Quotient#Metrology

File:Divide12by3.svg

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 of two numbers.{{Cite web|title=Quotient|website=Dictionary.com|url=http://dictionary.reference.com/browse/quotient}} The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in the case of Euclidean division){{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}} or a fraction or ratio (in the case of a general 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 $6+\tfrac{2}{3}=6.66...$ (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 quantities.{{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}}

{{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}}

Ratios is the special case for dimensionless quotients of two quantities of the same kind.{{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}}{{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}}

Quotients with a non-trivial dimension and compound units, especially when the divisor is a duration (e.g., "per second"), are known as rates.{{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}}

For example, density (mass divided by volume, in units of kg/m³) is said to be a "quotient", whereas mass fraction (mass divided by mass, in kg/kg or in percent) is a "ratio".{{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}}

Specific quantities are intensive quantities resulting from the quotient of a physical quantity by mass, volume, or other measures of the system "size".

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.

$\dfrac{1}{2} \quad
\begin{align}
& \leftarrow \text{dividend or numerator} \\
& \leftarrow \text{divisor or denominator}
\end{align}
\Biggr \} \leftarrow \text{quotient}$

Integer part definition

The quotient is also less commonly defined as the greatest 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.{{MathWorld|urlname=Quotient|title=Quotient}}

Quotient of two integers

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

A more detailed definition goes as follows:{{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}}

: 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 $r = \tfrac a b$ and $b \neq 0$ .

The existence of irrational numbers—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.{{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}}

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 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 into a number of similar cosets, while a quotient space may be formed in a similar process by breaking a vector space into a number of similar linear subspaces.

References

External links

{{Commonscatinline|Quotients}}