# 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. ### I Continuity of Quotient of Complex Values

Hey all, I have a very simple question regarding the quotient of complex values. Consider the function: $$f(a) = \sqrt{\frac{a-1i}{a+1i}}$$ where ##i## is the imaginary unit. When I evaluate f(0) in Mathematica, I get ##f(0) = 1i##, as expected. But if I evaluate at a very small value of ##a##...
2. ### A What is the definition of quotient Lie algebra?

Please, in the definition of quotient Lie algebra If ##I## is an ideal of ##\mathfrak{g}##, then the vector space ##\mathfrak{g} / I## with the bracket defined by: $$[x+I, y+I]=[x, y]+I, for all x, y \in \mathfrak{g}$$, is a Lie algebra called the quotient Lie algebra of ##\mathfrak{g}## by...
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### Given two random numbers in (0,1) what are odds that the quotient is closer to an odd integer?

Closer to odd number implies ##|y/x - (2n+1)| < 1/2## for ##n = 0,1,2...##. Then $$-\frac 1 2 < \frac y x - (2n+1) < \frac 1 2 \implies\\ y < (2n + 1.5)x,\\ y > (2n + 0.5)x$$ for each ##n##. We note ##x \in (0,1)## implies ##y## can be larger than 1 since the slope is greater than 1 (but we know...
4. ### Prove that |f| is bounded by a quotient

I just spent 15 minutes to re-type all the Latex again because I lost everything while editing. why does this happen?? This is a huge waste of time. ##f## is entire so ##f## is holomorphic on ##\mathbb{D}∪C##. Also, ##\mathbb{D}∪C## is a connected set. By the maximum principle, ##f## restricted...
5. ### B Is the Quotient Topology of Real Numbers Homeomorphic to Real Numbers?

Hello! I have two related exercises I need help with 1. Partition the space ##\mathbb{R}## into the interval ##[a,b]##, and singletons disjoint from this interval. The associated equivalence ##\sim## is defined by ##x\sim y## if and only if either##x=y## or ##x,y\in[a,b]##. Then...
6. ### B Open sets in quotient topology

Hello! Reading a textbook I found that authors use the same trick to show that subsets of quotient topology are open. And I don't understand why this trick is valid. Below I provide there example for manifold (Mobius strip) where this trick was used Quote from "Differential Geometry and...
7. ### I Time derivative using the quotient rule

Hi Guys Sorry for the rudimentary post. I am busy with a numerical solution to a mechanics problem, and the results are just not as expected. I am re-checking the mathematics to ensure that all is in order in doing so I am second guessing a few things Referring to the attached scan, is the...
8. ### Cartesian sum of subspace and quotient space isomorphic to whole space

Let ##n=\dim X## and ##m=\dim Y##. Define a basis for ##X: y_1,...,y_m,z_{m+1},...,z_n##. The first ##m## terms are a basis for ##Y##. The remaining ##n-m## terms are a basis for its complement w.r.t ##X##. Let's call it ##Z##. ##X## is the direct sum of ##Y## and ##Z##; denote it as ##X=Y+Z##...
9. ### Am I using quotient spaces correctly in this linear algebra proof?

%%% Assume that ##X/Y## is defined. Since ##\dim Y = \dim X##, it follows that ##\dim {X/Y}=0## and that ##X/Y=\{0\}##. Suppose that ##Y## is a proper subspace of ##X##. Then there is an ##x\in X## such that ##x\notin Y##. Let us consider the equivalence class: ##\{x\}_Y=\{x_0\in...
10. ### I Poincaré algebra and quotient group

I see that the first four equations are definitions. The problem is about the dimensions of the quotient. Why does the set Kx forms a six dimensional Lie algebra?
11. ### I Treating the Derivative like a Quotient

Good Morning I have read that it is not justified to split the "numerator" and "denominator" in the symbol for, say, dx/dt However, when I look at Wikipedia's discussion on the Principle of Virtual Work, they do just that. (See picture, below). I was told it is OK in 1D cases, but note the...
12. ### I Trying to get a better understanding of the quotient V/U in linear algebra

Hi! I want to check if i have understood concepts regarding the quotient U/V correctly or not. I have read definitions that ##V/U = \{v + U : v ∈ V\}## . U is a subspace of V. But v + U is also defined as the set ##\{v + u : u ∈ U\}##. So V/U is a set of sets is this the correct understanding...
13. ### Linear algebra, find a basis for the quotient space

Let V = C[x] be the vector space of all polynomials in x with complex coefficients and let ##W = \{p(x) ∈ V: p (1) = p (−1) = 0\}##. Determine a basis for V/W The solution of this problem that i found did the following: Why do they choose the basis to be {1+W, x + W} at the end? I mean since...
14. ### Question about the quotient of the charge and mass of an electron

hello Witch of these are certain sentences? a-\dfrac{e}{m_e}>\dfrac{H^{-}}{m_{H^{-}}}\cdot{1000} b-\dfrac{e}{m_e}>\dfrac{H^{+}}{m_{H^{+}}}\cdot{1000} The first accurate measurement of e/m was made by english physicist J.J. Thomson in 1897, who demostrated that the quotient charge-mass of the...
15. ### Manipulating a Laurent Series Equation

Not really a homework problem, just an equation from my textbook that I do not understand. I can't think of any way to even begin manipulating the right hand side to make it equal the left hand side. Just to confirm equality (thanks to another user for suggestion), I multiplied both sides by of...

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17. ### Approximate Intelligence Quotient Gauge

The following thread regards how I am to receive an accurate gauge of my IQ. Heretofore, I undertook several IQ tests, namely the Serebriakoff Advanced Culture Fair Test, Numerus Basic, Mensa Norway, Mensa Denmark, Tero 41 and Logica Stella. Additionally, I took the Mensa Luxembourg Online Test...
18. ### I Question about the quotient rule of derivatives

Now, I understand how to use the quotient rule for derivatives and everything. I do not struggle with using it, my question is mostly about the formula itself...I very much enjoy WHY we do things in math, not just “here’s the formula, do it”...Here is the formula for the quotient rule of...
19. ### Sturm-Liouville and Rayleigh Quotient Problem

By comparing the given equation to the equation for the Sturm-Liouville form, I see that p(x) must equal 1, q(x) must equal 0 and sigma(x) must equal 1. After this, I have no idea what I should be doing for part 2.
20. ### Limit with the quotient law

Homework Statement lim (1/x - 1/3) / (x-3) x->3 Homework EquationsThe Attempt at a Solution I tried to cancel the bottom (x-3) out by multiplying the top by 3/3 and x/x and then got ((3-x)/3x)/(x-3) but ended with 0/0 and the right answer is -1/9. The top part is confusing me.

48. ### FDQR ( Freshman Dream Quotient Rule)

Homework Statement I have been working with researching and writing a paper on the Freshman Dream Quotient Rule. This rule states , and I was wondering if anyone can come up with an example of 2 functions which make this work. Homework EquationsThe Attempt at a Solution I have looked at...
49. ### A What is the Freshman Dream Quotient Rule?

I have been interested in this idea of the FDQR. This idea states the following. I have been trying to see if there is some 2 functions which make this true, but have not found it on research or with trying functions. Does anyone have any insight on this. I think it is just neat and want to...
50. ### MHB Evaluate Quotient: $ab+cd \over ad+bc$

Given positive real numbers $a,\,b,\,c,$ and $d$ that satisfy the system below: $a^2+ d^2-ad = b^2+ c^2+ bc$ and $a^2+ b^2= c^2+ d^2$. Find all possible values of the expression $\dfrac{ab + cd}{ad + bc}$.