What is quotient: Definition and 355 Discussions

In arithmetic, a quotient (from Latin: quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a division (in the case of Euclidean division), or as a fraction or a ratio (in the case of proper division). For example, when dividing 20 (the dividend) by 3 (the divisor), the quotient is "6 with a remainder of 2" in the Euclidean division sense, and



6



2
3





{\displaystyle 6{\tfrac {2}{3}}}
in the proper division sense. In the second sense, a quotient is simply the ratio of a dividend to its divisor.

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  1. R

    Differentiation using the quotient rule

    Homework Statement Use the quotient rule to differentiate y=(〖2x〗^4-3x)/(4x-1) Homework Equations y=(v du/dx-u dv/dx)/v^2 The Attempt at a Solution Please also find attached attempt as jpeg for clarity, and textbook supplied answer...
  2. PhizKid

    Derivative of (-sqrt(1 + x^2)) / x: What Did I Do Wrong?

    Homework Statement Differentiate (-sqrt(1 + x^2)) / x Homework Equations [f'(x)g(x) - f(g)g'(x)] / [g(x)]^2 The Attempt at a Solution I need to simplify it further, but before I do I checked it into Mathematica and it says the derivative is incorrect. What did I do wrong?
  3. S

    Quotient Group is isomorphic to the Circle Group

    A portion of a homework problem was given me to solve for practice. I have solved some but not all of the homework problem and I hope you all can help. Here is the problem: 1. For each x \in R it is conventional to write cis(x) = cos(x) + i sin(x). Prove that cis(x+y) = cis(x) cis(y)...
  4. jtbell

    What's your JFQ (Junk Food Quotient)?

    How many of these have you eaten? http://www.time.com/time/specials/packages/article/0,28804,2104472_2104473_2104484,00.html?hpt=hp_t3 (time.com) I've eaten most of these more times than I can count. I've eaten Snickers and Moon Pies a few times. I've never eaten fried pork rinds, even though...
  5. V

    The Difference Quotient and Integral Calculus

    I'm just a high school senior who noticed that the derivative has a general formula that we all know is, \frac{f(x+h)-f(x)}{h} but that there is no general formula (at least I haven't heard of it yet) for the integral of a function. I know I cannot simply just take the inverse of the difference...
  6. J

    Prove that a retraction is a quotient map

    Homework Statement As in title. Homework Equations Described in my attempt. The Attempt at a Solution Where do I go from here? I need to show that those 2 unioned sets are open in A. I'm not seeing it
  7. A

    Intuition for Quotient Ring in Polynomials

    I just had a discussion with someone who said he thought about quotient rings of polynomials as simply adjoining an element that is a root of the polynomial defining the ideal. For example, consider a field, F, and a polynomial, x-a, in F[x]. If we let (x-a) denote the ideal generated by x-a...
  8. T

    Calculus quotient rule problem

    Homework Statement using the quotient rule, find the derivative of y = (2x-3)/(√(x^2-5)). Do not leave the answer in complex form.Homework Equations the quotient rule g(x)f'(x) - f(x)g'(x) --------------------- (g(x))^2 The Attempt at a...
  9. R

    Integration of a quotient with a factor in the denominator that has no real root

    Homework Statement $$\int \frac{x-1}{(x+1)(x^2+1)} dx$$ Homework Equations N/A The Attempt at a Solution I thought that I would use partial fractions, so: $$\frac{x-1}{(x+1)(x^2+1)} = \frac{A}{x+1} + \frac{B}{x^2+1}$$ $$x-1 = A(x^2+1) + B(x+1)$$ ##x=-1 \Rightarrow (-1)-1 = A((-1)^2+1) +...
  10. J

    How does one integrate a quotient?

    How do you integrate quotients? Let's practice with one like... \int\frac{3x + x^{2}}{5x - 1} dx Thanks for the help!
  11. T

    Product of Quotient Groups Isomorphism

    Homework Statement I have attached the problem below. Homework Equations The Attempt at a Solution I have tried to use the natural epimorphism from G x G x ... x G to (G x G x ... x G)/(K1 x K2 x ... x Kn), but I do not believe that this is an injective function. Then I tried...
  12. L

    Quotient of the Mutlplicative Monoid of a Ring

    In abstract algebra (ring theory specifically), when we are dealing with factorization, UFD's, and so on, we are often only interested in the multiplicative structure of the ring, not the additive structure. So here is the basic situation we face: we a start with an integral domain (R,+,*)...
  13. Z

    Difference Quotient - average rate of change

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  14. A

    Need help about Quotient Rings (Factor Rings)

    Suppose S is an Ideal of a Ring R I want to verify the multiplication operation in the Factor Ring R/S which is (S + a)(S + b) = (S + ab) for this i Need to show that : (S + ab) ≤ (S +a)(S + b) please give me some idea about it IT IS GIVEN AS DEFINITION IN THE BOOK I.N Herstein
  15. S

    Quotient Topology and Adjunction Space

    Does anyone have any good reference to exercises concerning these topics? I would like to understand them better. Thank you.
  16. S

    MHB Quotient rule with square roots

    I have the answer to this problem but I am stumped as how to get there. Here it is h(x)=e^x/5/sqrt2x^2-10x+17, I'm getting stuck moving the square root up. Help
  17. nukeman

    Simple difference quotient question. Function -> simplify answer

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  18. S

    Re-Expressing a Quotient of Polynomials

    This is a basic question but I don't think I've ever seen anything like it before. If Q(x) \ = \ b_0(x \ - \ a_1)(x \ - \ a_2)\cdot \ . \ . \ . \ \cdot (x \ - \ a_s) then \frac{P(x)}{Q(x)} \ = \ R(x) \ + \ \sum_{i=1}^s \frac{P(a_i)}{(x \ - \ a_i)Q'(a_i)}I just don't understand where the P(aᵢ)...
  19. F

    Differentiate a function using the quotient rule

    1. Differentiate 2. y = \frac{v^{3} - 2v\sqrt{v}}{v} 3. I am trying to use the quotient rule, but am having trouble understanding how to use the square root. Here's what I have \frac{v(3v^{2}-2*1/2\sqrt{v}) - v^{3}+2v*1/2\sqrt{v}}{v^{2}}
  20. U

    Quotient Test for Convergence of Series

    Homework Statement The quotient test can be used to determine whether a series is converging or not. The full description is in the attachment. Homework Equations The Attempt at a Solution ( i ) Why must they both follow the same behaviour? Even if p ≠ 0, it says nothing about...
  21. I

    Respiratory Quotient: Normal Value & Calculation

    Homework Statement A normal value of respiratory quotient in humans is 0,8. It means that when a person has done an external work of 2000J and consumed 2litres of O2 the volume of produced CO2 was? the energy produced in metabolic process was? Homework Equations So I take this RQ=Vco2/Vo2...
  22. M

    Calculating the Metric on Quotient Space of E

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  23. DryRun

    Finding quotient using while loop

    Suppose that the number ##\pi## is divided by 2. The resulting quotient is divided by 2 again. This process is continued till the current quotient is less than or equal to 0.01. What is the largest quotient that is greater than 0.01? Here is my attempt: r = 1; quo = 1; while (quo > 0.01)...
  24. S

    How to verify if a mapping is quotient.

    Prove or disprove that f is a quotient mapping. f:R^3\{(x1,x2,x3):x1=0}--->R^2 defined by (x1,x2,x3)|->(x2/x1,x3/x1)
  25. I

    Quotient set of equivalence class in de Rham cohomology

    Hi all, So the equivalence class X/\sim is the set of all equivalences classes [x]. I was wondering if there was a way of writing it in terms of the usual quotient operation: G/N=\{gN\ |\ g\in G\}? From what I've read, it would be something like X/\sim = X/[e]. But, since I'm looking at the de...
  26. V

    Algebra II Simplifying Radicals Using Product and Quotient Properties

    Homework Statement Simplify. \sqrt[3]{\frac{5}{4}} The answer according to the textbook is: \frac{\sqrt [3]{10}}{2} Homework Equations -- The Attempt at a Solution Separated numerator and deonominator into individual cube roots and multiplied both by \sqrt[3] {4} ...
  27. N

    Some basic question about a quotient ring

    It's been awhile since I studied ring theory but here's a question about it: Let R = C[x1, x2, x3, x4, x5, x6, x7, x8] be a complex polynomial ring in 8 variables. Let f1 = x1 x3 +x5 x7 and f2 = x2 x4 +x6 x8. How do \bar{f}1, \bar{f}2 in (f1,f2)R/(x1,x2)R look like? Is...
  28. H

    MHB Finding elements in a quotient ring

    I have to describle the elements of the quotient ring $$\frac{\mathbb{Z}[x]}{2\mathbb{Z}[x]+x^2\mathbb{Z}[x]}$$ do I use the division algorithm to write any element as $$f(x)=(x^2+2)q(x)+r(x)$$ where $\operatorname{deg}(r(x))<2$ so the elements of the ring are the linear polynomials over...
  29. N

    Nonzero divisor in a quotient ring

    How do you show that x is a nonzero divisor in C[x,y,z,w]/<yz-xw>? Here's how one can start off on this problem but I would like a nice way to finish it: If x were a zero divisor, then there is a function f not in <yz-xw> so that f*x = g*(yz-xw).Here's another question which is slightly...
  30. R

    Abstract algebra: proving an ideal is maximal, Constructing quotient rings

    Homework Statement M = {(pa,b) | a, b are integers and p is prime} Prove that M is a maximal ideal in Z x Z Homework Equations The Attempt at a Solution I know that there are two ways to prove an ideal is maximal: You can show that, in the ring R, whenever J is an ideal such...
  31. T

    Show Quotient Groups are isomorphic

    Homework Statement Show that Z18/M isomorphic to Z6 where m is the cyclic subgroup <6> operation is addition The Attempt at a Solution M = <6> , so M = {6, 12, 0} I figured I could show that Z18/M has 6 distinct right cosets if I wanted to do M + 0 = {6, 12, 0} M + 1 = {7, 13, 1}...
  32. J

    Quotient groups of permutations

    hey guys, I just want grasp the whole concept of quotient groups, I understand say, D8/K where K={1,a2} I can see the quotient group pretty clearly without much trouble however I start to get stuck when working with larger groups, say S4 For instance S4/L where L is the...
  33. D

    Question about wiki artical on Quotient Groups

    Hi I am trying to learn about quotient groups to fill the gaps on things I didn't quite understand from undergrad. Anyway I have a question regarding this: Can someone please explain how { 0, 2 }+{ 1, 3 }={ 1, 3 } in Z4/{ 0, 2 }? I would think since 0 + 1 = 1 and 2 + 3 = 1 under mod 4...
  34. F

    Quotient of First Order Ordinary Derivatives

    How do you solve (analytically or numerically) a differential equation of this form, $$\frac{\mathrm{d}y(x)/\mathrm{d}x}{\mathrm{d}z(x)/\mathrm{d}x} = a[1-y(x)-z(x)] + b$$ where a, b are constants. Also, $$y(0) = z(0) = 0$$
  35. P

    I have troubles simplifying this quotient of factorials

    Homework Statement I'm trying to self-study Mary L. Boas' book Mathematical Methods in the Physical Sciences. One of the exercices asks the reader to find the limit of n -> ∞ (n!)2 / (2n)! Homework Equations None The Attempt at a Solution Instinctively I know that (2n)! grows...
  36. I

    Mod or quotient remainder theorem (QRT)

    I have to prove this problem. For all n integers, if n mod 5 = 3, then n2 mod 5 = 4 Proof: Let n be an integer such that n mod 5 = 3. n = 5k+3 for some integer k by definition of MOD or QRT? Which one would be correct? Am I using the definition of MOD or QRT? I'm thinking its QRT because its...
  37. Δ

    Uniform convergence of a quotient

    Homework Statement Let f,g be continuous on a closed bounded interval [a,b] with |g(x)| > 0 for all x in [a,b]. Suppose that f_n \to f and g_n \to g uniformly on [a,b]. Prove that \frac{1}{g_n} is defined for large n and \frac{f_n}{g_n} \to \frac{f}{g} uniformly on [a,b]. Show that this is...
  38. H

    Finding the derivative, quotient rule with natural log function.

    Homework Statement Find y' of y= 1-3ln(7x)/x^4 Homework Equations The Attempt at a Solution I used the quotient rule and got: y'=x^4*d/dx(1-3ln(7x)-(1-3ln(7x)*d/dx(x^4)/(x^4)2 which is: x^4*(0-3*1/7x*7)-(1-3ln(7x))*4x^3/x^8 simplified to: 3x^4/x-1+3ln(7x)*4x^3 3x^3-4x^3+12x^3ln(7x)/x^8 take...
  39. D

    Trying to find the quotient of infinite sums

    i am trying to re-express the following in terms of a rational function: \frac{(0+x+2x^2+3x^3+...)}{1+x+x^2+x^3+...} . i know that this is supposed to be \frac{1}{x-1} but I can't figure out how to do it. I know the denominator is just \frac{1}{1-x}. so in order for this work out, the...
  40. N

    Quotient rule Differentiation

    Homework Statement a) Differentiate y=((x-1)/(x+1))^2 b (non calculus simplification question): How would i simplify 10(5x+3)(5x-1)+5(5x-1)^2 to get 25(5x-1)(3x+1) should i expand all terms then combine and factor? Homework Equations The Attempt at a Solution a) i tried...
  41. F

    What Are the Elements of the Quotient Group D4/N?

    Homework Statement Let D4 = { (1)(2)(3)(4) , (13)(24) , (1234) , (1432) , (14)(23) , (12)(34) , (13), (24) } and N=<(13)(24)> which is a normal subgroup of d4 . List the elements of d4/N . Homework Equations The Attempt at a Solution I computed the left and right cosets to...
  42. rustynail

    How to Use the Quotient Rule to Find Derivatives of Functions

    Homework Statement I want to prove that if y = \frac{u}{v} then \frac{dy}{dx} = \frac{ v \frac{du}{dx} - u \frac{dv}{dx} }{v²} u and v are functions of x. 2. The attempt at a solution y = uv^{-1} y + dy = ( u + du ) ( v + dv )^{-1} then I suppose I could use Newton's Binomial...
  43. DryRun

    Integration of quotient: Finding the Right Substitution

    Homework Statement \int \frac{r^3-r^5}{\sqrt{1-r^2}}\,.dr The attempt at a solution The presence of \sqrt{1-r^2} suggests that i use the substitution r=sinθ The integrand becomes: \frac{\sin^3\theta-\sin^5\theta}{\cos\theta} \frac{dr}{d\theta}=\cos\theta \int...
  44. N

    Example in topology: quotient maps and arcwise connected

    Just to make sure that I'm not overlooking anything, is the following an example of a quotient map p: X \to Y with the properties that Y is pathwise connected (i.e. connected by a continuous function from the unit interval), \forall y \in Y: p^{-1}(\{ y \}) \subset X also pathwise connected and...
  45. B

    What is the Basis of a Quotient Ring?

    In my Abstract Algebra course, it was said that if E := \frac{\mathbb{Z}_{3}[X]}{(X^2 + X + 2)}. The basis of E over \mathbb{Z}_{3} is equal to [1,\bar{X}]. But this, honestly, doesn't really make sense to me. Why should \bar{X} be in the basis without it containing any other \bar{X}^n...
  46. N

    [topology] compact, locally connected, quotient topology

    Homework Statement Let X be a compact and locally connected topological space. Prove that by identifying a finite number of points of X, one gets a topological space Y that is connected for the quotient topology. Homework Equations The components of a locally connected space are open...
  47. J

    Fundamental Group of Quotient Space

    Hi I don't know how to attack the following question, any hints would be appreciated: If G is a simply connected topological group and H is a discrete subgroup, then \pi_1(G/H, 1) \cong H .Thank you James
  48. F

    What are Quotient Spaces and How Are They Used in Algebra and Topology?

    I'm having some troubles understanding the concepts of quotient algebra. May someone explain me what exactly they are, giving some concrete examples? I know that a quotient set is the set of all equivalence classes, but it sounds very vague for me and i can't make the analogy with quotient...
  49. K

    Showing the quotient is not a UFD

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  50. P

    Looking for Explanation of a Quotient Group

    So I have an elementary understanding of group theory and the goals of studying sets and operations on them but I noticed that along the way, my understanding of a quotient group is severely flawed. I understand what a so called normal subgroup is, but could someone please give an in depth...
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