Fractions Definition and 604 Threads

I am seeking to understand Rings of Fractions and Fields of Fractions - and hence am reading Dummit and Foote Section 7.5 Exercise 3 in Section 7.5 reads as follows: Let F be a field. Prove the F contains a unique smallest subfield F_0 and that F_0 is isomorphic to either \mathbb{Q}...

Homework Statement Use partial fractions to find the sum of the series.Homework Equations \displaystyle \sum^{∞}_{n=1} \frac{8}{n(n+3)} The Attempt at a Solution I end up with: \displaystyle \frac{8}{3n} - \frac{8}{3(n+3)} I am stuck here.

Homework Statement This is an arc length problem in three dimensions. I was given the vector r(t)=<et, 1, t> from t=0 to t=1 Homework Equations Arc Length= \int |\sqrt{r'(t)}| dt from t1 to t2 where |\sqrt{r'(t)}| is the magnitude of the derivative of the vector The Attempt at a...

1. ∫[(x4 + x + 1)/(x(x2 + 1))]dx 2. When I first did this problem, I divided and got: ∫[x + (-x2 + x + 1)/(x3 + x)]dx (x3 + x) = x(x2 + 1) I then set up the fraction as: A/x + B/(x2 + 1) BUT, the solution to this problem says: A/x + [(Bx + C)/(x2 + 1)] How would I know to use...

1. So, i have the next integrand... 2. \int \frac{1}{(x-1)^2(x+1)^2}\,dx 3. I proceeded by resolving it by partial fraction and i came up with the next... \int \frac{1}{((x-1)^2)((x+1)^2)}\,dx = \int \frac{A}{(x-1)} + \frac{B}{(x-1)^2} + \frac{C}{(x+1)} + \frac{D}{(x+1)^2}\,dx The thing is...

Homework Statement Use the method of partial fractions to show that: $$\frac{2x^2}{(1-x(1+x)} $$ , may be written as: $$-2+\frac{1}{1-x}+\frac{1}{1+x}$$ , where $$\lvert x\rvert\neq1 $$. Homework Equations The Attempt at a Solution I obviously know how to do it but in the...

Homework Statement (x-2)/(x+3) less than (x+1)/(x) The Attempt at a Solution I broke it up into cases. When x+3 less than 0 and x less than 0 and then when both are positive, when one is positive and the other is negative and then the other way around. I'm not sure if that's the right way...

I've been trying to get out this question for a while now: ai) Show that (x,y,z) = (1,1,1) is a solution to the following system of equations: x + y + z = 3 2x + 2y + 2z = 6 3x + 3y +3z = 9 aii) Hence find the general solution of the system b) Express 2x^2 + 3/(x^2 + 1)^2 in partial...

(- 5x/3 + 2/3) (- 5x/3 + 2/3) If the above example was; (-5x + 2) (-5x + 2) = 25x^2 - 10x - 10x + 4 = 25x^2 - 20x + 4 The problem is I don't know how to deal with the denominators in this form? Anyone help

(This is my first post.) I can't seem to find a good way of solving this sort of congruence for x: x^2 / 3 + 11 \equiv 5 (mod x) Through trial and error it appears at least 3 and 6 are answers, but how can you reach them regularly? (I'm heard conflicting things about fractions being...

Whoa, this here is kicking me hard! Okay, so I've got everything pretty well down until... stuff like... \int \frac{3x + 32}{x^{2}-16x + 64}dx So, I get how to factor the denominator, but then what? The above won't factor... Also, I read that if the degree of the numerator is higher than the...

Homework Statement ∫ (x^3)/(x^2+2x+1) I think I could solve it if I knew how they did this operation: From the solution: ' (x^3)/(x^2+2x+1) = (x-2) + (3x+2)/(x+1)^2 ( After long division) Did they use polynomialdivision? x^3: x^2-2X+1= If so, how?

Homework Statement Hi I just have a problem in regards to setting up your partial fractions when doing nonhomogeneous differential equations using Laplace transforms. I’m at the stage of getting the inverse Laplace of: (1-625S^4)/(S^3 (25S^2+1) ) Homework Equations The Attempt...

Homework Statement A mineral slurry contains 3 components, solids, liquids and toluene. The slurry has 20 wt% solids and 80 wt% liquid. In the liquid portion of the slurry the mole ratio of toluene to water is 1:5.67 Determine the composition of the slurry (solids, toluene and water) in...

Hello friends, I am attempting to solve this problem for a sorting algorithm with a lot of elements in fraction form (I'm avoiding floating point operations). My question is: Given a sequence of increasing fractions, does adding 1 to the denominator affect the ordering of th sequence...

Hi! I am quite new at Maths in general, just recently started get an interest for it (Aswell as general physics/philosophy) and I am trying to learn on my own, so if this question is totally retarded, feel free to let your anger out haha. Anyway, I borrowed a book from a friend that looked...

From a fraction with infinite sum in denominator to partial fractions?? I am currently studying a course on Perturbation Methods and in particular an example considering the following integral \int_{0}^{\frac{\pi}{4}} \frac{d\theta}{\epsilon^2 + \sin^2 \theta}. There's a section of the...

Homework Statement Why is \sqrt{\frac{1}{-1}} \neq \sqrt{\frac{-1}{1}} when quite obviously \frac{1}{-1} = \frac{-1}{1} Homework Equations N/A The Attempt at a Solution By the above inequality, I mean when one calculates \sqrt{\frac{1}{-1}} as \frac{\sqrt{1}}{\sqrt{-1}}, and...

One last question to Integrate x^2/(1+4•x^2). I would assume you would do long division but 4x^2 is bigger than x^2. so would you either pull out a 1/4 and it would be 1/4 ∫ x^2/(1/4+•x^2) dx or would the first term when doing long division be 1/4? or am I just totally wrong and you...

Homework Statement Consider an object that is coasting horizontally subject to a drag force f = -bv = cv^2. Write down Newton's second law... The Attempt at a Solution So I did all of the steps leading up to this: m∫\frac{dv}{bv+cv^2}=-t dt Using partial fractions I get \frac{1}{bv+cv} =...

Homework Statement determine the indefinite integral: ∫ (4x+10)/(9x^2+24x+16) dx Homework Equations partial fractions technique The Attempt at a Solution i know it's partial fractions and i thought i did it right but i got the wrong answer. (4x+10)/(9x^2+24x+16) =...

Could someone show me the proof to why we use the reciprocal in fractions division. I ask this because it seem we are taught the how in math but never the why. Algebra proof would be best thanks.

Just stumbled upon something and I've never been taught it before and cannot see why its true... Hoping someone can help x/y + y/x = x^2 + y^2/xy Thanks

Homework Statement I need to integrate v(t) = V( \frac{1- e^{-2gt/V}}{1+ e^{-2gt/V}}) to show that the position function is given by s(t) = Vt + \frac{V^2}{g}ln(\frac{1 + e^{-2gt/V}}{2}) Homework Equations g is the acceleration due to gravity V is the terminal velocity The Attempt at...

Im reading Lang's first course in calculus and can't understand one step that he does when trying to integrate quotients with quadratic factors in the denominator. He's trying to find the integral of \int{\frac{1}{(x^2+1)^n}dx} but he's first starting with the case where n=1 Then while...

-2((1/((x^2 + 1)^2 )) - ((4x^2 )/(x^2 +1)^3 )) = 6x^2-2/(x^2+1) This is actually the last step of this problem. I understand everything they did up until here, but I'm a bit confused as to how they got from their last step, to the actual answer. Could someone explain?

Homework Statement \int\frac{1}{x\sqrt[3]{x+1}}dx (That's a cubic root in the denominator, by the way. Not an x cubed.) The Attempt at a Solution I thought possibly partial fractions, but I've never seen it done with a root in the denominator. Integration by parts was...

Homework Statement 1) ____1___ __ ___2___ __ ____3____ =0 3x-7 5x-5 3x+1 here's the other way of presenting the problem. 1/(3x-7) - 2/(5x-5) - 3/(3x+1) =0 ans: x=2/3, 3 2) _____2_____ + _____1_____ __...

A guy spends 1/6 of his life as a boy, 1/8 as a youth, 1/2 as a man and 15 years as a mature adult. I got about 70 by trial and error but I need to know exactly how to solve it.

Ok here's the problem. 2/5 of the children in Year 8 play the piano. 1/4 of the children in Year 8 play the violin No child plays more than one instrument. 1) Of the children in Year 8. What fraction do not play the piano or the violin? Easy i did 1-(2/5+1/4) = 7/20There are fewer than 50...

1.) 16 3/2 =16 1/2 =(2 square root)^3 =(2(4x4)^3 =(2 square root 4)^3

can someone help me (4a 3/2)(2a1\2) (3x5/6)(8x2/3) (27a6)-2/3 the fractions are powers

hello world, I've been doing some summertime training to brush up my math skills and have been struggling with this [dy]/[/dt]=(4exp(-y)+const*exp(-2y))^1/2 In fact this is the simplified version of a Bernouilli equation. I know that it is separable, I'm just struggling with the...

What is the procedure to integrate this kind of a fraction, i am guessing some trigonometric identity will apply but i am not quite sure how [SIZE="4"]∫[1/(a-bx^2)] dx can someone provide me a link which would help me learn about this kind of integration. Any help is Appreciated

Homework Statement Use partial fractions to find the sum of the series: \Sigman=1 to infinity \frac{5}{n(n+1)(n+2} Homework Equations Partial Fraction breakdown: \Sigma \frac{5}{2n}+\frac{5}{2(n+2)}+\frac{5}{(n+1)} The Attempt at a Solution When I tried to cancel terms out, it is...

2 4/7 + 1 3/5 = 4 6/35 This is easy on a calculator, but I would like to understand how to do it without using calculators. First I separate the integers from the fractions; 4/7 + 3/5 = Then find the LCM, which I get to be 35 2 4/7 + 1 3/5 = 4/7 + 3/5 = 21/35 2 + 1 20/35 = 3 41/35 Now I...

Why in partial fractions does the power of the denominator have to be one more than that of the numerator, when splitting up the expression. Skip to 5:30. Thanks.

Is it possible to solve any equation if we use continued fractions? I've heard that polynomial equations could be solved using continued fractions, and I used to obtain one of the several roots of a polynomial equation of low degrees using continued fractions in high school, but I read somewhere...

I've seen handouts that get distributed at the beginning of first-year calculus courses spelling out the rules of adding & multiplying fractions (among other things), and all the mistakes that are made by students coming out of high school. I think that was mostly based on instructors'...

Homework Statement ∫10x-2x2/((x-1)2(x+3)) Solve by partial fractions. The Attempt at a Solution ∫A/(x-1) +B/(x-1)2 + C(x+3) after setting up the partial fractions and multiplying each term by LCD: 10x-2x2= A(x-1)(x+3) + B(x+3) + C(x-1)2 10x-2x2= A(x2+2x-3) +Bx+3B +Cx-C 10x-2x2=...

Hi all, I'd appreciate a little help here. I'm making a video game, and one of the features is the ability to slow down time - a kind of Matrix-esque Bullet Time effect. I know there's a computing section, and a physics section, but I felt the issue is primarily mathematical, which is why...

Now yesterday I got help in realizing how to evaluate the sums of certain series, but while doing it I never got the reason behind why we take a series such as: \sum from k=1 to ∞ 1/k(k+3), I know how to solve the sum, but why do we have to convert it to a set of partial fractions in order to do it?

Not sure if this is the correct place to put this question, but here it goes (sorry about the vague title, but not sure how to describe it): We are asked to consider two rational fractoins, for example: a/b & x/y We are now asked to do the following: (a + x)/(b + y) Now, we are...

I have a fraction in the denominator of another fraction, and I'm trying to put a set of brackets around it. However, I can't seem to get them to size properly. Example below: Q_1 \left[ \frac{Q_2}{4\pi \left( r_2+\sqrt{ \dfrac{Q_2\gamma A}{4\pi}} \right)^2 } + Q_3 \right] which comes...

Suppose we can write a real number x as a continued fraction like this x=a0+1/(a1+1/(a2+1/(a2+...=[a0; a1, a2, a3, a4, ... an...]. Is there a binary operation f(i,x) so that f(i,x)=ai? I was wondering if there was a formula which gives the ith item in the sequence of integers which is...

Homework Statement I have this lowpass circuit which I have transformed to the S-domain. The circuit is to be exposed to a unit step, and then I shall convert the transient response to the time domain. Here's the transfer function of the lowpass circuit: H(s) = \frac{\frac{1}{LC}}{s^2 +...

In partial fractions, why \frac{3x+5}{(1-2x)^2} = \frac{A}{(1-2x)^2} + \frac{B}{(1-2x)} and not \frac{3x+5}{(1-2x)^2} = \frac{A}{(1-2x)} + \frac{B}{(1-2x)} Why exists the exponent on the denominator in the right hand side of the equation?

First of all, thank you, everyone, for all your help with my math conundrums! I've really appreciated the patience and helpful hints - because I do want to learn it myself, so this method is just enough for me to cut the math problems down to size. I'm thankful. And, I have noticed my scores...

Homework Statement Hi. My first post! I'm trying to solve for where a is a constant: ∫ (x/a)1/2*(x/(x-a)) dx Homework Equations See above The Attempt at a Solution I've tried integration by parts by setting u=(x/a)1/2 but I end up having to solve ∫ (x/a)1/2ln(x-a) - which I...

Homework Statement Ti-6Al-4V is a metal alloy used to make biomaterials. Its composition is 90% Ti; 6% Al and 4% V (mass percentages). (a) What are the mass fractions of Ti, Al and V? (b) What are the mole fractions of Ti, Al and V? Homework Equations Mass fraction = m/mtotal...