fractional Definition and 221 Threads

Evaluate the integral $$\int_0^1 x\left\{\frac{1}{x}\right\}\, dx$$ where ##\{\frac{1}{x}\}## denotes the fractional part of ##1/x##.

If emw spectrum is continuous, possible wavelengths should be infinite and there should be fraction of frequencies like 25,2 hertz. Well is there a fractional frequency of light? In high school when we are teaching interference of light we say "only the same wavelength of lights interfere with...

The actual problem that i was looking at with my students was supposed to be ##x^\frac{2}{3} - x^\frac{-2}{3}-6=0##(which is easy to solve using quadratic equations) of which i wanted them to solve, ...then i realized then that i had erronously posted ##x^\frac{2}{3} - x^\frac{-3}{2}-6=0## on...

I am trying to compute the Peebles equation as found here: I am doing so in Python and the following is my attempt: However, I'm unable to solve it. Either my solver is not enough, or I have wrongly done the function for calculating the Equation. # imports from scipy.optimize import fsolve...

I have tried manipulating this to $$1-\frac{8}{(\pi k_B T)^2}\mu^\frac12(\epsilon^\frac32-\mu^\frac32)=0\Leftrightarrow\left[1+\mu^\frac12(\epsilon^\frac32-\mu^\frac32)\right]^{-\frac{8}{(\pi k_B T)^2}}=0$$ but this doesn't seem to lead anywhere. any hints please? the solution is one of these...

I have to solve a certain numerical problem without using calculator and furthermore, there is a time limit for solving this problem. The answer I have got so far is ## \sqrt{\frac{100}{99}}## I know I can reduce the numerator to 10 but then I am stuck with square root of denominator which is...

Let $p$ be a prime number. Find the fractional part of $\dfrac{(p+1)!}{p^2}$.

My attempt: ##E^2 = p^2c^2 + m_0^2c^4## ##2E dE = 2pc^2 dp ## ##\frac{dE}{E} = \frac{pc^2}{E^2}dp=\frac{p^2c^2}{E^2}## % (dp/p = 1%) ##=\frac{E^2-m_0^2c^4}{E^2}## % ##=1-\frac{m_0^2c^4}{E^2}## % ##=1-\frac{m_0^2c^4}{m^2c^4}## % ##=1-\frac{1}{\gamma ^2}## % ##=\frac{v^2}{c^2} ##% =0.81...

Hi, PF This is the quote: "If ##m## is an integer and ##n## is a positive integer, then 6. Limit of a power: ## \displaystyle\lim_{x \to{a}}{\left[f(x)\right]^{m/n}} ## whenever ##L>0## if ##n## is even, and ##L\neq{0}## if ##m<0##" What do I understand? -whenever ##L>0## if ##n## is even: ##m##...

Hello! (Wave) I found the following algorithm for the fractional knapsack problem. Why at the case else, we do not change the variable w to w+(W-w)/S.weight? (Thinking)

(If I should have posted this in the Math thread instead of the Homework thread, please let me know.) I have three questions which I will ask in sequence. They all relate to each other. I've typed my questions and solutions attempts below. I've also attached a hand-written version of this...

So I got the answer through a little addition i.e 9^(1/2) multiplied by 9^(1/2) = 9^1 or 9 3 x 3 = 9 so 3 is the answer to what is 9^(1/2) I've tested this out with a few other numbers and have made this generalization, x^(1/2) = √x It seems to make the equations orderly and consistent but is...

Why do fractional errors have a different error value from subbing in the raw values? e.g. 10 +- 1 divided by 10 +- 1 fractional error yields 20% 11/9 - 9/11 yields 40/99 In cases like these do we use the original values and attempt to find the maximum error distance or use fractional errors?

Does anyone know any good research on this topic? I'm basically looking for information on what would be solving integral and differential equations in which the unknown you need to solve for is the level of a integral or derivative in the equation. For example F'1/2(u)+F'x(u)=F'1/3(u) where the...

I know how to find integral solutions of linear equations like x+y=C or x+y+z=C where C is a constant. But I don't have any idea how to solve these type of questions.I am only able to predict that both x and y will be greater than 243554.Please help.

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So the problem I’m attempting to solve is ##\lim_{x\to a} I_{\alpha}f(x)=\zeta (\alpha )## for f, and a, where ##\zeta (\cdot )## is the Riemann zeta function and ##I_{\alpha}## is the Riemann-Liouville left fractional integral operator, namely the integral equation $$\lim_{x\to...

Could someone please explain fractional distillation? I find it really confusing to understand. Thanks!

I was just thinking about this earlier and couldn't come up with a good enough resolution. I'm guessing it's a matter of convention more than anything. If we have ##x^{2} = a##, taking the principle root of both sides gives ##\sqrt{x^{2}} = \sqrt{a} \implies |x| = \sqrt{a}##. Yet evidently if...

Hi. I would like to check that my understanding is correct. For ##f(x)=x^{1/n}## where n is an integer. If n is odd then f(x) is an odd function while if n is even then f(x) is neither odd or even as it involves the square root function which is only defined for non-negative x. For ## f(x) =...

Hi , I'm looking at the argument in David Tongs notes (http://www.damtp.cam.ac.uk/user/tong/qhe/three.pdf) for ground state degeneracy on depending on the topology of the manifold (page 97, section 3.2.4). I follow up to getting equation 3.31 but I'm stuck on the comment after : ' But such an...

Summary: Apparently an ice cube gains mass when it melts So I'm asked to "Find the fractional increase in inertial mass when an ice cube melts ". All I've got off the top of my head right now is that a cube has energy = mc^2, and then when the cube melts, energy Q = (Heat of fusion)(m) is...

https://www.chemguide.co.uk/physical/phaseeqia/idealpd.html#top I learned about phase diagrams involving partial Vapour composition, temperature and composition of binary solutions from this website. You can find it if you scroll down to a little above the end First it considers a binary...

Suppose we use fractional derivatives (https://en.m.wikipedia.org/wiki/Fractional_calculus) in GR, hence we have a local group symmetry ##SO(3-\epsilon,1+\epsilon)## does any reference exist about an equation for ##\epsilon## ?, since it could depend on coordinates too.

$$10π \left( \frac V {4π} \right)^{2/3} = 5\sqrt[3] {{V^2}\frac π 2}$$Not sure how to deal with the $$10π$$ and how we get $$\frac π 2$$.

Kindly help me with this. Solve i^(-21/2) Note: i means iota.

Homework Statement Determine whether there exist ##A## and ##B## such that: $$\frac{1}{3x^2-5x-2} = \frac{A}{3x+1} + \frac{B}{x-2}$$Homework Equations None The Attempt at a Solution [/B] First I divided the polynomial ##3x^2-5x-2## by ##3x+1## and got ##x-2## as a result without a...

Why is fractional uncertainty not affected by systematic error? For example à vernier calipers measures the diameter of a coin: (5.06+-0.04) mm Can taking more readings, say 6, and taking average, reduce fractional error?

Have been referring to this forum a lot and I think I finally understand it. However I couldn't find a discussion specific to applying the Laws of Ohm, Poiseuille and Bernoulli's to why we use a pressure wire to study coronary blood flow through a lesion site in the coronary artery. So I made...

Homework Statement Homework Equations The Attempt at a Solution it looks like I got too big numbers for the uncertainty

In the case of spin 1/2 the state has to be rotated twice 180° to recover the initial one. If we consider a square made of arcs of equators on a sphere. The interior angle on the sphere is chosen to be 108°. Then the sphere is rolled along those arcs on a plane. The square hence draws...

Homework Statement What would have caused humans to come up with fractional exponent notations? Homework EquationsThe Attempt at a Solution I understand that it makes sense to use the exponent notation when we have to multiply the same number a number of times. For example, 10^8 is the short...

Please help with this algebra II question. It needs to be simplified. Thank you.

Most QFT texts, such as Peskin&Schroeder and D. Tong's lecture notes, contain a mention that the renormalizability of an interacting theory requires the coupling constants to have correct dimensions, making scalar fields with ##\phi^5 , \phi^6, \dots## interactions uninteresting. Maybe there are...

Homework Statement a3/2a5/4 Homework EquationsThe Attempt at a Solution I'm hoping you can help. My solution to this problem would be: a3/2+5/4=a8/6=a4/3 But the answer in the back of my book is given as a11/4 I'm confused!

Homework Statement D+8/D-2 = 9/4 See image, original equation in black. Homework EquationsThe Attempt at a Solution See image. Having a little trouble with this. Ive attempted to solve it two ways. The first was to multiply both sides by ##d-2## which gave me the correct answer of...

Factor (x^3/8) - (512/x^3). LCD = 8x^3 (x^6 - 8(512))/8x^3 (x^6 - 4096)/8x^3 [(x^3 - 64)(x^3 + 64)]/8x^3 In the numerator, the expression (x^3 - 64) is the difference of cubes, right?

Factor [(a + b)^2]/4 - [a^2b^2]/9 Can someone get me started?

How can this code be coded by another way? ## \frac {300-270.1}{T-130}## = ## \frac {313.-270.1}{135-130}## And it is also very strange that the right side O.K but left side is not but everything is the same for left and right. #,# \frac {300-270.1}{T-130}## = ## \frac...

I've been thinking about it since yesterday and have noticed this pattern: We have, the first order derivative of a function ##f(x)## is: $$f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h} ...(1)$$ The second order derivative of the same function is: $$f''(x)=\lim_{h\rightarrow...

<Moderator note: Thread moved from General Physics hence no formatting template shown> The fractional uncertainty is defined as: uncertainty/measured value. So for 2 cm +/- 1 cm we have 50%. For 9 cm +/- 1 cm we have 11.1%. My question is what if the measured value is 0 cm? How is the...

Homework Statement Find critical numbers of the function: F(x)=t^3/4 - 2t^1/4 Derivative I got: F'(x)=3/4 t^-1/4 - 1/2 t^-3/4 Homework EquationsThe Attempt at a Solution I have found the derivative and I understand I must pull out a t in order to find critical numbers, and run across this...

Hello! We know from 'Binomial Expansion' that (1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+... for \left| x \right|<1 . Why doesn't it work for other values of x? I can't understand this condition. I would be really grateful for clear explanation!

Say I have got a super cooled cylinder of half hydrogen and half helium. This cylinder has a pressure of 100,000 Pascal's. At this pressure, hydrogen boils at 20 Kelvin and helium at 4.21 Kelvin. I hope to separate helium by cooling gas down to 4.21 Kelvin but I know that even at the...

For most of the algebra questions I'm working on, I'm doing okay, with an exception here and there. The good news for those who have helped me, I'm getting a handle on the addition and associative properties and finally learned I needed to subtract on both sides of equation. Feeling better about...

Hello, I have a question regarding "polynomials" that have terms with interger and fractional powers. Homework Statement I want to solve: $$ x+a(x^2-b)^{1/2}+c=0$$ Homework Equations The Attempt at a Solution My approach is to make a change of variable x=f(y) to get a true polynomial (integer...

I am trying to work through a simplication of this factorial with variables: (n/2)!/[(n+2)/2]! I get, 2[n(n-1)]/2[(n+2)(n+1)n(n-1)] cancelling the 2[n(n-1)] leaves me with 1/[(n+2)(n+1)] However, Wolfram Alpha tells me this can be simplified as 2/(n+2) and I don't see that. Thanks

Homework Statement If ##\{ x \}## denotes the fractional part of x, then solve: ## \{ x \} + \{ -x \} = x^2 + x -6## It's provided that there are going to be 4 roots of this equation. And two of them will be integers. Homework Equations ## 0 \lt \{ x \} \lt 1~~\text{if}~~x \not\in I## ## \{...

Homework Statement So, I'm solving a dipole thing and I have these vectors: |r + d - r'| = (r² + d² - r'²)(1/2) Homework Equations I want to expand this but I have no idea how! I know I may have an infinite power series, but I may expand at the square terms tops... Before I needed to do the...

why single phase induction motor is commonly used for low HP rather than integral HP like 1,2,3HP... i understand that single phase gives low power...but say you have designed such that your motor delivers 10HP for rated voltage and current...why such single phase machines are not common?can...