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.

View More On Wikipedia.org
Recent contents Top users
  1. L

    Bilinear mapping between quotient spaces

    Problem: Let L and M be finite dimensional linear spaces over the field K and let g: L\times M \rightarrow K be a bilinear mapping. Let L_0 be the left kernel of g and let M_0 be the right kernel of g. a) Prove that dim L/L_0 = dim M/M_0. b) Prove that g induces the bilinear mapping g': L/L_0...
  2. W

    Proof: Quotient and Remainder involving floor function

    Homework Statement Show that if a (is in) Z and d (is in) Z+, d>1 then the quotient and remainder when a is divided by d are a/d and a-d(floor function(a/d)) Homework Equations The Attempt at a Solution solution (that i have from handout - that i don't understand) by thm 2 p202 (? i am not...
  3. K

    Quotient Field of the Positive Rationals

    So earlier this year I came here to discuss about having fun with groups, rings and isomorphisms and such. I fell upon the idea of finding an isomorphism of the positive rationals to the sequence of the exponents found in their prime factorization. I didn't know what much to do with it since I...
  4. N

    What is the correct derivative for the quotient \frac{\sqrt{x}}{x^3+1}?

    EDIT: I found the mistake, question is answered! Its funny because I spent 40+ minutes trying to get the right answer and looking for the mistake but typing it all out in latex helped me to find it! Homework Statement \frac{\sqrt{x}}{x^3+1} The Attempt at a Solution...
  5. L

    Orders of Quotient Groups (Abstract Algebra)

    Homework Statement Let H be a subgroup of K and K be a subgroup of G. Prove that |G:H|=|G:K||K:H|. Do not assume that G is finite Homework Equations |G:H|=|G/H|, the order of the quotient group of H in G. This is the number of left cosets of H in G. The Attempt at a Solution I...
  6. L

    Proving Linear Isomorphism: Quotient Spaces in Vector Subspaces

    Hey all, We have not covered quotient vector spaces in class, but my homework (due before next lecture) covers a few proofs regarding quotient spaces. I've done some reading on them and some of their aspects, but as it is still a new concept, I am struggling with how to go about this proof...
  7. E

    Simplifying after applying quotient rule

    Homework Statement http://images.calcchat.com/solutionart/etf5e/03/c/se03c01047.png Homework Equations The Attempt at a Solution What am i missing, because i get: 6/4 [(-1 + tanx*secx- (tanx)^2]
  8. F

    Is the Quotient Space of Identifying Rational Numbers Compact and Hausdorff?

    Let X be the quotient space obtained of \mathbb{R} identifiying every rational number to a point. Is X a Hausdorff space? Is X a compact space?
  9. K

    Holomorphic Quotient Maps on Projective Space

    Homework Statement Let \mathbb{CP}^n be n-dimensional complex projective space, and let \pi: \mathbb C^{n+1}\setminus\{0\} \to \mathbb{CP}^n be the quotient map taking \pi(z_1,\ldots,z_{n+1}) = [z_1,\ldots,z_{n+1}] where the square brackets represent the equivalence class of lines through...
  10. A

    Confusion about quotient spaces

    If X is a topological vector space and Y is a subspace, we can define the quotient space X/Y as the set of all cosets x + Y of elements of X. There is an associated mapping \pi, called the quotient map, defined by \pi(x) = x + Y. If I'm not mistaken, there is an equivalence relation lurking...
  11. nukeman

    Difference quotient question

    Homework Statement I am having trouble with a couple difference quotient questions. Here is a question I can't seem to solve. Evaluate the difference quotient for the given funtion and simplify your answer. f(x) = 4 + 3x - x^2 , f(3 + h) - f(3) / h Homework Equations...
  12. T

    Prove Quotient Topology: Lee's Introduction to Smooth Manifolds

    Homework Statement This is from Lee's Introduction to Smooth Manifolds. Suppose π : X → Y is a quotient map. Prove that the restriction of π to any saturated open or closed subset of X is a quotient map. Homework Equations Lee defines a subset U of X to be saturated if U = π-1(π(U)). π...
  13. C

    Double derivative of a quotient

    Find d^2y/dx^2 when dy/dx = (3x^2 - 24x - 45) ÷ 2y i tried by (6x-24) ÷ 2y. Unsure what to do about the y on the denominator.
  14. J

    Arithmetic Question for Finding Derivative using Quotient Rule

    Homework Statement Find dy/dx for the following function: y = (11-cos(x))/(2+cos(x)) Homework Equations Quotient Rule: y'= ((g(x))(f'(x)) - (f(x))(g'(x)))/ (g(x)^2) The Attempt at a Solution I used the quotient rule to come up with this: y'= ((2+cos(x))(sin(x)) -...
  15. K

    What's the motivation for coset and quotient structure?

    I'm curious why people develop these objects. Although I've seen some proofs of theorems using coset(or quotient space somtimes), it remains mysterious to me how people come up with these in the first palce. So what's the motivation for inventing coset or quotient space, logical and historical?
  16. H

    Proving Normality of a Quotient Group: A Shortcut Method

    I have a question I need to resolve before my exam on thursday. It relates to the following result: Let N be a normal subgroup of G, and let K be any subgroup of G containing N. Then K/N is a subgroup of G/N. Furthermore, K/N is normal in G/N if and only if K is normal in G. The first part...
  17. D

    Differentiation of a Quotient check

    Homework Statement S = (8x^2 - 7x + 125)/(8x+7) Where the value of x is unknown The Attempt at a Solution u = 8x^2 - 7x + 125 du/dx = 16x - 7 v = 8x + 7 dv/dx = 8 ds/dx = (v*du/dx - u*dv/dx)/v^2 = (8x+7*16x-7 -...
  18. J

    Calculus: Composite and Quotient rules. HELP

    Homework Statement Remember to show your working explicitly throughout your answer to this question. (a) (i) Use the Composite Rule to differentiate the function f(x) = (x^2− 6x + 23)^(3/2) (ii) Use the Quotient Rule and your answer to part (a)(i) to show that the function: g(x) = (x −...
  19. S

    Proving the parametrization of a Torus imbedded in R3 is a Quotient map

    Homework Statement Let b > a > 0. Consider the map F : [0, 1] X [0, 1] -> R3 defined by F(s, t) = ((b+a cos(2PIt)) cos(2PIs), (b+a cos(2PIt)) sin(2PIs), a sin(2PIt)). This is the parametrization of a Torus. Show F is a quotient map onto it's image. Homework Equations Proving that any subset...
  20. E

    Product Rule vs. Quotient Rule: A Common Mistake in Differentiating Equations

    When differentiating the following equation, y=\frac{d}{dx}\frac{e^x}{x}, why is it wrong to rearrange the equation to y=\frac{d}{dx}\frac{1}{x}e^x and apply the product rule? Doing so gives me a different result than using the chain rule in conjunction with the quotient rule. Sorry about...
  21. Y

    A question regarding quotient space

    Does quotient space V/N include N itself? I think it does because: V/N includes all the cosets of N, and a coset of N is defined as N+\alpha={\epsilon+\alpha: \epsilon \in N}. This definition is from Advanced Calculus by Loomis & Sternberg and it does not say \alpha cannot be 0. So V/N...
  22. F

    How is the quotient of two constants calculated in a given equation?

    Homework Statement Part of a larger problem. I know that F_{1}^2+2F_{1}F_{2}-F_{2}^2=0 where F_{1} and F_{2} are x and y components of a force. Hence \frac{F_{1}}{F_{2}}=1\pm\sqrt{2} I can't see how that step is done. Homework Equations The Attempt at a Solution...
  23. G

    Quotient Ring of a Polynomial Ring

    Hi, given a polynomial ring R=\mathbb{C}[x_1,\ldots,x_n] and an ideal I=\langle f_1, f_2 \rangle, \quad f_1, f_2 \in R, is it always true that R/I \cong (R/\langle f_1 \rangle)/\phi(\langle f_2 \rangle), with \phi: R \rightarrow R/I being the quotient map? That is, is quotienting by I always...
  24. W

    Factoring 'h' out in difference quotient

    Hi all, I'm a beginner in calculus so my question might be stupid. When a function is differentiable, then in difference quotient one can always factor 'h' out in the numerator, even if the function is exponential and 'h' is in the exponent. Is some magic behind this or something else? I've read...
  25. J

    Quotient Rule (In what order do you choose which to differentiate)

    Homework Statement The Derivative of x-2/(x^2-1) Can you show me how to get the derivative using the quotient rule and tell me if the order matters? I always thought that the order never mattered until I got a wrong graph because it seems I got the wrong derivative. Homework...
  26. E

    Discrete quotient group from closed subgroup

    Hi All, I've come across a theorem that I'm trying to prove, which states that: The quotient group G/H is a discrete group iff the normal subgroup H is open. In fact I'm only really interested in the direction H open implies G/H discrete.. To a lesser extent I'm also interested in the H...
  27. F

    Quotient Map Theorem: Topology Induced by f

    Here is theorem 9.2 from Stephen Willard's General Topology: If X and Y are topological spaces and f:X\to Y is continuous and either open or closed, then the topology \tau on Y is the quotient topology induced by f. So f has to be onto doesn't it? Otherwise there will be multiple...
  28. P

    Derivative of product & quotient of functions - method of increments

    In my calculus book, the method of increments is used to find the derived function of the product and quotient of two functions. For example for the derivative of the product of functions u and v where u0 and v0 are the values of u and v at x = x0: y = uv y' = u0(dv/dx) + v0(du/dx) +...
  29. A

    Dy/dx - a limit or a quotient?

    dy/dx - a limit or a quotient?? Hello Friends. I have a confusion in differentiation. We know that lim \Deltax->0 ; \Deltay/\Deltax = dy/dx. Then how can dy and dx be treated as independent variables, and dy/dx as their quotient, if dy/dy is actually a limit? Also, since we won't be able to...
  30. R

    I finding the derivative of this quotient

    Homework Statement (x)/(x^2+1)^1/2 Homework Equations The Attempt at a Solution I got to this (x^2+1)^-1/2[(x^2+1)-x^2/(x^2+1)
  31. P

    How antidifferentiate quotient without using advanced methods?

    How do I find the indefinite integral \int\frac{dx}{x^{2}+16} without using partial integration or variable change? I have no clue how this can be done.
  32. E

    Confusion on the definition of a quotient map

    Let X and Y be topological spaces; let p:X -> Y be a surjective map. The map p is said to be a quotient map provided a subset U of Y is open in Y if and only if p^-1(U) is open in X. Let X be the subspace [0,1] U [2,3] of R, and let Y be the subspace [0,2] of R. The map p:X -> Y defined by...
  33. E

    Help w/ the difference quotient

    Homework Statement Show that if f(x)=sinx then (f(x+h)-f(x))/h=((sin(h/2))/(h/2))(cos(x+h/2) Homework Equations Trig identities, possibly the half angle formulas? The Attempt at a Solution (f(x+h)-f(x))/h = (f(x+ h/2 + h/2)-f(x))/(h/2 + h/2) = (sin(x+ h/2 + h/2)-sin(x))/(h/2 +...
  34. E

    Exploring the Difference Quotient for f(x) = sinx

    Homework Statement Show that if f(x)=sinx then (f(x+h)-f(x))/h=((sin(h/2))/(h/2))(cos(x+h/2) Homework Equations Trig identities, possibly the half angle formulas? The Attempt at a Solution (f(x+h)-f(x))/h = (f(x+ h/2 + h/2)-f(x))/(h/2 + h/2) = (sin(x+ h/2 + h/2)-sin(x))/(h/2 +...
  35. A

    Derive Quotient Rule: Find the Mistake

    Homework Statement derive f(x)=x2+2x+1/ x2-3x+2The Attempt at a Solution (x2-3x+2)(d/dx)(x2+2x+1)-(x2+2x+1)(d/dx)(x2-3x+2)/(x2-3x+2)2 =(x2-3x+2)(2x+2)-(x2+2x+1)(2x-3) =(2x3-4x2-2x+4)-(2x3+x2-4x-3) =5x2+2x+7/(x2-3x+2)2 that's what I'm getting... but according to the online derivative...
  36. Rasalhague

    Quotient of infinitesimals indeterminate?

    In Lectures on the hyperreals: an introduction to nonstandard analysis, pp. 50-51, Goldblatt includes among his hyperreal axioms that the sum of two infinitesimals is infinitesimal, that the product of an infinitesimal and an appreciable (i.e. nonzero real) number is infinitesimal, and that the...
  37. K

    Need Help in Direct Product and Quotient (factor) Group

    Hi All. I have several questions on abstract algebra. Here are my questions and the attempts I had done so far. (1)Let denote Z as the integer.According to theorem, the direct product of Z3 X Z7 = Z21. Hence, is Z4 X Z2 is equal to Z8? Z2X Z2 is equal to Z4? (2)For Z4 ={0,1,2,3 }, and Z2...
  38. L

    Gibbs Energy and Reaction Quotient

    I have been putting some thought into understanding Gibbs energy but I can't quite figure one thing out. Here is my dilemma: Say that someone wants to react A with B to form C, and they mix pure A with pure B. At the moment the reaction starts, there is no C in the mixture (is this...
  39. I

    Tricky difference quotient problem i am stuck

    Homework Statement find f(a+h)-f(a)/h for the f (x)= 2x+3x2 Homework Equations f(a+h)-f(a)/h The Attempt at a Solution f(a+h)-f(a)/h =2(a+h)-3(a+h)2-2x+3x2 =2a+2h-3a2-3h2-2a+3a2 =2h-3h2/h that is where i am stuck help please
  40. I

    Difference quotient (i know this is right) but it is telling me otherwise

    Homework Statement Given the function , calculate the following values f(x) = 2-5x f(a) f(a+h) f(a+h)-f(a)/h The Attempt at a Solution f(a)= 2-5x f(a+h)= 2-5(x-h) f(a+h)-f(a)/h = -5 i am 100% sure but the darn computer is telling me that f(a) and f(a+h) is wrong. my calc professor is using...
  41. E

    Simplify using Factoring after Quotient Rule

    I am taking an online Introductory Calculus course. I have a decent understanding thus far, however, the problem I'm working on gets somewhat messy and I am having a difficult time simplyifing the answer. f"(x) = (x^2 + 9)^2 (-2x) - [(9 - x^2)(2)(x^2 + 9)(2x)]/(x^2 + 9)^4 the solutions manual...
  42. T

    Difference Quotient Isn't working.

    Homework Statement Find the derivative of f using the difference quotient and use the derivative of f to determine any points on the graph of f where the tangent line is horizontal. f(x)=3x^3-9x Homework Equations The Attempt at a Solution \lim_{\Delta...
  43. W

    SO(3) as a quotient group of SU(2)?

    we know there is a two to one homomorphism from SU(2) to SO(3) suppose u is an element in SU(2) then u and -u map into the same element in SO(3) the question is, maybe SO(3) is a quotient group of SU(2)? with respect to the subgroup {I,-I}?
  44. S

    Is the Quotient Rule Necessary?

    If you can write, for example \frac{cosx}{x+1} = (cosx)(x+1)^{-1} then what is the point of the quotient rule? Can't you just use the product rule for everything?
  45. P

    Is there a way to prove the quotient rule using differentials

    Specifically, how do you prove the quotient rule using a similar method that Leibniz used for the product rule?: http://en.wikipedia.org/wiki/Product_rule#Discovery_by_Leibniz I've tried it once for d(u/v) but I keep getting a vdv term in the denominator.
  46. Z

    Quotient groups related problem

    Homework Statement Let G be a finite group and N\triangleleftG such that |N| = n, and gcd(n,[G:N]) = 1. Proof that if x^{n} = e then x\inN. Homework Equations none. The Attempt at a Solution I defined |G| = m and and tried to find an integer which divides both n and m/n. I went for some X...
  47. P

    Quotient Groups and their Index

    As a way to keep busy in between semesters I decided to work my way through Algebra by Dummit and Foote in order to prepare for the fall. Working my way through quotient groups is proving to be quite difficult and as a result I'm stuck on an exercise that looks simple, but I just don't know...
  48. B

    What is the/a Quotient Bundle.?

    Hi, everybody: I have read different sources for quotient bundle: Milnor and Stasheff, Wikipedia Wolfram, and I still cannot figure out how it's defined. All I know is that it involves a space X, a subspace Y, and a restriction. Let Tx be a bundle p:M-->X for X, and...
  49. T

    Quotient rule integration problem

    \int \frac{1}{2x+3}=\frac{\ln |2x+3|}{2}+c so why is \int \frac{1}{x^2+x}\neq \frac{\ln |x^2+x|}{2x+1}+c ? is it because in general , \int \frac{1}{x}=\ln |x|+c the denominator is meant to be only linear function ?
  50. S

    Quotient Spaces/Homeomorphic spaces?

    Hi, Problem: Let X=\{x\times y|x^2+y^2\leq1\}, \mbox{ in } R^2. \mbox{ Let } X^{\star} \mbox{ be the partition of X consisting of all the one point sets } \{x\times y\}, x^2+y^2<1, \mbox{ along with the set } S^1=\{x\times y | x^2+y^2=1\}. \mbox{ Then it continues by saying...
Back
Top