Numbers Definition and 1000 Threads

In [this post][1] user William Ryman asked what would happen if we try to build "complex numbers" with shapes other than circle or hyperbola in the role of a "unit circle". [Here][2] I proposed three shapes that could work. The common principle behind them being that if the unit curve is...

Closer to odd number implies ##|y/x - (2n+1)| < 1/2## for ##n = 0,1,2...##. Then $$-\frac 1 2 < \frac y x - (2n+1) < \frac 1 2 \implies\\ y < (2n + 1.5)x,\\ y > (2n + 0.5)x$$ for each ##n##. We note ##x \in (0,1)## implies ##y## can be larger than 1 since the slope is greater than 1 (but we know...

Hi PF! Given three random numbers between 0 and 1, how to evenly populate a sphere of radius ##R## (assuming we use every point). I think it's similar to the 2D circle equivalent described here. Does this imply the PDF is ##4 x^2##, where the remaining analysis holds? Then one point is the...

From the wikipedia page for Fibonacci numbers, I got that the matrix representation for closed-form expression for Fibonacci numbers is: \begin{pmatrix} 1 & 1 \\ 1 & 0\\ \end{pmatrix} ^ n = \begin{pmatrix} F_{n+1} & F_n \\ F_n & F_{n-1}\\ \end{pmatrix} That only works...

Proof: Suppose ## p ## and ## p^{2}+8 ## are both prime numbers. Since ## p^{2}+8 ## is prime, it follows that ## p ## is odd, so ## p\neq 2 ##. Let ## p>3 ##. Then ## p^{2}\equiv 1 \mod 3 ##, so ## p^{2}+8\equiv 0 \mod 3 ##. Note that ## p^{2}+8 ## can only be prime for ## p=3 ##. Thus ##...

For some reason the 3-dimensional hypercomplex numbers were not touched in the most of overviews of hypercomplex numbers. But I think this is not deseved. Intuition. Basically, if you add two complex dimensions to reals, say ##i## and ##j##, you automatically get a fourth dimension ##ij##...

Needing Direction I have sets of number: week 9 27 turned in 25 good 2 bad = 93% & 7% week 10 56 turned in 55 good 1 bad = 98% & 2% week 11 75 turned in 74 good 1 bad = 99% & 1% week 12 6 turned in 5...

I just saw that one of the ways of calculating Pi uses the set of prime numbers. This must sound crazy even to people who understand it, is it possible that this can be explained in terms that I, a mere mortal can understand or it is out of reach for non mathematicians?

(z-3)3=-8, solve for z. I'm new to complex numbers, so I'm stuck on this basic problem: how do you find all real and non-real solutions in the equality, (z-3)^3=-8? Thanks a bunch.

Hello All. This is my first post on the Physics Forums. I have started to self-study calculus and based on the feedback from this site and others, I have chosen Elementary Calculus: An Infinitesimal Approach by Jerome Keisler. I am working through the problems for section 1.5 (page 34/35)...

Proof: Note that all primes less than 50 will divide 50!, because each prime is a term of 50!. Applying the Fundamental Theorem of Arithmetic produces: Each term k of 50! that is non-prime has a unique prime factorization. Since 48, 49 and 50 are not primes, it follows that all primes...

Proof: Suppose n is an integer such that ## n>11 ##. Then n is either even or odd. Now we consider these two cases separately. Case #1: Let n be an even integer. Then we have ## n=2k ## for some ## k\in\mathbb{Z} ##. Consider the integer ## n-6 ##. Note...

For part (a), ##z##=##\dfrac {3+i}{3-i}## ⋅##\dfrac {3+i}{3+i}## ##z##=##\dfrac {4}{5}##+##\dfrac {3}{5}i## part (b) no problem as long as one understands the argand plane... For part (c) Modulus of ##z=1## and Modulus of ##z-z^*##=##\frac{6}{5}i##

Playing around with my calculator, I realized that if I do successive rooting operations on any positive non-zero number, I always get the number one. Can I conclude that the infinite root of any positive number will always be zero? If the statement is true, is there any synthesized formula to...

Hi, Would any member of Math Help Board explain me the highlighted area in the following paragraphs? Generating Random Distributions Now the only missing thing in previous cases is how would one generate a Uniform random, Normal random distributions. We therefore look to cover algorithms to...

A Giuga Number is a positive integer x > 1 with prime factors p1,p2,p3,...pi that satifies the relationship 1/p1 + 1/p2 + 1/p3 +...+ 1/pi - 1/x = k, where k is a positive integer, k=1 in this case. The first few Giuga numbers are 30, 858, 1772, and 66,198. For example, for x = 30 the prime...

Was fooling around and wrote down these two equations today that appear to work. I'm not all that bright and I'm positive these either have some proof or restate some conjecture--probably something in a textbook. Could somebody help me out? \forall n \in \mathbb{N}_0\smallsetminus\{0\} n^2 =...

1) Two sets have the same cardinality if there exists a bijection (one to one correspondence) from ##X## to ##Y##. Bijections are both injective and surjective. Such sets are said to be equipotent, or equinumerous. (credit to wiki) 2) ##|A|\leq |B|## means that there is an injective function...

OK, here once a sketch is done, we have two circles ##c_1## and ##c_2## with centre's ##c_1 (3,2)## and ##c_2 (7,5)## having radius ##2## and ##1## respectively. It follows that the distance between the the two centre's is given by ##L=\sqrt {(7-3)^2+(5-2)^2}##=##5## Now, the least possible...

Find the question below; note that no solution is provided for this question. My approach; Find part of my sketch here; * My diagram may not be accurate..i just noted that, ##OP## takes smallest value of ##12## when ##|z+5|=|z-5|## i.e at the end of its minor axis and greatest value ##13##...

Easy questions, just a lot of computation... $$\frac {z}{z^*}=\frac {a+bi}{a-bi} ×\frac {a+bi}{a+bi}$$ $$c+di=\frac {a^2-b^2}{a^2+b^2}+\frac {2abi}{a^2+b^2}$$ $$⇒c^2= \frac {a^4-2a^2b^2+b^4}{(a^2+b^2)^2}$$ $$⇒d^2= \frac {4a^2b^2}{(a^2+b^2)^2}$$ Therefore...

$$(1+i)z+(2-i)w=3+4i$$ $$iz+(3+i)w=-1+5i$$ ok, multiplying the first equation by##(1-i)## and the second equation by ##i##, we get, $$2z+(1-3i)w=7+i$$ $$-z+(-1+3i)w=-5-i$$ adding the two equations, we get ##z=2##, We know that, $$iz+(3+i)w=-1+5i$$ $$⇒2i+(3+i)w=-1+5i$$...

Using the inequality of arithmetic and geometric means, $$\frac {x+y}{2}≥\sqrt{xy}$$ $$6^2≥xy$$ $$36≥xy$$ I can see the textbook answer is ##36##, my question is can ##x=y?##, like in this case.

Hi, I'd like to clarify the following terminology (Fradkin, Quantum Field Theory an integrated approach) "carry the quantum numbers of the representation of the gauge group": Does the author basically mean that the wilson loop is a charged operator, in a sense that it transforms non-trivially...

I have been trying to find a pattern in this sequence of numbers: 72.1, 25.2, 35.1, 58.3, 164.14, 99.8, 23.1, 51.5, 13.2 I have tried every method I can think of like finding the mean, putting them in numerical order, trying to see if the differences between each number has any correlation (It...

Find a. Subtracting 4 is the same as adding $\boxed{(-4)}$ b. Subtracting -7 is the same as adding $\boxed{(7)}$ c. Subtracting a positive number is the same as adding a \boxed{negative} number, where that $\boxed{?}$ is opposite of the original number d. Subtracting a negative number is...

Hello folks, I am currently studying from Griffiths' Introduction to Quantum Mechanics and I've got a doubt about good quantum numbers that the text has been unable to solve. As I understand it, good quantum numbers are the eigenvalues of the eigenvectors of an operator O that remain...

I went ahead and tried to prove by induction but I got stuck at the base case for ## N =1 ## ( in my course we don't define ## 0 ## as natural so that's why I started from ## N = 1 ## ) which gives ## \sum_{k=0}^1 z_k = 1 + z = 1+ a + ib ## . I need to show that this is equal to ## \frac{1-...

I found this article about Alan Turing and his concept of Turing machines on the AMS website. Since we often get questions about countability and computability I thought it is worth sharing. https://blogs.ams.org/featurecolumn/2021/12/01/alan-turing-computable-numbers/ It also contains a Python...

Hi, I was watching the following video. I found some points confusing. Could you please help me to understand the gaps? Thanks, in advance! Question 1: Around 4:22, the video says the following. So for those mathematicians, negative numbers didn't exist. You could subtract, that is find...

https://www.physicsforums.com/attachments/11378

Show sqrt(3), sqrt(5), sqrt(7), sqrt(24), and sqrt(31) are not rational numbers.

Hi Guys Finally after a great struggle I have managed to develop and solve my Lagrangian system. I have checked it numerous times over and over and I believe that everything is correct. I have attached a PDF which has the derivation of the system. Please ignore all the forcing functions and...

If you take the cube of 2, you get 8. If you take the square of 3, you get 9. That is, 3 squared minus 2 cube equals 1. Are there any other examples of this? Where the difference between two powered numbers is equal to one? And is there some kind of theorem that says when this is...

So I think I have the principles mixed up here because I'm getting kind of "circular" answers. ## N = N_1 + N_2## ##dN## = 0 bc/ particle number fixed so ##dN_1 = -dN_2## ##F = cN^2 = c(N_1 + N_2)^2## In diffusive equilibrium, free energy would be minimized and chemical potentials equal... $$...

I have done the working on the attached sheet, i got ##3## as the minimum. I do not have the mark scheme for this worksheet. Apple trees will have ##22## trees per row ( 3 rows in total). Bananas will have ##22## trees per row (4 rows in total). Mango trees will have ##22## trees per row (5...

Kindly see attached: Since n belongs to the class of Natural numbers, then we may have, if ##n=1, hcf (A,B)= hcf(15,8)=1## ## n=2, hcf(17,9)=1## . . . ##n=8, hcf (29,15)=1##... Therefore in my reasoning the correct solution is b. Is there a different...

This is the problem, i think its not possible to get the lcm from the options given, i need a second opinion on this: lcm ought to be## 22×23×48=24,288## lcm[{22, 23, 32, 33}]=24,288## ok my initial thinking here was not correct. I was finding the lcm without first finding the product... The...

Hey! 😊 The algorithm of the Booth for multiplying signed numbersrs of fixed complement representation decimal point by 2 is implemented by multi-consecutive digit stain control of the multiplier, so after each test the algorithm proceeds by $N$ digits (e.g. for $3$-bit control step $N$ is...

I need help actually creating the proof. I've done the scratch needed for the problem, it's just forming the proof that I need help in. Bar(a+bi/c+di)= (a-bi) / (c-di) Bar ((a+bi/c+di)*(c-di/c-di)) = ((a-bi/c-di)*(c+di/c+di)) Bar((ac+bd/c^2 +d^2)+(i(bc-ad)/c^2+d^2)) =...

w = {0,0 | 1,1 | 2,2...} Let x = number of primes up to w+1 Let y = number of primes up to w-1 Now there's an empty prime box in the 0,0 slot not connected to anything. So I let x = p-1 and y = p+1 p = [p0, p1, p2...] Now p0 becomes 1,0/1 It can be either on or off. For the sake of...

I screenshot https://math.stackexchange.com/q/4185842.

I think that real number is countable. Because there is one to one correspondence from natural numbers to (0,1) real numbers. 0.1 - 1 0.2 - 2 0.3 - 3 ... 0.21 - 12 ... 0.123 - 321 ... 0.1245 - 5421 ... I think that is a one-to-one corresepondence. Any errors here?

Determine whether the number is a natural number, an integer, a rational number, or an irrational number. (Some numbers fit in more than one category.) The following facts will be helpful in some cases: Any number of the form sqrt{n} where n is a natural number that is not a perfect square, is...

Give an example of irrational numbers a and b such that the indicated expression is (a) rational and (b) irrational. 1. a +b 2. a•b 3. a/b 4. a - b What exactly is this question asking for? Can someone rephrase the statement above? Thanks

Two numbers add up to 72. One number is twice the other. Find the numbers. Let x and y be our two numbers. Two numbers add to 72. x + y = 72 One number is twice the other. I can use x or y for this next set up. x = 2y Here is the system: x + y = 72 x = 2y You say?

Hello, I've been using "Guide to Essential Math" by S.M. Blinder from time to time to stay on top of my basic mathematics. I'm currently on the section on Bernoulli Numbers. In that section he has the following (snippet below). Is the transition to equation 7.61 just wrong? The equation just...

Hey! :giggle: How can we calculate the number of natural numbers between $2$ and $n$ that have primitive roots? Let $m$ be a positive integer. Then $g$ is a primitive root modulo $m$, with $(g,m)=1$, if the modulo of $g\in (Z/m)^{\star}$ is a generator of the group. We have that $g$ is a...

Hi all, I have a project to code in 8051 series, DS80C320-ECG (data source as reference): "Division of two 16 bit unsigned integers being in the internal memory, quotient and remainder should be stored". I find a way to do it but there is a part of the program that i don't understand, I attach...