View source for Quotient

{{Short description|Mathematical result of division}}
{{Other uses}}
[[File:Divide12by3.svg|thumb|alt=12 apples divided into 4 groups of 3 each.|The quotient of 12 apples by 3 apples is 4.]]
{{Calculation results}}

In [[arithmetic]], a '''quotient''' (from {{langx|la|quotiens}} 'how many times', pronounced {{IPAc-en|ˈ|k|w|oʊ|ʃ|ən|t}}) is a quantity produced by the [[division (mathematics)|division]] of two numbers.<ref>{{Cite web|title=Quotient|website=Dictionary.com|url=http://dictionary.reference.com/browse/quotient}}</ref> The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in the case of [[Euclidean division]])<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Integer Division|url=https://mathworld.wolfram.com/IntegerDivision.html#:~:text=Integer%20division%20is%20division%20in,and%20is%20the%20floor%20function.|access-date=2020-08-27|website=mathworld.wolfram.com|language=en}}</ref> or a [[Fraction (mathematics)|fraction]] or [[ratio]] (in the case of a general [[Division (mathematics)|division]]). For example, when dividing 20 (the ''dividend'') by 3 (the ''divisor''), the ''quotient'' is 6 (with a remainder of 2) in the first sense and <math>6+\tfrac{2}{3}=6.66...</math> (a [[repeating decimal]]) in the second sense.

{{anchor|Metrology}}In [[metrology]] ([[International System of Quantities]] and the [[International System of Units]]), "quotient" refers to the general case with respect to the [[units of measurement]] of [[physical quantity|physical quantities]].<ref name="ISO 80000-1"/><ref>{{Cite book |last=James |first=R. C. |url=https://books.google.com/books?id=UyIfgBIwLMQC&dq=dictionary+ratio&pg=PA349 |title=Mathematics Dictionary |date=1992-07-31 |publisher=Springer Science & Business Media |isbn=978-0-412-99041-0 |language=en}}</ref>
<ref name="International Electrotechnical Vocabulary e707">{{cite web | title=IEC 60050 - Details for IEV number 102-01-22: "quotient" | website=International Electrotechnical Vocabulary | url=https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=102-01-22 | language=ja | access-date=2023-09-13}}</ref> 
''[[Ratio]]s'' is the special case for [[dimensionless]] quotients of two quantities of the same [[kind of quantity|kind]].<ref name="ISO 80000-1">{{cite web | title=ISO 80000-1:2022(en) Quantities and units — Part 1: General | website=iso.org | url=https://www.iso.org/obp/ui/#iso:std:iso:80000:-1:ed-2:v1:en | ref={{sfnref | iso.org}} | access-date=2023-07-23}}</ref><ref name="International Electrotechnical Vocabulary g891">{{cite web | title=IEC 60050 - Details for IEV number 102-01-23: "ratio" | website=International Electrotechnical Vocabulary | url=https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=102-01-23 | language=ja | access-date=2023-09-13}}</ref>
Quotients with a non-trivial [[dimension (physics)|dimension]] and [[compound unit]]s, especially when the divisor is a duration (e.g., "[[per second]]"), are known as [[rate (mathematics)|''rates'']].<ref name="International Electrotechnical Vocabulary x558">{{cite web | title=IEC 60050 - Details for IEV number 112-03-18: "rate" | website=International Electrotechnical Vocabulary | url=https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=112-03-18 | language=ja | access-date=2023-09-13}}</ref>
For example, [[density]] (mass divided by volume, in units of [[kilogram per cubic metre|kg/m<sup>3</sup>]]) is said to be a "quotient", whereas [[mass fraction (chemistry)|mass fraction]] (mass divided by mass, in kg/kg or in percent) is a "ratio".<ref>{{cite book | title=Special Publication 811 {{!}} The NIST Guide for the use of the International System of Units |chapter=NIST Guide to the SI, Chapter 7: Rules and Style Conventions for Expressing Values of Quantities | first1=A. |last1=Thompson |first2=B. N. |last2=Taylor |date=March 4, 2020 | access-date=October 25, 2021 |chapter-url=https://www.nist.gov/pml/special-publication-811/nist-guide-si-chapter-7-rules-and-style-conventions-expressing-values |publisher=[[National Institute of Standards and Technology]]}}</ref> 
''[[Specific quantity|Specific quantities]]'' are [[intensive quantity|intensive quantities]] resulting from the quotient of a physical quantity by mass, volume, or other measures of the system "size".<ref name="ISO 80000-1"/>

==Notation==
{{Main|Division (mathematics)#Notation}}

The quotient is most frequently encountered as two numbers, or two variables, divided by a horizontal line.  The words "dividend" and "divisor" refer to each individual part, while the word "quotient" refers to the whole.

<math display="block">
\dfrac{1}{2} \quad
\begin{align} 
& \leftarrow \text{dividend  or  numerator} \\ 
& \leftarrow \text{divisor  or  denominator} 
\end{align}
\Biggr \} \leftarrow \text{quotient}
</math>

==Integer part definition==
The quotient is also less commonly defined as the greatest [[Natural number|whole number]] of times a divisor may be subtracted from a dividend—before making the [[remainder]] negative. For example, the divisor 3 may be subtracted up to 6 times from the dividend 20, before the remainder becomes negative:
: 20 − 3 − 3 − 3 − 3 − 3 − 3 ≥ 0,
while
: 20 − 3 − 3 − 3 − 3 − 3 − 3 − 3 < 0.
In this sense, a quotient is the [[integer part]] of the ratio of two numbers.<ref>{{MathWorld|urlname=Quotient|title=Quotient}}</ref>

==Quotient of two integers==
{{Main|Rational number}}

A [[rational number]] can be defined as the quotient of two [[integer]]s (as long as the denominator is non-zero).

A more detailed definition goes as follows:<ref>{{Cite book |title=Discrete mathematics with applications |last=Epp |first=Susanna S. |date=2011-01-01 |publisher=Brooks/Cole |isbn=9780495391326 |oclc=970542319 |pages=163}}</ref>

: A real number ''r'' is rational, if and only if it can be expressed as a quotient of two integers with a nonzero denominator. A real number that is not rational is irrational.

Or more formally: 
: Given a real number ''r'', ''r'' is rational if and only if there exists integers ''a'' and ''b'' such that <math>r = \tfrac a b</math> and  <math>b \neq 0</math>.

The existence of [[irrational number]]s—numbers that are not a quotient of two integers—was first discovered in geometry, in such things as the ratio of the diagonal to the side in a square.<ref>{{Cite web|title=Irrationality of the square root of 2.|url=https://www.math.utah.edu/~pa/math/q1.html|access-date=2020-08-27|website=www.math.utah.edu}}</ref>

== More general quotients ==
Outside of arithmetic, many branches of mathematics have borrowed the word "quotient" to describe structures built by breaking larger structures into pieces. Given a [[Set (mathematics)|set]] with an [[equivalence relation]] defined on it, a "[[quotient set]]" may be created which contains those equivalence classes as elements. A [[quotient group]] may be formed by breaking a [[Group (mathematics)|group]] into a number of similar [[cosets]], while a [[Quotient space (linear algebra)|quotient space]] may be formed in a similar process by breaking a [[vector space]] into a number of similar [[linear subspace]]s.

==See also==
* [[Product (mathematics)]]
* [[Quotient category]]
* [[Quotient graph]]
*[[Division (mathematics)#Of integers|Integer division]]
* [[Quotient module]]
* [[Quotient object]]
* [[Quotient of a formal language]], also left and right quotient
* [[Quotient ring]]
* [[Quotient set]]
* [[Quotient space (topology)]]
* [[Quotient type]]
* [[Quotition and partition]]

==References==
{{Reflist}}

==External links==
*{{Commonscatinline|Quotients}}

{{Fractions and ratios}}
{{Authority control}}

[[Category:Quotients| ]]