Editing Slash (punctuation) (section)

=== Mathematics <span class="anchor" id="Arithmetic"></span><span class="anchor" id="Fraction"></span><span class="anchor" id="Ratio"></span><span class="anchor" id="Ratios"></span><span class="anchor" id="Math"></span><span class="anchor" id="Maths"></span>===

==== Fractions ====
The slash is used between two numbers to indicate a [[fraction]] or [[ratio]]. Such formatting developed as a way to write the horizontal [[fraction bar]] on a single line of text. It is first attested in [[Kingdom of England|England]] and [[Viceroyalty of Mexico|Mexico]] in the 18th century.<ref name="jeff">{{cite web |last=Miller |first=Jeff |title=Fractions |url=http://jeff560.tripod.com/fractions.html |work=Earliest Uses of Various Mathematical Symbols |via=Tripod.com |date=22 December 2014 |access-date=15 February 2016 |url-status=live |archive-url=https://web.archive.org/web/20230602010313/https://jeff560.tripod.com/fractions.html |archive-date=2 June 2023}}</ref> This notation is known as an online, solidus,<ref name="Eckersley et al">{{cite book |last1=Eckersley |first1=Richard |last2=Angstadt |first2=Richard |last3=Ellertson |first3=Charles M. |last4=Hendel |first4=Richard |last5=Pascal |first5=Naomi B. |last6=Walker Scott |first6=Anita |title=Glossary of Typesetting Terms |publisher=University of Chicago Press |date=1994 |ref={{harvid|Eckersley & al.|1994}} |isbn=0226183718 |pages=[https://books.google.com/books?id=oeTnynRiN8AC&pg=PA93 93], [https://books.google.com/books?id=oeTnynRiN8AC&pg=PA97 97]}}</ref> or shilling fraction.<ref name="Eckersley et al" /> Nowadays fractions, unlike inline division, are often given using smaller numbers, [[superscript]], and [[subscript]] (e.g., {{sup|23}}/{{sub|43}}). This notation is responsible for the current form of the [[percent sign|percent]] {{char|%}}, [[permille]] {{char|‰}}, and [[permyriad]] {{char|‱}} signs, developed from the horizontal form {{sfrac|0|0}} which represented an early modern corruption of an Italian abbreviation of ''per cento''.<ref>{{cite book |last=Smith |first=D. E. |title=Rara Arithmetica |date=1908 |location=Boston |publisher=Ginn & Co. |url= https://archive.org/details/67224711 |via=Internet Archive}}</ref>

[[File:123 fraction slash 456.svg|thumb|A fraction automatically generated by the font from basic digits and the Unicode fraction bar, 123⁄456.]]
Unicode provides for a dedicated fraction slash {{char|⁄}} that is distinct from the ASCII solidus {{char|/}}. Many typefaces draw this fraction slash (and the division slash) at a less vertical angle than the solidus. The separate encoding permits automatic formatting of the preceding and succeeding digits by glyph substitution with numerator and denominator glyphs, which are in turn distinct from superscript and subscript glyphs (e.g., display of "1, fraction slash, 2" as {{notatypo|"½"}}, and similarly "123, fraction slash, 456" as "123⁄456").<ref>{{cite book |title=The Unicode Standard |edition=6.0 |page=192 |chapter=Writing Systems and Punctuation: General Punctuation: Fraction Slash |chapter-url=https://www.unicode.org/versions/Unicode6.0.0/ch06.pdf#G12861 |date=2011 |isbn=9781936213016 |editor1-first=Julie D. |editor-last=Allen |ref={{harvid|Unicode|2011}} |publisher=Unicode Consortium |access-date=30 May 2018 |archive-date=30 July 2015 |archive-url=https://web.archive.org/web/20150730233934/http://www.unicode.org/versions/Unicode6.0.0/ch06.pdf#G12861 |url-status=live}}</ref> This is supported by an increasing number of environments and [[computer font]]s. Because support is not yet universal, some authors still use [[Unicode subscripts and superscripts#Uses|Unicode subscripts and superscripts]] to compose fractions, and many computer fonts design these characters for this purpose. In addition, [[precomposed character|precomposed fractions]] of the multiples less than 1 of {{sup|1}}/{{sub|n}} for 2 ≤ n ≤ 6 and n = 8 (e.g. {{notatypo|⅔}} and {{notatypo|⅝}}, as well as {{notatypo|⅐}}, {{notatypo|⅑}}, and {{notatypo|⅒}}, are found in the Unicode [[Number Forms]] or [[Latin-1 Supplement (Unicode block)|Latin-1 Supplement]] blocks.<ref>{{cite web |work=The Unicode Standard |edition=12.1 |publisher=Unicode Consortium |title=Number Forms |date=2019 |url=https://unicode.org/charts/PDF/U2150.pdf |access-date=22 November 2019 |archive-date=24 November 2019 |archive-url=https://web.archive.org/web/20191124140205/http://unicode.org/charts/PDF/U2150.pdf |url-status=live}}</ref>

This notation can also be used when the concept of fractions is extended from numbers to arbitrary rings by the method of [[localization of a ring]].

==== Division <span class="anchor" id="division"></span>====
The division slash {{char|∕}} is used between two numbers to indicate [[division (math)|division]].{{efn|The [[ISO 80000]] standard says that the [[division sign]] {{char|÷}}, used in elementary schools in many [[Anglophone]] countries, "should not be used" to indicate division because in other countries it is used to indicate a range of values or negation.<ref name=ISO>ISO 80000-2, Section 9 "Operations", 2-9.6</ref>}} This use developed from the [[fraction slash]] in the late 18th or early 19th century.<ref name="jeff" /> The formatting was advocated by [[Augustus De Morgan|De Morgan]] in the mid-19th century.<ref>{{cite book |last=De Morgan |first=Augustus |author-link=Augustus De Morgan |contribution=The Calculus of Functions |title=Encyclopædia Metropolitana |date=1845 |location=London |publisher=B. Fellowes et al.}}</ref>{{full citation needed|date=September 2023|reason=Volume and page number needed.}},<ref name="DeM">{{cite web |last1=Morgan |first1=Augustus De |title=A Treatise on the Calculus of Functions (Extracted From The Encyclopædia Metropolitana) |url=https://www.google.com/books/edition/A_Treatise_on_the_Calculus_of_Functions/GoM_AAAAcAAJ?hl=en&gbpv=1&pg=PA84&printsec=frontcover&dq=division |publisher=Baldwin and Cradock |language=en |date=1836}} Page 84 in this version</ref> who wrote:
:The occurrence of fractions, such as {{sfrac|a|b}}, {{sfrac|a+b|c+d}}, in the <!-- the previous two words are difficult to decipher in the scan, but it's hard to imagine them being anything beside these two words --> verbal part of mathematical works is a source of considerable loss of room, and creates an inelegant and even confused appearance in the printed page. It is very desirable, in every point of view, except the strictly mathematical one, that some method of representation should be adopted which does not require a larger space than is usual between two successive lines. At the same time, it is by no means of very great importance that the verbal part should entirely coincide with the mathematical part in notation, so long as the latter remains to preserve the usual conventions. The symbol ÷ has been disused for a sufficient reason, namely, the number of times which the pen must be taken off to form it. This has been, and we imagine always will be, the cause either of abandonment or abbreviation. The question is, whether a new and easy notation could not be substituted; and it is desirable that it should be derived from analogy, such as (accidentally, we believe) does exist in >, =, and <. If we look at × and +, and observe that the first is made by turning the second through half a right angle, denoting multiplication, which is primarily an extension of addition in like manner as division is an extension of subtraction, we may thus invent the symbol / or \ to denote division, which is also the symbol of subtraction turned through half a right angle. If a/b were used to denote a divided by b, and (a+b)/(c+d) to denote a + b divided by c + d, all necessity for increased spacing would be avoided; but this alteration should not be introduced into completely mathematical expressions, though it would be convenient in particular cases.<ref name="DeM" />
<!-- (The proceeding is an interesting elaboration on the same theme, but probably too much detail for this one topic on this encyclopedia page:) --><!--
:A complicated exponent might be avoided by the use of the symbol λ⁻¹. The student would soon learn to consider λ⁻¹{(a+bx)/(c+ex)×λa} as meaning the same thing as <center><math>a^{\frac{a+bx}{c+ex}}</math>.</center>
:If, in the course of investigation, some plan of this kind be not adopted, the expense of printing will place a limit to analysis. A work entirely devoted to the consideration of such expressions as the preceding might easily be doubled in size and price by the frequent occurrence of them in the text, or else rendered confused and unintelligible by successive abbreviations. Considering, however, that λ is an inverse symbol in the sense of (124.), and λ⁻¹ a direct one, it would, perhaps, rather be advisable that some method of denoting an exponent should be adopted which does not raise the exponent above the symbol of the root. Either of the following might be proposed, the defect of them all being that they are not derived from analogy:
:<center>a Λ {(a+bx)/(c+ex)} a:{(a+bx)/(c+ex)},</center>
:or the like. But we do not advocate the introduction of these into purely symbolical expressions, any more than in the former case.
-->

==== Quotient of set ====
{{see also|Set (mathematics)}}
A ''quotient of a set'' is informally a new set obtained by identifying some elements of the original set. This is denoted as a fraction <math>S / R</math> (sometimes even as a built fraction), where the numerator <math>S</math> is the original set (often equipped with some algebraic structure). What is appropriate as denominator depends on the context.

In the most general case, the denominator is an [[equivalence relation]] <math>\sim</math> on the original set <math>S</math>, and elements are to be identified in the quotient <math>S/{\sim}</math> if they are equivalent according to <math>\sim</math>; this is technically achieved by making <math>S/{\sim}</math> the set of all [[equivalence class]]es of <math>\sim</math>.

In [[group theory]], the slash is used to mark [[quotient group]]s. The general form is <math>G/N</math>, where <math>G</math> is the original group and <math>N</math> is the normal subgroup; this is read "<math>G</math> mod <math>N</math>", where "mod" is short for "[[modulo operation|modulo]]". Formally this is a special case of quotient by an equivalence relation, where <math>g \sim h</math> iff <math>g = hn</math> for some <math>n \in N</math>. Since many algebraic structures ([[Ring (mathematics)|ring]]s, [[vector space]]s, etc.) in particular are groups, the same style of quotients extend also to these, although the denominator may need to satisfy additional [[Closure (mathematics)|closure]] properties for the quotient to preserve the full algebraic structure of the original (e.g. for the quotient of a ring to be a ring, the denominator must be an [[Ideal (ring theory)|ideal]]).

When the original set is the set of [[integer]]s <math>\mathbb{Z}</math>, the denominator may alternatively be just an integer: <math>\mathbb{Z}/n</math>. This is an alternative notation for the set <math>\mathbb{Z}_n</math> of [[modular arithmetic#Integers modulo m|integers modulo ''n'']] (needed because <math>\mathbb{Z}_n</math> is also notation for the very different [[P-adic number|ring of ''n''-adic integers]]). <math>\mathbb{Z}/n</math> is an abbreviation of <math>\mathbb{Z}/n\mathbb{Z}</math> or <math>\mathbb{Z}/(n)</math>, which both are ways of writing the set in question as a quotient of groups.

==== Combining slash ====
Slashes may also be used as a [[combining character]] in mathematical formulae. The most important use of this is that combining a slash with a [[binary relation|relation]] negates it, producing e.g. 'not equal' <math>\neq</math> as negation of <math>=</math> or 'not in' <math>\notin</math> as negation of <math>\in</math>; these slashed relation symbols are always implicitly defined in terms of the non-slashed base symbol. The graphical form of the negation slash is mostly the same as for a division slash, except in some cases where that would look odd; the negation <math>\nmid</math> of <math>\mid</math> (divides) and negation <math>\nsim</math> of <math>\sim</math> (various meanings) customarily both have their negations slashes less steep and in particular shorter than the usual one.

The [[Feynman slash notation]] is an unrelated use of combining slashes, mostly seen in [[quantum field theory]]. This kind of combining slash takes a vector base symbol and converts it to a matrix quantity. Technically this notation is a shorthand for contracting the vector with the [[gamma matrix|Dirac gamma matrices]], so <math>A\!\!\!/ = \gamma^\mu A_\mu</math>; what one gains is not only a more compact formula, but also not having to allocate a letter as the contracted index.