt at a Solution In 5(c),the summation notation stated...

Homework Statement Homework Equations see picture above The Attempt at a Solution I can follow most of the steps, but not all. I got confused with finding ##|\frac{dz}{dt}|##. It is easy to derive ##\frac{dz}{dt}## from ##z##. Normally, I would square the two components of ##dz/dt## and...

Homework Statement I don't understand example 2. For part a, I got a slightly different answer. Homework Equations see picture The Attempt at a Solution ##|z-1|=2=\sqrt{x^2+y^2-1}## ##4=x^2+y^2-1 \neq (x-1)^2+y^2##

Hi, I was just wondering if you have a cross product can you multiply out the constants and put them to one side. So ik x ik x E is equal to i^2(k x k x E) therefore is equal to -k x k x E. Is that correct?