Editing Quotient (section)

{{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"/>