Numbers Definition and 1000 Threads

To find √(-2)√(-3). Method 1. √(-2)√(-3) = √( (-2)(-3) ) = √(6). Method 2. √(-2)√(-3) = √( (-1)(2) )√( (-1)(3) ) = √((-1)√(2)√(-1)√(3) = i√(2)i√(3) = (i)(i)√(2)√(3) = -1√( (2)(3) ) =-√6. Why don't the two methods give the same answer? Thanks for any help.

Prove that $\dfrac{y^2z}{x}+y^2+z\ge\dfrac{9y^2z}{x+y^2+z}$ for all positive real numbers $x,\,y$ and $z$.

From @fresh_42's Insight https://www.physicsforums.com/insights/10-math-things-we-all-learnt-wrong-at-school/ Please discuss! They are not. They are equivalence classes. My favorite example is, that it makes a huge difference whether you carry home a pie from the bakery or ##12/12## pieces of...

Hello! Assume I have a system containing intrinsic spin and orbital angular momentum and before coupling the two, ##|J,J_z,S,S_z,L,L_z>## is a good basis (i.e. all these quantum numbers are good), with ##J=L+S## and ##J_z = S_z + L_z##. If I add a term of the form ##S\cdot L##, ##L_z## and...

Is it possible to make subsets of rational numbers in Mathematica using the plot command, or any other command? Ie., say I want to graph the set of rational numbers from 0 to 1.

Hi everyone, I have a quick question about Aleph numbers. Are they even possible? By containing infinity to a finite set, isn’t that essentially disproving the infinity in the first place? Can they be used in an actual scenario’s, or are they just purely hypothetical? Can they be used to...

[Mentor Note -- Thread moved from the technical forums, so no Homework Template is shown] Hi all, Recently I 'did' (a virtual lab) a drag laboraty experiment that used a wind tunnel to measure drag coefficient of 3 different shapes (cylinder, airfoil, triangular prism) and I'm not convinced...

Summary:: probability, Probability and Combinatorics Hello, I need someone to check If I correctly analized the probability of the following event: A group of younglings formed by 5 girls and 5 boys, Is going to divide in two teams of 5 elements each to play a game. a) suposing that the...

The number of even natural numbers less than 100000 that can be formed from the digits of the set (0,1,2,3,4,5,6) so that the digits in the number are not repeated is? Here I understand that the even number in the last place is an even number, that is, it has 4 possibilities, but won't the...

Hello, I have this (I am solving scholarship exams)math problem and I don't quite know what to do with it , Could You please help? The exercise is about complex numbers and it says: Calculate in the algebraic form(a+bi) I thought on applying substitution since -1=i^2 and z is the real part but...

I am reading the textbook Magnetism and Magnetic Materials by Coey and I am confused about how they grouped the terms and how they ended up getting the sums of L and S. My confusion lies in the two red boxes. Also, how is D even considered here when we have up to $2p_1$? And why would the spin...

Dear PF Forum, Can someone help me with the algorithm for finding a very large prime number? In RSA Encryption (1024 bit? 2048?, I forget, should look it up at wiki for that), Private Keys is a - two prime number packet. Now, what I wonder is, what algorithm that the computer use to find those...

Reals $x,\,y$ and $z$ satisfies $3x+2y+z=1$. For relatively prime positive integers $p$ and $q$, let the maximum of $\dfrac{1}{1+|x|}+\dfrac{1}{1+|y|}+\dfrac{1}{1+|z|}$ be $\dfrac{q}{p}$. Find $p+q$.

This is not a contextual question, but a stylistic one; hence it doesn't seem to belong in the other threads. I am proof-reading a paper, and I am unsure about the way the author uses square brackets for the indication of (numbered) sources. In order not to be quoting a source without...

How do we get ##\epsilon(2p+\epsilon)<\epsilon(2p+1)<2-p^2## from ##0<\epsilon<1## and ##\epsilon<\dfrac{2-p^2}{2p+1}##? Answer: As we have ##\epsilon<1##, we've got ##2p+\epsilon<2p+1##; therefore, ## \epsilon(2p+\epsilon)<\epsilon(2p+1) ##; -as we have ##\epsilon<\dfrac{2-p^2}{2p+1}##, we...

I have tried to solve this question for a while but I am not able to get the logic right.

Prove for all $n\in N$ $\dfrac{|a_1+...a_n|}{1+|a_1+...+a_n|}\leq\dfrac{|a_1|}{1+|a_1|}+...\dfrac{|a_n|}{1+|a_n|}$

Hi PF community, I'm reading about complex numbers and i have some questions about the argument of a complex number that i can't solve with Google or reading again the same page. Well, my first doubt is about , i can't understand where come this and why there is some random integer, i...

Assume a transformer as above, with 230V L-N, and I want to work out the L-L voltage. A phasor diagram will show me that the voltages are 120° out of phase. (230∠0°) + (230∠120°) = (230cos0 + j230sin0) + (230cos120 + j230sin120) = 230 + (-115 + j199.2) 115 + j199.2 = 230∠60 What I’m looking...

Let us suppose we have an equation such that $$N = \sum_{i=1}^N ix_i = x_1 + 2x_2 + 3x_3 + ...+Nx_N$$ and we also know that the solutions (i.e ##x_i##) ranges from ##\{0, N\}##. For example, if ##N=4## we would have $$x_1 + 2x_2 + 3x_3 + 4x_4 = 4$$ and ##x_1,x_2,x_3,x_4## will range from...

If $x$ and $y$ are positive real numbers, prove that $4x^4+4y^3+5x^2+y+1\ge 12xy$.

For any natural number $n$, let $S(n)$ denote the sum of the digits of $n$. Find the number of all 3-digit numbers $n$ such that $S(S(n))=2$.

I get 2^5 x 3^80 Am I correct?

Hi, I'm new to PF, but was hoping that there might be people on this forum with an interest in Taxicab numbers, particular on the "structure" of such integer sequences. If yes, would be delighted to hear from you.

Let $a$ and $b$ be positive real numbers such that $a+b=1$. Prove that $a^ab^b+a^bb^a\le 1$.

I have a question related to the following passage in the quantum mechanical scattering textbook by Taylor, Here Taylor makes the choice to use a basis of total angular momentum eigenvectors instead of using the simple tensor product given in the first equation above (6.47). I understand that...

The impedance Z = R -j/wC + ##\frac{1}{\frac{1}{R} - \frac{\omega C}{j}}## But,1/wC=R So, solving this, I find: Z= 3R/2(1-j) |Z| =##\frac{3R}{\sqrt 2}## I =##\frac{V_i \sqrt 2} {3R}## Vi - IR-IXc =Vo Solving this, ##Vo = V_i -\frac {V_i \sqrt 2}{3} - \frac{V_i \sqrt 2}{3R} \frac{-j}{wC}##...

If (u,v) = 1, prove that (u+v,u-v) is either 1 or 2. Where (,) means: $$ux_1 + vx_2 = 1$$ $$u + v(x_2/x_1) = 1/x_1, u(x_1/x_2) + v = 1/x_2$$ $$u + v = 1/x_1 + 1/x_2 - v x_2/x_1 - u x_1/x_2$$ $$u - v = 1/x_1 - 1/x_2 + u x_1/x_2 - v x_2/x_1$$ Now we can express (u+v,u-v). But i am not sure if...

Prove/Disprove: There exists ## a \in \mathbb{N} ## such that for all ## n,m \in \mathbb{Z} ## that satisfy ## n \cdot m = a ## then ## n > 0 \text{ or } m > 0 ##. My attempt (The statement's false, here's proof by contradiction ): Suppose There exists ## a \in \mathbb{N} ## such that for all...

I think the solution should be: METHOD #1: \begin{align} (\sqrt[4] {-1})^4 & = (-1)^{\frac 4 4} \nonumber \\ & = (-1)^1 \text{, can reduce 4/4 since base is a constant and not a variable in ℝ} \nonumber \\ & = -1 \nonumber \end{align} Alternatively, METHOD #2 for same answer is...

$\tiny{gre.al.13}$ For which of the following conditions will the sum of integers m and n always be an odd integer.? a. m is an odd integer b. n is an odd integer c. m and n both are odd integers d. m and n both are even integers e. m is an odd integer and n is an even integerI chose e just...

I have seen a solution for this question which was as follows, first out of 15 elements, take away 5, thus there are 11 gaps created for the remaining 10 numbers (say N) as, _N_N_N_N_N_N_N_N_N_N_ now, now we can insert back the 5 to comply with the non-consecutive stipulation for which, number...

I'm trying to write a C++ program to generate random numbers using the acceptance-rejection method. To plot the graphs, I'm using ROOT by CERN. I am checking if y values taken from the rectangular boundary are less than or equal to the function ##f(x_{i}) = e^{-k(x_{i} - x_{0})^{2}}##. void...

Hi PF, this is just for fun...Or not; I don't know. In 1777 Euler set up the notation ##i## to identify any roots of ##x^2-1##, which are indistinguishable, and verified ##i^2=-1##. This way, the set of real numbers grew larger, to a bigger set called complex numbers. This is a translation made...

If Imaginary numbers do exist and have real applications, then why do we call imaginary numbers "imaginary numbers"? . They exist. They're used all the time. What makes them "imaginary"?

Summary:: Problem interpreting a vector space of functions f such that f: S={1} -> R Hello, Another question related to Jim Hefferon' Linear Algebra free book. Before explaining what I don't understand, here is the problem : I have trouble understanding how the dimension of resulting space...

Below is the problem and the correct answer for this algebra problem is 7√2. But I cannot get to the correct answer.

Hello Can some one explain how you work out the combinations of quantum numbers for infinite wells in higher dimensions? For example if i have an energy level $$E_4$$ In a 2D well, then for quantum numbers does this mean the combinations allowed must be: $$4^2 + 1^2$$ $$1^2 + 4^2$$ So then...

Formula used : arc length = radius × angle (in radian). I interpreted this as: •Taking a unit circle, we get "angle (in radian) = arc length". This means radian measure of an angle is arc length, which can be represented on a real number line. Hence, it is a real number. Is this way to...

since ##|z|≤3## →##z=0+0i##, therefore we shall have centre##(0,0)## and radius ##3##, find my sketch below,

Hi, I used to use MATLAB for this kind of thing, but now my pc broke and I need to run some scripts. I have a .txt file structured like this 10 -2.34454 12 -2.34566 14 -2.34677 ... ... and I want to store the data in two variables: the first is the "counting" (10, 12, ...) and the second is...

Anyone know what the simplest possible self-contained numeral system for complex numbers would be, analogous to signed ternary for integers? My guess would be quarter-imaginary base (https://en.wikipedia.org/wiki/Quater-imaginary_base.)

This is just an editing, not a conceptual, question. (Hence I don't put it in the other forums.) In a text, when one refers to a particular equation by number, as in "we see in Equation (12) that...", the "equation" is capitalized (upper case). When it is not named, of course, not :"we see in...

I want to generate random numbers in C++. I do not want to use C library function (`<cstdlib> <ctime> (time.h)` ) and class. So I cannot use `rand()` function in C. I want to generate random integer numbers and I guess I can use `<random>` library in C++11. How can I use this generate random...

I'm trying to grok what an algebraic number could look like. Yes, I understand that an algebraic number is any number that could be a solution (root) to a polynomial having integer coefficients (or rational coefficients, since any set of rational coefficients can be made into integers by...

Hello there.We know that we have sets of numbers like the real numbers, complex numbers, quaternions, octonions.Could we find a set of numbers more general than that of real numbers that has basic properties of the real numbers like commutativity, order, addition, multiplication, division and...

n = 3, l = 2, me = -2, -1, 0, 1, 2. ms = -1 or 1. The correct answer is that ms can also be 0. Why?

"The dual space is the space of all linear maps from the original vector space to the real numbers." Spacetime and Geometry by Carroll. Dual space can be anything that maps a vector space (including matrix and all other vector spaces) to real numbers. So why do we picked only a vector as a...

Commutative property of addition. If a & b are integers then, a+b = b+a 2+3 = 3+2 5. Does not work for subtraction. 2-3 = -1 3-2= 1 Having said that, what about the special case with negative numbers (when we also move their respective signs) -5 + 7 = 2 & 7 + (-5) = 2. 15 -7 = 8 & -7 + 15...

Here is my attempt When we raise both sides to the power (p-1)/2, we get x^(p-1)= -1^[(p-1)/2](modp) Looking at p=3(mod4), the possible values of p are {3, 7, 11, 19, 23, 31...}. Putting these values of p into (p-1)/2 we get odd integers. {1, 3, 5, 9, 11, 15...}. So we have x^(p-1) =...