Numbers Definition and 1000 Threads

I've read that it is unsatisfactory to consider infinitely many basis vectors to span an infinite dimensional space. For example, for the infinite dimensional Hilbert space, {e1,e2,e3...} we could use this to make an arbitrary infinite tuple (a,b,c,...). If this is looked down upon, then why are...

The total number of numbers less than 1000 and divisible by 5 formed with 0,1,2,...9 such that each digit does not occur more than once in each number is what? Solution:Divisible by 5 ==> number ending in 0 or 5. Number of ways with no repeated digit: [0, 9]---> ends in 5 = 1 way (only 5...

Determine all prime numbers $p$ such that the total number of positive divisors of $A = p^2 + 1007$ (including $1$ and $A$) is less than $7$.

Hello I have used a random number generator to create a list of uniformly random numbers, between 0 and 1. The usual check that I do is sorting the list, and histograming the difference between the following and the previous one. The shape of the histogram should follow an negative...

Why should a person prefer irrational coordinate system over rational? My friend stated that its because most lines such as ##y=e## cannot be plotted on a rational grid system. But that cannot be true since ##e## does have a rational number summation ##2+1/10+7/100...## which can be utilised to...

Hi, I just have a few questions I'm struggling to find straightforward answers to online. The 4 quantum numbers of an electron in an atom describe the energy level, shape and suborbital of the orbital, and the fourth assigns a value to the electron's spin. Question 1) why is it in lone atoms...

Homework Statement Prove that algebraic numbers are denumerable Homework EquationsThe Attempt at a Solution This is a very standard exercise, but I haven't looked at its proof and want to see if I can prove it myself. With each element ##(a_n, a_{n-1},...,a_1,a_0 ) \in \mathbb{N}^n## we can...

Homework Statement ##\mathbb{R} \setminus C \sim \mathbb{R} \sim \mathbb{R} \cup C##. Homework EquationsThe Attempt at a Solution I have to show that all of these have the same cardinality. For ##\mathbb{R} \cup C \sim \mathbb{R}##, if ##C = \{c_1, c_2, ... c_n \}## is finite we can define ##...

Homework Statement (1) Prove that there exists no smallest positive real number. (2) Does there exist a smallest positive rational number? (3) Given a real number x, does there exist a smallest real number y > x? Homework EquationsThe Attempt at a Solution (1) Suppose that ##a## is the...

Hello everyone. Iam reading about complex numbers at the moment ad Iam quite confused. I know how to use them but Iam not getting a real understanding of what they actually are :-( What exactly is the imaginary part of a complex number? I read that it could in example be phase... Thanks in...

My background is QM as done in Griffiths( So yes I have a background of operators, observables and scattering matrix), Classical fields as done in Goldstein and Particle physics as in Griffiths. Griffiths actually works out Feynman rules for QED and QCD. I've started QFT with Peskin and...

The sum of two numbers is 13 and their product is 40. Find the numbers. Is this the correct set up? Let x and y be the two numbers. x + y = 13 xy = 40

Homework Statement Homework EquationsThe Attempt at a Solution I attempted to use the formula zj = xj + iyj to substitute both z's. Further simplification gave me (x1 + x2)cosθ + (y2 - y1)sinθ or, Re(z2 + z1)cosθ + Im(z2 - z1)sinθ. Is this a valid answer? Or are there any other identities...

Say you have an orange and a banana. You can say that they are two fruits. But this pertains to the categorization of fruit, which could be considered a mental construct of a category. You cannot say that you have two yellow objects, because you really don't. Relative to the category of color...

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with Exercise 2.1.10 Part (c) ... ... Exercise 2.1.10 Part (c) reads as follows:I am unable to make a meaningful start on...

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with Exercise 2.1.10 Part (b) ... ... Exercise 2.1.10 Part (b) reads as follows: I am unable to make a meaningful start on...

Ordering on the Set of Real Numbers ... Sohrab, Exercise 2.1.10 (a) ... I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with Exercise 2.1.10 Part (a) ... ... Exercise 2.1.10...

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with Exercise 2.1.12 Part (1) ... ... Exercise 2.1.12 Part (1) reads as follows: I am unable to make a meaningful start on...

Homework Statement I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with Exercise 2.1.12 Part (1) ... ... Exercise 2.1.12 Part (1) reads as follows: I am unable to make a...

How is it possible to ignore the addition sign and imaginary number without contradicting fundamental Mathematics? I find it difficult to understand.

Hello, would like to derive a length of list of random numbers in which I may find some special sequence of few numbers with some probability. For clearness I give an example: I have two generator of (pseudo) random numbers with same range of numbers, let's say (1-k). First generator give a...

we can find two irrational numbers $x$ and $y$ to make $x^y$ rational,true or false statement? if true then find else prove it .

Hey guys, I am seriously confused by this problem. 3/4 [ 5/6 ( -18/25 ) + 1/2 ] I would appreciate it if someone shows me the step by step process. Thanks in advance :)

Find all natural numbers $n$, for which: \[\sqrt{1 \underbrace{4...4}_{n \: times}} \in \Bbb{N}\]

I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ... I am currently focused on Section 1.5: Constructing the Rational Numbers ... I need help with Exercise 1.5.9 (3) ...Exercise 1.5.9 reads as follows: We are at the point in Bloch's book where he has just...

Homework Statement Although the basic properties of inequalities were stated in terms of the collection P of all positive numbers, and < was defined in terms of P, this procedure can be reversed. Suppose that P10-P12 are replaced by (P'10) For any numbers a and b one, and only one, of the...

I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ... I am currently focused on Chapter 1: Construction of the Real Numbers ... I need help/clarification with an aspect of Theorem 1.3.7 ... Theorem 1.3.7 and the start of the proof reads as follows: n the above proof we...

I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ... I am currently focused on Chapter 1: Construction of the Real Numbers ... I need help/clarification with an aspect of Theorem 1.3.7 ... Theorem 1.3.7 and the start of the proof reads as...

Is there a set of $2015$ consecutive positive integers containing exactly $15$ prime numbers?

I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ... I am currently focused on Chapter 1: Construction of the Real Numbers ... I need help/clarification with an aspect of Theorem 1.2.10 ... Theorem 1.2.10 reads as follows: Towards the end (second last line) of the...

I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ... I am currently focused on Chapter 1: Construction of the Real Numbers ... I need help/clarification with an aspect of Theorem 1.2.10 ... Theorem 1.2.10 reads as follows: Towards the end (second last line) of the...

I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ... I am currently focused on Chapter 1: Construction of the Real Numbers ... I need help/clarification with an aspect of Theorem 1.2.7 (1) ... Theorem 1.2.7 reads as follows: https://www.physicsforums.com/attachments/6976...

I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ... I am currently focused on Chapter 1: Construction of the Real Numbers ... I need help/clarification with an aspect of Theorem 1.2.7 (1) ... Theorem 1.2.7 reads as follows: In the above proof of (1) we read the...

The Bell Inequality tests are only valid for positive numbers, which is reasonable because counts and probabilities cannot be negative. CHSH generates a negative number, which means CHSH experiments are invalid. Bell's Inequality can be violated by having a negative value. For example...

Hello all, Three consecutive elements of a geometric series are: m-3i, 8+i, n+17i where n and m are real numbers. I need to find n and m. I have tried using the conjugate in order to find (8+i)/(m-3i) and (n+17i)/(8+i), and was hopeful that at the end I will be able to compare the real and...

Hello all, I am trying to find the algebraic representation of the following numbers: \[rcis(90^{\circ}+\theta )\] and \[rcis(90^{\circ}-\theta )\] The answers in the book are: \[-y+ix\] and \[y+ix\] respectively. I don't get it... In the first case, if I take 90 degrees (working with...

Hello all, I wish to plot and following complex numbers on a plane, and to find out which shape will be created. I find it hard to figure out the first one, I believe that the others will follow more easily (the forth is also tricky). \[z_{1}=\frac{2}{i-1}\] \[z_{2}=-\bar{z_{1}}\]...

Homework Statement Let 0≤p≤1. Let there be k distinct numbers (they can be natural numbers) a1, a2, ... , ak, each repeating respectively b1, b2, ... , bk times. Let q < ∑r=1k br Determine the minimal values of b1 ... bk such that the probability of q numbers chosen out of ∑r=1k br numbers...

Hiya all, I need your assistance with the following problem: A) Show that the equation \[z^{2}+i\bar{z}=(-2)\] has only two imaginary solutions. B) If Z1 and Z2 are the solutions, draw a rectangle which has the following vertices: Z1+3 , Z2+3 , Z1+i , Z2+i I do not know how to even...

Hello everyone, I have a complex number problem that i would greatly appreciate some help with. Thanks in advance to anyone offering their time to make a contribution. Q) Write the following in polar form: I have attempted the question (please see my working below) and have been advised that i...

Homework Statement as listed above the question is how many and which three digit NIP can be formed whit the use of prime numbers[/B]Homework Equations nothing currently trying to understand[/B]The Attempt at a Solution well i have found at least 168 primer numbers below 1000 i mean in the...

By using oxidation numbers can someone show me what is oxidised and reduced Fe203+3co->2Fe+3co2

Given any finite set of natural numbers, it seems evident that the odd numbers form a subset of the natural numbers. But what happens "at infinity"? I mean, if we account for all infinitely many natural numbers, there would be also infinitely many odd numbers. In such case, is it still true that...

(sorry, the thread title got mangled. It should be "why are irrational and transcendental so commonly used to describe numbers") Is this simply out of the most common ways of how one would try to describe a number? (e.g. first try ratios, then polynomials) Or is there a deeper reason for this...

I was investigating the number of unique grid points in a Cartesian coordinate system if I were to start at a corner (say coordinate 1,1,1), and make one step in each of the three positive directions (coordinates 1,2,1; 2,1,1; and 1,1,2). Now I went from 1 point to 3 points. I repeat the same...

Recently, I was intrigued by the summations of finite powers and therefore by the formula which generalizes the summations. "Faulhaber's formula". However, I didn't find an intuitive simple meaning of "bernoulli numbers", only meaning by their applications, which, of course, I can't understand...

This isn't original or anything, but I was thinking about how would one go about formalizing (in a general sense) an informal wikipedia picture such as this: https://upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Omega-exp-omega-labeled.svg/487px-Omega-exp-omega-labeled.svg.png For example...

Hi, Can anyone please tell me any easy way of prime factorizing 5-digit composite numbers from 10,000 to 99,999 with little writing or mentally? Thanks.

It's not a homework question. I just thought up a method of finding answers to problems where a number is raised to a complex number and I need to know if I am right. If we have to find e^(i), can we do it by; first squaring it to get, e^(-1) which is 1/e and then taking its square root to get...

Homework Statement exp(z)=-4+3i, find z in x+iy form Homework Equations See attached image. The Attempt at a Solution See attached image. exp(z)=exp(x+iy)=exp(x)*exp(iy)=exp(x)*[cos(y)+isin(y)] ... y=inv(tan(-3/4)=-.6432 ... mag(-4+3i)=5, x= ln (5)..exp(ln(5))=5 ...