Editing Addition (section)

== Generalizations ==
There are many binary operations that can be viewed as generalizations of the addition operation on the real numbers. The field of algebra is centrally concerned with such generalized operations, and they also appear in [[set theory]] and [[category theory]].

=== Algebra ===
{{main|Vector addition|Matrix addition|Modular arithmetic|Linear combination}}
In [[linear algebra]], a [[vector space]] is an algebraic structure that allows for adding any two [[coordinate vector|vectors]] and for scaling vectors. A familiar vector space is the set of all ordered pairs of real numbers; the ordered pair <math> (a,b) </math> is interpreted as a vector from the origin in the Euclidean plane to the point <math> (a,b) </math> in the plane. The sum of two vectors is obtained by adding their individual coordinates:
<math display="block"> (a,b) + (c,d) = (a+c,b+d). </math>
This addition operation is central to [[classical mechanics]], in which [[velocity|velocities]], [[acceleration]]s and [[force]]s are all represented by vectors.{{sfnp|Gbur|2011|p=1}}

Matrix addition is defined for two matrices of the same dimensions. The sum of two ''m'' × ''n'' (pronounced "m by n") matrices '''A''' and '''B''', denoted by {{nowrap|'''A''' + '''B'''}}, is again an {{nowrap|''m'' × ''n''}} matrix computed by adding corresponding elements:<ref>Lipschutz, S., & Lipson, M. (2001). Schaum's outline of theory and problems of linear algebra. Erlangga.</ref><ref>{{cite book |title=Mathematical methods for physics and engineering |url=https://archive.org/details/mathematicalmeth00rile |url-access=registration |first1=K.F. |last1=Riley |first2=M.P.|last2=Hobson |first3=S.J. |last3=Bence |publisher=Cambridge University Press |year=2010 |isbn=978-0-521-86153-3}}</ref>
<math display=block>\begin{align}
\mathbf{A}+\mathbf{B} &= 
\begin{bmatrix}
 a_{11} & a_{12} & \cdots & a_{1n} \\
 a_{21} & a_{22} & \cdots & a_{2n} \\
 \vdots & \vdots & \ddots & \vdots \\
 a_{m1} & a_{m2} & \cdots & a_{mn} \\
\end{bmatrix} +
\begin{bmatrix}
 b_{11} & b_{12} & \cdots & b_{1n} \\
 b_{21} & b_{22} & \cdots & b_{2n} \\
 \vdots & \vdots & \ddots & \vdots \\
 b_{m1} & b_{m2} & \cdots & b_{mn} \\
\end{bmatrix}\\[8mu]
 &= \begin{bmatrix}
 a_{11} + b_{11} & a_{12} + b_{12} & \cdots & a_{1n} + b_{1n} \\
 a_{21} + b_{21} & a_{22} + b_{22} & \cdots & a_{2n} + b_{2n} \\
 \vdots & \vdots & \ddots & \vdots \\
 a_{m1} + b_{m1} & a_{m2} + b_{m2} & \cdots & a_{mn} + b_{mn} \\
\end{bmatrix} \\

\end{align}</math>

For example:
: <math>
\begin{align}
  \begin{bmatrix}
    1 & 3 \\
    1 & 0 \\
    1 & 2
  \end{bmatrix}
+
  \begin{bmatrix}
    0 & 0 \\
    7 & 5 \\
    2 & 1
  \end{bmatrix}
&= \begin{bmatrix}
    1+0 & 3+0 \\
    1+7 & 0+5 \\
    1+2 & 2+1
  \end{bmatrix}\\[8mu]
&=
  \begin{bmatrix}
    1 & 3 \\
    8 & 5 \\
    3 & 3
  \end{bmatrix}
\end{align}
</math>

In [[modular arithmetic]], the set of available numbers is restricted to a finite subset of the integers, and addition "wraps around" when reaching a certain value, called the modulus. For example, the set of integers modulo 12 has twelve elements; it inherits an addition operation from the integers that is central to [[set theory (music)|musical set theory]]. The set of integers modulo&nbsp;2 has just two elements; the addition operation it inherits is known in [[Boolean logic]] as the "[[exclusive or]]" function. A similar "wrap around" operation arises in [[geometry]], where the sum of two [[angle|angle measures]] is often taken to be their sum as real numbers modulo&nbsp;2π. This amounts to an addition operation on the [[circle]], which in turn generalizes to addition operations on many-dimensional [[torus|tori]].

The general theory of abstract algebra allows an "addition" operation to be any [[associative]] and [[commutative]] operation on a set. Basic [[algebraic structure]]s with such an addition operation include [[commutative monoid]]s and [[abelian group]]s.

[[Linear combination]]s combine multiplication and summation; they are sums in which each term has a multiplier, usually a [[real numbers|real]] or [[complex numbers|complex]] number. Linear combinations are especially useful in contexts where straightforward addition would violate some normalization rule, such as [[mixed strategy|mixing]] of [[strategy (game theory)|strategies]] in [[game theory]] or [[quantum superposition|superposition]] of [[quantum state|states]] in [[quantum mechanics]].{{sfnp|Rieffel|Polak|2011|p=16}}

=== Set theory and category theory ===
A far-reaching generalization of the addition of natural numbers is the addition of [[ordinal number]]s and [[cardinal number]]s in set theory. These give two different generalizations of the addition of natural numbers to the [[transfinite number|transfinite]]. Unlike most addition operations, the addition of ordinal numbers is not commutative.{{sfnp|Cheng|2017|pp=124–132}} Addition of cardinal numbers, however, is a commutative operation closely related to the [[disjoint union]] operation.

In [[category theory]], disjoint union is seen as a particular case of the [[coproduct]] operation,{{sfnp|Riehl|2016|p=100}} and general coproducts are perhaps the most abstract of all the generalizations of addition. Some coproducts, such as [[direct sum]] and [[wedge sum]], are named to evoke their connection with addition.