Numbers Definition and 1000 Threads

Problem Statement: Given in the "Attempt at a solution section". Relevant Equations: Given in the "Attempt at a solution section". Problem Statement: Given in the "Attempt at a solution section". Relevant Equations: Given in the "Attempt at a solution section". I am having some serious...

1) Can I say that humans in history named, recognized countable set of numbers (1,2..Pi,sqrt(2) ... etc) the count of this set is named aleph-0 ? 2) Do mathematicians investigate the "other" continuum set of numbers which is infinity bigger ? Do we suspect, knows anything about that set like...

So we know that all the energy originates from the spring: E(spring) = (1/2)kd^2 As the block moves up the ramp, friction does work on the block over a distance of 2d: W = μmgcos(θ)* 2d So subtracting the work done by friction from the spring energy, gives us the energy left, so we'll set it...

I'm aware of the axioms of real numbers, the constructions of real number using the rational numbers (Cauchy sequence and Dedekind cut). But I can't relate the arithmetic of irrational numbers to real world usage. I can think the negative and positive irrational numbers to represent...

I am trying to write an algorithm that generates two random numbers in a given interval such that their ratio is an irrational number. I understand that all numbers stored on a computer are rational, so it is not possible to have a truly irrational number in a simulation. So, instead I am...

I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume I: Foundations and Elementary Real Analysis ... ... I am focused on Chapter 3: Convergent Sequences ... ... I need some help to fully understand the proof of Corollary 3.2.7 ...Garling's statement and proof of...

I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume I: Foundations and Elementary Real Analysis ... ... I am focused on Chapter 3: Convergent Sequences I need some help to fully understand the proof of Corollary 3.2.7 ...Garling's statement and proof of...

For simplicity, let ##N=1##. The following histograms show my results. The generated random numbers are initially exponentially distributed. But after re-scaling they become almost uniformly distributed. What is the cause of that, and is there a solution? P.S. Here is my code in Matlab...

I consider two sequences of numbers $A=\{a_1,...,a_n\}$ and $B=\{k-a_1,...,k-a_n\}$, where $a_1 \le a_2 \le ... \le a_n \le k$. I am looking for such conditions under which: $gcd(a_1,...,a_n) = gcd(k-a_1,...,k-a_n)=1$. In more general form: $gcd(a_1,...,a_n) = gcd(k-a_1,...,k-a_n) \ge 1$. I...

If someone can straighten out my logic or concur with the presence of a mistake in the proof (even though the conclusion is correct, of course), I would be much obliged. I’m looking at the proof of the corollary near the middle of the page (image of page attached below). I simply don’t find...

Hello everyone! I was going through a simple high school level mathematics book and got to the following question: n2 - n + 41 is a prime for all positive integers n. You're supposed to find a counter-example and prove the statement false. You could of course sit and enter different...

Consider the function sqrt(x). What is the domain of this function? Is it all real positive numbers? This is what I was taught in high school, but I was also taught that plugging in -1 would give an answer of i. So if the function takes negative inputs, shouldn’t they be part of the domain?

The definition of a number on the decimal number line from British Standard 1959 (if my memory serves me right) says that a number occupies a range of values, on the number line, between -50% and +49% of the least significant digit and so each number is separated from its neighbour by 1% of the...

I am trying to understand Aubry-Andre model. It has the following form $$H=∑_n c^†_nc_{n+1}+H.C.+V∑_n cos(2πβn)c^†_nc_n$$ This reference (at the 3rd page) says that if ##\beta## is irrational (rational) then the period of potential is quasi-periodic incommensurate (periodic commensurate) with...

First, I would like to say any response needs to be explained like I’m 5 years old. A degree in psychology didn’t require advanced math :) That said, here is my question: If any quantifiable number can be squared, does this mean light can also be squared. I read the forum about light being a...

Homework Statement Prove that if ##p## is a prime number and if ##p>5## then ##p^2-37## is divisible by ##12## Homework EquationsThe Attempt at a Solution So I think that the number ##p^2-37## should be expressed in a way that we can clearly see that it is divisible by 3 and by 2 twice...

[Note from mentor: This was split off from another thread, which you can go to by clicking the arrow in the quote below] Actually they are not. See https://www.amazon.com/dp/3319658662/?tag=pfamazon01-20 Sec. 5.1.

Hi, I'm working on an assignment for circuit theory, and I'm wondering if someone could let me know if I'm heading in the right direction? 1) I have a voltage value of 120 /_0 (polar form), from this can I assume that Arctan (a/b) =0, so voltage =120 in phase? Therefore, V =120+J0, where V...

Good morning, I am working on a problem where I am finding the 4th Coefficient in a sample of 4 discrete time Fourier Series coefficients. I got the sum but now I have to solve for a_3 which consists of a real and imaginary part. Any assitance on how to solve for the a_3? Thank you. $a_k =...

Hello I thought is would be fun to try a problem in which I had a complex number elevated to a complex power. To do this, I first tried to manipulate the general equation ## z^{w} ## (where ##z ## and ##w## are complex numbers) to look a bit more approachable. My work is as follows: ##z^{w}##...

Hey, So I need to know what number is 75% between 2 lines? Line 1 = 124.464 and Line 2 = 124.444. Currently I use Line 1 + Line 2 / 2 = 50% or (124.454) Then (124.454 + Line 1) / 2 = 75% or (124.459). Is there a easier or better way to get these results, cause this formula can't get percents...

Homework Statement Prove that the set of positive rational numbers is is countable by showing that the function K is a 1-1 correspondence between the set of positive rational numbers and the set of positive integers if K(m/n) = p_1^{2a_1}p_2^{2a_2}...p_s^{2a_s}q_1^{2b_1-1}...q_t^{2b_t-1}...

Give an example where a proposition with a quantifier is true if the quantifier ranges over the integers, but false if it ranges over rational numbers. I do not know where to go about when answering this, I know that an integer can be a rational number, for example 5 is an integer but can also...

Homework Statement Prove that if a set A of natural numbers contains n_0 and contains k+1 whenever it contains k, then A contains all natural numbers ≥ n_0Homework EquationsThe Attempt at a Solution I'm just confused by the question, please don't answer it. Logically it makes sense that if n_0...

I have been reading the book _Able Seamen: The Lower Decks of the Royal Navy 1850-1939_ by Brian Lavery, Naval Institute Press 2011, and I came across an interesting paragraph regarding engine efficiency (and labor requirements) as steam began to replace sail. I reproduce the relevant...

Homework Statement (1/x) + (1/(1-x)) > 0 Homework EquationsThe Attempt at a Solution 1+x-x/(x-x^2) > 0 1/(x-x^2) > 0 x-x^2 > 0 x> x^2 only occurs when 0<x<1 but in the solutions Spivak tells me "x>1 or 0<x<1"

Homework Statement Express the sum as a fraction of whole numbers in lowest terms: ##\frac{1}{1⋅2}+\frac{1}{2⋅3}+\frac{1}{3⋅4}+...+\frac{1}{n(n+1)}## Homework EquationsThe Attempt at a Solution Please see attached image for my work. The reason I am posting the image rather than typing this...

Homework Statement Homework Equations - The Attempt at a Solution Here's my work : When I added 0000 0001 to 0000 0000 , I didn't get the correct answer . Could someone check where is my mistake please ?

Homework Statement Let ##a## and ##b## be positive numbers with ##a \gt b##. Let ##a_1## be their arithmetic mean and ##b_1## their geometric mean: ##a_1 = \frac {a + b} 2## and ##b_1 = \sqrt{ab}## Repeat this process so that, in general ##a_{n + 1} = \frac {a_n + b_n} 2## and ##b_{n + 1} =...

Homework Statement Find the number of natural numbers n such that (n2-900)/ (n-100) is an integer.Homework EquationsThe Attempt at a Solution I have done the following: (n2-900-9100+9100)/ (n-100) or,{ (n2 - (100)2) + 9100 }/ (n-100) or, (n+100) + (9100)/(n-100) I hope you can understand this...

$\textsf{ If $z$ and $u$ are complex numbers, show that}$ $$\displaystyle\bar{z}u=\bar{z}\bar{u} \textit{ and } \displaystyle \left(\frac{z}{u} \right)=\frac{\bar{z}}{\bar{u}}$$ok couldn't find good example on what this is and I'm not good at 2 page proof systemsso much help is mahalo

Hello! I am a bit confused about fermions in QFT when they are considered grassmann numbers. If you have 2 grassmann numbers ##\theta_1## and ##\theta_2##, something of the form ##\theta_1\theta_2\theta_1\theta_2## gives zero. However, a term in a QED lagrangian of the form...

How many ways we can order numbers? Is there infinity ways to order numbers? Is there a proof of it?

I was reading an old thread about multiplying successive prime numbers adding 1 to obtain another prime number. I have worked with prime numbers for several years now and have developed what I best call a bi-linear advancement. It is an open-ended sieve of Eratosthenes. After many, many hours...

1. The complex number are not ordered. Which else number are not ordered? 2. Are the infinitesimally numbers are ordered numbers? It there a difference between infinitesimally number to another infinitesimally number?

Homework Statement Using 8 bit representation , subtract the unsigned binary numbers shown by each of the following methods ; 101012 - 10112 1) Binary subtraction 2) 2's complement Homework Equations - The Attempt at a Solution Using binary subtraction : 101012 - 10112 = 0000 10102 Using 2's...

When writing the number 3, my hand seems to stutter or stall slightly before i complete the 3. This only happens some of the time though. I am only 16 years old but I have also suffered a minor concussion. Does this happen to anyone else, could it correlate with a deeper issue?

Homework Statement Prove that ##\forall n \in \mathbb{N}## $$\frac{n}{2} < 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{2^n - 1} \leq n \text{ .}$$ Homework Equations Peano axioms and field axioms for real numbers. The Attempt at a Solution Okay so my first assumption was that this part...

aa3.2 Let Q be the group of rational numbers under addition and let $Q^∗$ be the group of nonzero rational numbers under multiplication. In $Q$, list the elements in $\langle\frac{1}{2} \rangle$, In ${Q^∗}$ list elements in $\langle\frac{1}{2}\rangle $ ok just had time to post and clueless

Homework Statement If ##\lim_{n \rightarrow \infty} x_n = L## then ##\lim_{n\rightarrow\infty}cx_n = cL## where ##x_n## is a sequence in ##\mathbb{C}## and ##L, c \epsilon \mathbb{C}##. Homework Equations ##\lim_{n\rightarrow\infty} cx_n = cL## iff for all ##\varepsilon > 0##, there exists...

Hello! (Wave) Let $A$ be a $n \times n$ complex unitary matrix. I want to show that the eigenvalues $\lambda$ of the matrix $A+A^{\star}$ are real numbers that satisfy the relation $-2 \leq \lambda \leq 2$. I have looked up the definitions and I read that a unitary matrix is a square matrix...

Homework Statement $$\frac{1}{z}+\frac{1}{2-z}=1$$ Homework Equations Quadratic-formula and algebra The Attempt at a Solution Been struggling with this one.. I keep getting the wrong answer, but that isn't the worst part, I can live with a wrong answer as long as the math behind it is...

Homework Statement So the problem I have is this silly little equation.. $$\frac {z - 7}{z + 3} = i $$Homework Equations This is the thing, I don't think you need anything more advanced than basic algebra to solve this problem. The Attempt at a Solution And I've tried solving it doing the...

Hi, Mostly i work with Octave / Matlab but i am trying to get into python also. Lately i have a couple problems where my numbers can't be represented in binary64 or float64 default format because they exceed the max of 1.8x10^308. Is there a common way people deal with this (without toolkits)...

The exponential of a complex number is a complex number. Does this extent to the quaternions and the octonions? Does the exponential of a quaternion give a quaternion? Does the exponential of an octonion give an octonion? Thanks.

Hi there. I am working with a problem where a sequence of numbers arises. This sequence reads: ##\{0,1,3,5,10,15,21,28\}## as far as I have worked it. I am trying to figure out the underlying relation that gives this sequence. These are related to indexes in a matrix, and I am trying to...

Why are higher magic numbers not accurately predicted if nuclear potential is assumed to be a central potential? Nuclei with magic numbers have a higher stability that those without. If we think of the nuclear potential as a central potential though these magic numbers aren't predicted...

I am considerably confused about conservation laws like lepton number (L), baryon number (B) and comparable. Unlike the conservation laws for energy, momentum, angular momentum and electric charge, the conservations of L and B are not rigorously covered in textbooks. So my questions -...