Numbers Definition and 1000 Threads

Homework Statement "Put each of the following into the form Acos(ωt+θ)..." (a.) 4ejt+4e-jt Homework Equations Euler's Identity: ejθ = cos(θ)+jsin(θ) Phasor Analysis(?): Mcos(ωt+θ) ←→ Mejθ j = ej π/2 Trignometric Identities The Attempt at a Solution I attempted to use phasor analysis to...

Without any rough work

Find the sum of all real numbers $a$ such that $5a^4-10a^3+10a^2-5a-11=0$.

Hi, I'm trying to figure out how to compute probability related to a problem I am tackling for work, and I think I have a handle on how to do it with smaller numbers, but no idea how to approach it for larger numbers. (And I need to explain the answers to a judge in simple terms). So here is...

Homework Statement Suppose one extracts a ball from a box containing ##n## numbered balls from ##1## to ##n##. For each ##1 \leq k \leq n##, we define ##A_k=\{\text{the number of the chosen ball is divisible by k}\}.## Find ##P(A_k)## for each natural number which divides ##n##. The Attempt...

Find all irrational numbers $k$ such that $k^3-17k$ and $k^2+4k$ are both rational numbers.

I know this post is in the topology thread of this forum, for group theory, this seemed like the reasonable choice to post it in. I realize group theory is of great importance in physics and I'm trying to eventually understand Emmy Noether's theorem. I'm learning group theory on my own, and...

This is, perhaps, more a question of philosophy of math rather than math itself. While it may be trivial to most people fluent in math, it was a bit surprising for me to learn recently that the set of algebraic numbers is countable. However, I quickly realized why that is: Each algebraic number...

Is it purely coincidental that the internal symmetry related flavor quantum numbers(like isospin and weak isospin) and the spacetime symmetry related spin quantum number have SU(2) as underlying group? They refer to seemingly unrelated things but it is remarkable how ubiquitous SU(2) is.

Do we use imaginary numbers just in the intermediary steps of a predictive theory? For example, in QM, in order to make predictions in the real world, you square the wave function. The wave function might have have all the information, but in order to predict something you must operate on it to...

Hello! I am very unsure of how to solve this question. The question states z^2=a+bi, where a and b belong to real numbers. Find all possible solutions for z. I think that the solution includes the De Moivre's formula, however I am very confused by how to do this or what the formula means...

Hi! Can you tell me how the theta changes into theta minus pi/2? Can you show me, please?

Homework Statement A) how many numbers have distinct digits from 1000 to 9999? B) how many odd numbers have distinct digits from 1000 to 9999? The Attempt at a Solution a) the first place value has choices from 1-9, the second has choices from 0-9 but one number was used in...

after it was found out that the first 9 pi digits 141592653 result in the end sum of 9, i searched for its iteration in the large digit chain of pi. after scanning stuff.mit.edu/afs/sipb/contrib/pi/pi-billion.txt it was found that .141592653 occurs at the 427238911 place and ends on the...

Definition/Summary A perfect number is a number which is the sum of its proper divisors (half the sum of its total divisors). Even perfect numbers are a Mersenne prime times a power of two; odd perfect numbers are not known to exist. Equations Sum-of-divisors function...

Definition/Summary Let \mathbb{R} be the set of all real numbers. We can extend \mathbb{R} by adjoining two elements +\infty and -\infty. This forms the extended real number system. In notation: \overline{\mathbb{R}}:=\mathbb{R}\cup \{+\infty,-\infty\} The extended real numbers are...

Why are perfect numbers important? What is the best way of introducing these numbers to a first course on number theory? I could not find any application apart from the relation to Mersenne primes. Are there any other applications of perfect numbers?

Why are perfect numbers important? What is the best way of introducing these numbers on a first course in number theory? I could not find any application apart from their connection to Mersenne primes. Are there any applications of such numbers?

Hello, I am trying to calculate the surface cooling rate from the sides of a round tin can using Rayleigh, Prandtl and Grashof numbers but I'm getting a ridiculously high result, and I'm hoping someone could run through my numbers to tell me where I'm going wrong. Posting my question is...

I have a list of numbers x = [1, 11, 21, 1211, 111221,...], I need to find the 31st number in that list.

Evaluate the following expressions, expressing answers in rectangular form. 1. $\cos(1+j)$ 2.$\sinh(4-j3)$ can you help me on how to solve these problems. thanks in advance!

for about first 429407636 pi digits where can i get such software or how can i calculate it? (i have 1b pi digits)

Ive had this problem with the calculator since I bought it. It might be that the calculator does not have enough implemented functions to pull it off or I am missing something. It happens when I am solving for imaginery roots. Example: y^2 + y^1+y = 0. I go to mode, and select 6(equation)...

Basically I want to take an unknown amount of variables and sum them up. I'm sure it's simple. I know there is some way to tell VS to keep reading the opened stream from my numbers.txt file. #include <iostream> #include <fstream> using namespace std; ifstream data_input...

We know that i^3 is -i . But I am getting confused, because I thought that i can be written as √(-1) and i^3 = √(-1) × √(-1) × √(-1) = √(-1 × -1 × -1) = √( (-1)^2 × -1) = √(1× -1) = √(-1) = i ( and not -i ). Please help.:rolleyes: Sorry I couldn't use superscript because I was using my phone.

Please forgive me as I may have to edit this post to get the equations to show properly. I am doing some work with AC circuits and part of one of my phasor equations has this in it: \frac {2i} {1+cos(θ) + i sin(θ)} - i , where i is the imaginary number \sqrt{-1}. However, knowing the...

Find the number between 1000 and 2000 that cannot be expressed as sum of (that is >1) consecutive numbers.( To give example of sum of consecutive numbers 101 = 50 + 51 162 = 53 + 54 + 55 ) and show that it cannot be done

Hey again! (Rock)(Wasntme) I want to show that there are infinitely many composite numbers $n$,for which stands: $$a^{n-1} \equiv 1 \pmod n$$ So,it must be: $n \mid a^{n-1}-1$ But how can I show that there are infinitely many such $n$ ?? (Thinking)

Homework Statement My textbook says any integer greater than 1 is a product of primes. Wouldn't that mean that there are no prime numbers? What is the product of primes that create the integer 23? Homework Equations The Attempt at a Solution

Hello! (Wave) I am looking at the following exercise: Find how many numbers $k$ with $1 \leq k \leq 3600$ exist,that have at least one common factor $>1$ with $3600$. I thought that the number we are looking for is equal to: $$3600-\phi(3600)=3600-\phi(2^4 \cdot 3^2 \cdot 5^2...

Suppose that k is a cardinal number. I want to show this. A set of all sets that their cardinal number is k doesn't exist.

What's a good book on split-complex numbers? If it also covers dual numbers or the relation between split-complex numbers and special relativity or Minkowski 4-space or some analysis of split-complex numbers then all the better, but that's just gravy. I really just want a good reference for...

is there a way to solve this without performing the tedious expansion of $(1+j)^8$? here's the problem $\text{Im}[(1+j)^8(x+jy)]$

An irrational number is any real number which cannot be expressed as the ratio of two real numbers. Then is 3.62566 is also an irrational number? I thought all irrational numbers are uncountable. I am not sure that the above is an irrational number :confused:

Solved~Rearranging numbers changes answers? Homework Statement A car starts from rest and travels with a constant acceleration of 3ms^{-2}, while a bike which is at a distance of 100m away from the car starts with an initial velocity of 5ms^{-2} travels with a constant acceleration of...

Homework Statement Prove that the set of all algebraic numbers is a countable set. Solution: Algebraic numbers are solutions to polynomial equations of the form a_0 x^n + a_1 x^(n – 1) + . . . + a_n = 0 where a_0, a_1, . . . , a_n are integers. Let P = |a_0| + |a_1| + . . . + |a_n| + n...

I just wanted to check something. If I have a complex number of the form a = C * \exp(i \phi) where C is some non-complex scalar constant. Then the phase of this complex number is simply \phi. Is that correct?

The problem H(e^{j0.2\pi}) = {\frac{1 - 1.25e^{-j0.2\pi}}{1 - 0.8e^{-j0.2\pi}}} Solves to H(e^{j0.2\pi}) = 1.25e^{j0.210\pi} Attempts I'm really not sure how to get that answer, but I've tried a number of different approaches Multiplying by complex conjugate Multiplying by...

I am writing this in C#. Here is the code. using System; namespace ConsoleApplication3 { class Program { static void Main(string[] args) { int sum = 0; int uservalue; Int32.TryParse(Console.ReadLine(),out uservalue)...

Dear All, Hi! I am about to begin a Diploma in Aeronautical Engineering and would like to know if anyone could help me understand if in my future career of being an Aeronautical Engineer I would at any time be required to use Complex numbers to solve problems. If yes can you suggest examples...

I'm having trouble solving the equation m2 - n2 = 707, where n and m are natural numbers. Because there are 2 variables, even though they are discrete, the obvious thing to do would be to use another equation to solve for one of the variables and then insert the new form to the original...

Hello, please I need help. I have no idea how to start this. Can someone guide me? This is not homework, I'm just studying on my own and I really don't know how to begin this.

So i was messing around with egyptian fractions and i eventually decided to try adding them together and then reducing the one single fraction that formed into its lowest terms. Anyway i set them up with having 1 in the numerator ALWAYS (or else it won't work) and proceeded to add them up...

Let's say we have a result with the number 2624,499 and want to round it off to a certain number of significant figures. Some examples with different number of significant figures: a) 3 significant figures: 2420 or 2430? b) 4 significant figures: 2624 or 2625? c) 5 significant figures...

Homework Statement 8i = ( 2x + i ) (2y + i ) + 1 The final answers is [x =0, 4] [y=4, 0] Homework Equations The Attempt at a Solution The final answer in the book is stated as above but if I follow the solution I will get the real parts which would...

Can I get the link to the prime number theorem?

Find all values of $x$ such that $f(x)=x^2-19x+99$ is a perfect square for all $x\in N$.

Does anyone know of a reference work that lists natural numbers with unique properties? Like 26, for example, being the only natural number sandwiched between a square (25) and a cube (27). Does such a reference book exist? IH

I've been learning about polygonal numbers, and one of the exercises in this book ask me to show that 9t_{n}+1 [Fermat], 25t_{n}+3, and 49t_{n}+6 [both from Euler] are triangular numbers. I don't know how to approach these proofs, I've tried to show that they have some form similar to...

This problem is from Boas Mathermatical Methods 3ed. Section 16, problem 1. Show that if the line through the origin and the point z is rotated 90° about the origin, it becomes the line through the origin and the point iz. Use this idea in the following problem: Let z = ae^iωt be the...