Mathematics
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 England and Mexico in the 18th century. This notation is known as an online, solidus, or shilling fraction. Nowadays fractions, unlike inline division, are often given using smaller numbers, superscript, and subscript (e.g., 23/43). This notation is responsible for the current form of the percent %, permille ‰, and permyriad ‱ signs, developed from the horizontal form 0/0 which represented an early modern corruption of an Italian abbreviation of per cento.
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.
The division slash ∕ is used between two numbers to indicate division. This use developed from the fraction slash in the late 18th or early 19th century. The formatting was advocated by De Morgan in the mid-19th century, who wrote:
The occurrence of fractions, such as a/b, a+b/c+d, in the 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.
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
S
/
R
{\displaystyle S/R}
(sometimes even as a built fraction), where the numerator
S
{\displaystyle S}
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
∼
{\displaystyle \sim }
on the original set
S
{\displaystyle S}
, and elements are to be identified in the quotient
S
/
∼
{\displaystyle S/{\sim }}
if they are equivalent according to
∼
{\displaystyle \sim }
; this is technically achieved by making
S
/
∼
{\displaystyle S/{\sim }}
the set of all equivalence classes of
∼
{\displaystyle \sim }
.
In group theory, the slash is used to mark quotient groups. The general form is
G
/
N
{\displaystyle G/N}
, where
G
{\displaystyle G}
is the original group and
N
{\displaystyle N}
is the normal subgroup; this is read "
G
{\displaystyle G}
mod
N
{\displaystyle N}
", where "mod" is short for "modulo". Formally this is a special case of quotient by an equivalence relation, where
g
∼
h
{\displaystyle g\sim h}
iff
g
=
h
n
{\displaystyle g=hn}
for some
n
∈
N
{\displaystyle n\in N}
. Since many algebraic structures (rings, vector spaces, etc.) in particular are groups, the same style of quotients extend also to these, although the denominator may need to satisfy additional 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).
When the original set is the set of integers
Z
{\displaystyle \mathbb {Z} }
, the denominator may alternatively be just an integer:
Z
/
n
{\displaystyle \mathbb {Z} /n}
. This is an alternative notation for the set
Z
n
{\displaystyle \mathbb {Z} _{n}}
of integers modulo n (needed because
Z
n
{\displaystyle \mathbb {Z} _{n}}
is also notation for the very different ring of n-adic integers).
Z
/
n
{\displaystyle \mathbb {Z} /n}
is an abbreviation of
Z
/
n
Z
{\displaystyle \mathbb {Z} /n\mathbb {Z} }
or
Z
/
(
n
)
{\displaystyle \mathbb {Z} /(n)}
, which both are ways of writing the set in question as a quotient of groups.
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 relation negates it, producing e.g. 'not equal'
≠
{\displaystyle \neq }
as negation of
=
{\displaystyle =}
or 'not in'
∉
{\displaystyle \notin }
as negation of 'element of'
∈
{\displaystyle \in }
; 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
∤
{\displaystyle \nmid }
of
∣
{\displaystyle \mid }
(divides) and negation
≁
{\displaystyle \nsim }
of
∼
{\displaystyle \sim }
(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 Dirac gamma matrices, so
A
/
=
γ
μ
A
μ
{\displaystyle A\!\!\!/=\gamma ^{\mu }A_{\mu }}
; what one gains is not only a more compact formula, but also not having to allocate a letter as the contracted index.