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. K

    MHB Difference Quotient of f(x)=1/(x-3)

    Simply and find the difference quotient for f(x)= 1/(x-3) I know the difference quotient formula is f(x+h)-f(x)/h but when I try solving it I keep getting it wrong.
  2. JR Sauerland

    I What am I missing here in this difference quotient problem?

    I've highlighted the part in yellow I don't understand. He apparently 'drops' Δx in the last line, but doesn't display how. What I do know is that he is taking the limit as Δx→0, or as Delta x approaches zero. I'm simpy missing what he did though to drop the Δx. I mean, I understand he...
  3. M

    MHB Statements about subrings and quotient

    Hey! :o I want to check if the following statement are true. Let $R$ be a ring, $S$ a subring and $I$ an ideal. If $R$ is Noetherian then $S$ is also. If $R$ is Noetherian then $R/I$ is also. If $R$ is Artinian then $S$ is also. If $R$ is Artinian then $R/I$ is also. If $R$ is...
  4. M

    MHB Show that G is nilpotent when the quotient group is nilpotent

    Hey! :o I want to show that if $H\subseteq Z(G)$ and $G/H$ is nilpotent then $G$ is also nilpotent. I have done the following: Since $G/H$ is nilpotent there is a series of normal subgroups $$1\leq N_1\leq N_2\leq \cdots \leq N_k=G/H$$ with $N_{i+1}/N_i\subseteq Z((G/H)/N_i)$. From the...
  5. S

    I Why only normal subgroup is used to obtain group quotient

    Hello! As far as I know any subgroup can, in principle, be used to divide group into bundle of cosets. Then any group element belongs to one of the cosets (or to the subgroup itself). And such division still is not qualified as a quotient. Yes, the bundle of cosets in this case will be...
  6. mnb96

    Quotient of group by a semidirect product of subgroups

    Hello, if we consider a group G and two subgroups H,K such that HK \cong H \times K, then it is possible to prove that: G/(H\times K) \cong (G/H)/K Can we generalize the above equation to the case where HK \cong H \rtimes K is the semidirect product of H and K? Clearly, if HK is a semidirect...
  7. M

    MHB Quotient Groups & how to interpret notation?

    Hello, I am having some trouble truly interpreting what certain notation means when defining quotient groups, etc. (My deepest apologies in advance, with my college workload I simply have not had the time to really sit down and master latex.) Here are a few random examples I've seen in...
  8. T

    MHB What is meant by the unique integers Q and R in the quotient remainder theorem?

    Given any integer A, and a positive integer B, there exist unique integers Q and R such that $$A= B * Q + R$$ where $$ 0 ≤ R < B$$. When is says that $$Q$$ and $$R$$ are unique, what does that mean? That they are different from each other?
  9. T

    MHB Quotient remainder theorem problem.

    For any int $$n $$ , prove that $$ 4 | n (n^2 - 1) (n + 2)$$. I know I have to use the quotient remainder theorem, but I'm wondering how to go about this problem. I'm not sure how to phrase this problem in English.
  10. Saracen Rue

    Derivative of a fractional function without quotient rule

    Homework Statement The displacement of a particle can be modeled by the function x(t)=\frac{2x-5}{4x^2+2x}, where t is in seconds, x is in meters, and t ∈ [1,10] a) Determine the derivative of the function without using the quotient rule. b) Hence, find exactly when the particle is...
  11. G

    Cyclic Quotient Group: Is My Reasoning Sound?

    Hi everyone. So it's apparent that G/N cyclic --> G cyclic. But the converse does not seem to hold; in fact, from what I can discern, given N cyclic, all we need for G/N cyclic is that G is finitely generated. That is, if G=<g1,...,gn>, we can construct: G/N=<(g1 * ... *gn)*k> Where k is the...
  12. DeldotB

    Principle Ideals of a Polynomial Quotient Ring

    Homework Statement Let A be the algebra \mathbb{Z}_5[x]/I where I is the principle ideal generated by x^2+4 and \mathbb{Z}_5[x] is the ring of polynomials modulo 5. Find all the ideals of A Let G be the group of invertible elements in A. Find the subgroups of the prime decomposition.Homework...
  13. Math Amateur

    MHB Arithmetic for Quotient Groups - How exaclty does it work

    I have just received some help from Euge regarding the proof of part of the Correspondence Theorem (Lattice Isomorphism Theorem) for groups ... But Euge has made me realize that I do not understand quotient groups well enough ... here is the issue coming from Euge's post ... We are to consider...
  14. Math Amateur

    MHB Solutions to Irreducible Polynomials & Quotient Rings

    I have just finished a post entitled: http://mathhelpboards.com/linear-abstract-algebra-14/irreducible-polynomials-quotient-rings-rotman-proposition-3-116-a-16163.htmlon the Linear and Abstract Algebra Forum ... I want to have the following code recognised: k(z) \subseteq \text{ I am } \phi...
  15. Math Amateur

    MHB Irreducible Polynomials and Quotient Rings - Rotman Proposition 3.116

    I am reading Joseph J. Rotman's book: A First Course in Abstract Algebra with Applications (Third Edition) ... I am currently focused on Section 3.8 Quotient Rings and Finite Fields ... I need help with an aspect of the proof of Proposition 3.116 Proposition 3.116 and its proof reads as...
  16. nusaiba mustafa

    Exploring the Usual Respiratory Quotient for Humans: 0.7-1.0

    Why is usual respiratory quotient for humans is between 0.7 and1.0
  17. J

    MHB Find Difference Quotient & Evaluate/Approximate Limits

    I understand how to find a difference quotient, but afterwards it asks me to then evaluate or approximate each limit, is that just by plugging in the given limit or is there another step?
  18. HaLAA

    Show the range of f is isomorphic to a quotient of z

    Homework Statement Let G be any group and a in G, define f: Z → G by f(n) = a^n Apply any isomorphism theorem to show that range of f is isomorphic to a quotient group of Z Homework EquationsThe Attempt at a Solution The range of f is a^n , then quotient group of Z is Z/nZ Apply the first...
  19. K

    Are Projection Mappings considered Quotient Maps?

    The book I am using for my Introduction to Topology course is Principles of Topology by Fred H. Croom. Problem: Prove that if ##X=X_1\times X_2## is a product space, then the first coordinate projection is a quotient map. What I understand: Let ##X## be a finite product space and ##...
  20. J

    Two quotient groups implying Cartesian product?

    Assume that G is some group with two normal subgroups H_1 and H_2. Assuming that the group is additive, we also assume that H_1\cap H_2=\{0\}, H_1=G/H_2 and H_2=G/H_1 hold. The question is that is G=H_1\times H_2 the only possibility (up to an isomorphism) now?
  21. K

    MHB Proving that two quotient modules are isomorphic

    Given a ring $R$ and $R$-modules $A,B,C,D$ such that $$\sigma:A \rightarrow B, \tau: C \rightarrow D, \rho: A \rightarrow C, \kappa: B \rightarrow D, \ \mathrm{and} \ \kappa \circ \sigma = \tau \circ \rho,$$ where $\sigma, \tau, \rho, \kappa$ are homomorphisms and $\rho, \kappa$ are...
  22. W

    Is the Quotient of Two Operators AB-1 or B-1A?

    HI, Suppose there are two operators A and B , We have to find A /B - Will it equal to AB-1 OR B-1 A , Because i have read that it equals to AB-1 , BUT i could not find reason for that. thanks
  23. C

    Quotient field of the integral closure of a ring

    This is probably a stupid question. Let R be a domain, K its field of fractions, L a finite (say) extension of K, and S the integral closure of R in L. Is the quotient field of S equal to L ? I believe that not, but I have no counter-example.
  24. snoopies622

    Finding the remainder of an algebraic quotient

    I'm tutoring a pupil for a CLEP exam and her book includes the following algebra problem: What is the remainder when 9x^{23} - 7x^{12} - 2x^{5} +1 is divided by x+1 ? I know how to find the answer by computing the quotient of these two expressions, but in this case doing that is so tedious I...
  25. Z

    MHB Difference quotient for quadratic function

    Find the difference quotient f(x+h)-f(x)/h Where h\ne 0, for the function below f(x)=5x^2+4 Simplify your answer as much as possible. How do I do this?
  26. H

    Quotient space of the unit sphere

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  27. T

    MHB Simplifying the Difference Quotient for f(x) = 1/x

    I have a problem: f(x) = 1/x, [f(x) - f(a)] / x- a I am wondering how to approach this problem. I have so far. (1/x - 1/a) / (x-a) ([a-x] / xa) / (x-a) How would I simplify this? By the way, the answer is -1 / ax
  28. Math Amateur

    MHB Quotient Modules and Module Homomorphisms - Cohn - Corollary 1.16

    I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series) In Chapter 1: Basics we find Corollary 1.16 on module homomorphisms and quotient modules. I need help with some aspects of the proof. Corollary 1.16 reads as follows: In the above text...
  29. Math Amateur

    MHB Quotient Modules and Homomorphisms

    I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series) In Chapter 1: Basics we find Theorem 1.15 on module homomorphisms and quotient modules. I need help with some aspects of the proof. Theorem 1.15 reads as follows: In the proof of the...
  30. N

    Simplifying Difference Quotients with Variable Expressions - How to Solve?

    Homework Statement Find f(a), f(a + h),and the difference quotient f(a + h) − f(a), h where h ≠ 0. f(x) = 5x/x-1 f(a) = ? f(a + h) = ? f(a + h) − f(a)/(h) = ? The Attempt at a Solution As I understand it I must substitute the values but I don't understand the question...
  31. M

    How can the difference quotient be used to simplify the expression for y=2^x?

    1. Simplify the algebraic expression you get for Δy and Δy/Δx for the equation y=2^x 2. Use the difference quotient (f(x+h)-f(x))/h. No use of chain rule or other shortcuts. 3. I've tried a host of things, including raising terms to a natural log power (I.e. e^(ln2)*x*h), using...
  32. A

    Quotient modules and rings

    Let R be a local ring with maximal ideal J. Let M be a finitely generated R-module, and let V=M/JM. Then if \{x_1+JM,...,x_n+JM\} is a basis for V over R/J, then \{x_1, ... , x_n\} is a minimal set of generators for M. Proof Let N=\sum_{i=1}^n Rx_i. Since x_i + JM generate V=M/JM, we have...
  33. S

    Quotient Space Linear Operator

    Could anyone help me solve this problem? Let A,B be two subspace of V, a \in A, b \in B. Show that the following operation is linear and bijective: (A + B)/B → A/(A \cap B): a + b + B → a + A \cap B I really couldn't understand how the oparation itself works, i.e, what F(v) really is in...
  34. Math Amateur

    MHB Vector Spaces and Their Quotient Spaces - Simple Clarification Requested

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  35. Greg Bernhardt

    What is the Quotient Rule and how is it used to find derivatives?

    Definition/Summary The quotient rule is a formula for the derivative of the quotient of two functions, for which derivatives exist. Equations f(x) = \frac{g(x)}{h(x)} Then, f'(x) = \frac{h(x)g'(x)-g(x)h'(x)}{(h(x))^2} here, h(x) \: \neq \: 0 Extended explanation Even...
  36. Greg Bernhardt

    What is the Definition and Explanation of a Quotient Group?

    Definition/Summary A quotient group or factor group is a group G/H derived from some group H and normal subgroup H. Its elements are the cosets of H in G, and its group operation is coset multiplication. Its order is the index of H in G, or order(G)/order(H). Equations...
  37. S

    MHB Quotient rings and homorphic images

    am given that ϕ is a function from F(R) tp RxR defined by ϕ(f)=(f(0),f(1)) i proved that ϕ is a homomorphism from F(R) onto RxR. i showed that 1) ϕ(f) +ϕ(g)=ϕ(f+g) [for all f,g in F(R)] 2)ϕ(f)*ϕ(g)= ϕ(f*g) how do i show that ϕ is onto and define the kernal??(Wasntme)
  38. O

    MHB Understanding Finite Quotient Groups: G/H with G=Z6 and H=(0,3)

    G is a group and H is a normal subgroup of G. where G=Z6 and H=(0,3) i was told to list the elements of G/H I had: H= H+0={0,3} H+1={14} H+2={2,5} now they are saying H+3 is the same as H+0, how so?
  39. G

    Proof of Quotient Rule using Product Rule

    Hey guys, just trying to understand how the quotient rule is derived, so I head over to wikipedia and saw this: But I'm having some difficulty understanding what goes on between these two steps: Could someone shed some light on this?
  40. W

    Compute the Quotient: [Z+Z]/[2Z+2Z]

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  41. hideelo

    Understanding Saturated Sets in Quotient Maps

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  42. Math Amateur

    Mobius Band as a Quotient Topology

    I am reading Martin Crossley's book, Essential Topology. I am at present studying Example 5.55 regarding the Mobius Band as a quotient topology. Example 5.55 Is related to Examples 5.53 and 5.54. So I now present these Examples as follows: I cannot follow the relation (x,y) \sim (x', y')...
  43. Sudharaka

    MHB Annihilator of a Quotient Ring

    Hi everyone, :) I think I need to refresh my memory about annihilators and quotient rings. Hope you can help me with the following example. I want to find the annihilator of $a'$ and $b'$ of the quotient ring $R=\mathbb{Z}/(a'b')$ where $a',\,b'>1$. So if I go by the definition...
  44. Chacabucogod

    Quotient Spaces and Hyperplanes

    Hi, I'm currently reading Shilov's Linear Algebra and he mentions that Hyperplanes are planes that don't pass through the origin. Wouldn't that be a quotient space? Thank you.
  45. andrewkirk

    Is a state space with indistinguishable particles a quotient? Of what?

    I’ve been trying to understand how having indistinguishable particles in a system changes the nature of the state space. The QM texts I have gloss over this. A typical approach is to define the symmetric and anti-symmetric kets that serve as a basis for the eigenspace containing...
  46. S

    Munkres HW Problem Help Regarding Quotient Spaces and Collapsing

    This is how the problem appears in my book(Munkres 2nd edition Topology, sect. 22 pg 145) 6. Recall that R_K denotes the real line in the K-topology. Let Y be the quotient space obtained from R_K by collapsing the set K to a point; let p : R_K → Y be the quotient map. (a) Show that Y...
  47. AntSC

    Quotient & Product Rule: Same funtion, different answers?

    Can someone check my working. I don't understand why i am getting different answers? u(x,t)=\frac{{e}^{-\frac{x^2}{4Dt}}}{\sqrt{4Dt}} Differentiate w.r.t 't' by quotient rule: \frac{\partial u}{\partial t}=\left[ \frac{1}{\sqrt{4Dt}}\cdot \frac{x^2}{4Dt^2}\cdot...
  48. C

    Metric tensor after constructing a quotient space.

    Suppose we have some two-dimensional Riemannian manifold ##M^2## with a metric tensor ##g##. Initially it is always locally possible to transform away the off-diagonal elements of ##g##. Suppose now by choosing the appropriate equivalence relation and with a corresponding surjection we construct...
  49. W

    Factor (Quotient) Space definitions.

    I'm learning algebra by myself and this concept is confusing me. Please excuse me if I define anything wrong... I've never expressed myself in this language before. Lets say we have a group G and a group G' and there exists a homomorphism R: G → G' and for any element g \in G, the...
  50. Sudharaka

    MHB Finding Basis of the Quotient Space

    Hi everyone, :) This seems like a pretty simple question, but up to now I haven't found a method to solve it. Hope you can provide me a hint. :) Problem: Let \(V\) be a space with basis \(B=\{b_1,\,b_2,\,b_3,\,b_4,\,b_5\},\,U\) the subspace spanned by \(u_1=b_1+b_2+b_3+b_4+b_5\)...
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