Editing Addition (section)

== Related operations ==
Addition, along with subtraction, multiplication, and division, is considered one of the basic operations and is used in [[elementary arithmetic]].

=== Arithmetic ===
[[Subtraction]] can be thought of as a kind of addition—that is, the addition of an [[additive inverse]]. Subtraction is itself a sort of inverse to addition, in that adding <math> x </math> and subtracting <math> x </math> are [[inverse function]]s.{{sfnp|Kay|2021|p=[https://books.google.com/books?id=aw81EAAAQBAJ&pg=PA44 44]}} Given a set with an addition operation, one cannot always define a corresponding subtraction operation on that set; the set of natural numbers is a simple example. On the other hand, a subtraction operation uniquely determines an addition operation, an additive inverse operation, and an additive identity; for this reason, an additive group can be described as a set that is closed under subtraction.<ref>The set still must be nonempty. {{harvtxt|Dummit|Foote|1999}}, p. 48 discuss this criterion written multiplicatively.</ref>

[[Multiplication]] can be thought of as [[Multiplication and repeated addition|repeated addition]]. If a single term {{mvar|x}} appears in a sum <math> n </math> times, then the sum is the product of <math> n </math> and {{mvar|x}}. Nonetheless, this works only for [[natural number]]s.{{sfnp|Musser|Peterson|Burger|2013|p=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA101 101]}} By the definition in general, multiplication is the operation between two numbers, called the multiplier and the multiplicand, that are combined into a single number called the product.

[[File:Csl.JPG|thumb|A circular slide rule]]
In the real and complex numbers, addition and multiplication can be interchanged by the [[exponential function]]:{{sfnp|Rudin|1976|p=178}}
<math display="block"> e^{a+b} = e^a e^b. </math>
This identity allows multiplication to be carried out by consulting a [[mathematical table|table]] of [[logarithm]]s and computing addition by hand; it also enables multiplication on a [[slide rule]]. The formula is still a good first-order approximation in the broad context of [[Lie group]]s, where it relates multiplication of infinitesimal group elements with addition of vectors in the associated [[Lie algebra]].{{sfnp|Lee|2003|p=526|loc=Proposition 20.9}}

There are even more generalizations of multiplication than addition.<ref>{{harvtxt|Linderholm|1971}}, p. 49 observes, "By ''multiplication'', properly speaking, a mathematician may mean practically anything. By ''addition'' he may mean a great variety of things, but not so great a variety as he will mean by 'multiplication'."</ref> In general, multiplication operations always [[distributivity|distribute]] over addition; this requirement is formalized in the definition of a [[ring (mathematics)|ring]]. In some contexts, integers, distributivity over addition, and the existence of a multiplicative identity are enough to determine the multiplication operation uniquely. The distributive property also provides information about the addition operation; by expanding the product <math> (1+1)(a+b) </math> in both ways, one concludes that addition is forced to be commutative. For this reason, ring addition is commutative in general.<ref>{{harvtxt|Dummit|Foote|1999}}, p. 224. For this argument to work, one must assume that addition is a group operation and that multiplication has an identity.</ref>

[[Division (mathematics)|Division]] is an arithmetic operation remotely related to addition. Since <math> a/b = ab^{-1} </math>, division is right distributive over addition: <math> (a+b)/c = a/c + b/c </math>.<ref>For an example of left and right distributivity, see {{harvtxt|Loday|2002}}, p. 15.</ref> However, division is not left distributive over addition, such as <math> 1/(2+2) </math> is not the same as <math> 1/2 + 1/2 </math>.

=== Ordering ===
[[File:XPlusOne.svg|right|thumb|[[Log-log plot]] of {{nowrap|1={{mvar|x}} + 1}} and {{nowrap|1=max ({{mvar|x}}, 1)}} from {{mvar|x}} = 0.001 to 1000<ref>Compare {{harvtxt|Viro|2001}}, p. 2, Figure 1.</ref>]]
The maximum operation <math> \max(a,b) </math> is a binary operation similar to addition. In fact, if two nonnegative numbers <math> a </math> and <math> b </math> are of different [[orders of magnitude]], their sum is approximately equal to their maximum. This approximation is extremely useful in the applications of mathematics, for example, in truncating [[Taylor series]]. However, it presents a perpetual difficulty in [[numerical analysis]], essentially since "max" is not invertible. If <math> b </math> is much greater than <math> a </math>, then a straightforward calculation of <math> (a + b) - b </math> can accumulate an unacceptable [[round-off error]], perhaps even returning zero. See also ''[[Loss of significance]]''.

The approximation becomes exact in a kind of infinite limit; if either <math> a </math> or <math> b </math> is an infinite [[cardinal number]], their cardinal sum is exactly equal to the greater of the two.<ref>Enderton calls this statement the "Absorption Law of Cardinal Arithmetic"; it depends on the comparability of cardinals and therefore on the [[Axiom of Choice]].</ref> Accordingly, there is no subtraction operation for infinite cardinals.{{sfnp|Enderton|1977|p=[http://books.google.com/books?id=JlR-Ehk35XkC&pg=PA164 164]}}

Maximization is commutative and associative, like addition. Furthermore, since addition preserves the ordering of real numbers, addition distributes over "max" in the same way that multiplication distributes over addition:
<math display="block"> a + \max(b,c) = \max(a+b,a+c).</math>
For these reasons, in [[tropical geometry]] one replaces multiplication with addition and addition with maximization. In this context, addition is called "tropical multiplication", maximization is called "tropical addition", and the tropical "additive identity" is [[extended real number line|negative infinity]].{{sfnp|Mikhalkin|2006|p=1}} Some authors prefer to replace addition with minimization; then the additive identity is positive infinity.{{sfnp|Akian|Bapat|Gaubert|2005|p=4}}

Tying these observations together, tropical addition is approximately related to regular addition through the [[logarithm]]:
<math display="block">\log(a+b) \approx \max(\log a, \log b),</math>
which becomes more accurate as the base of the logarithm increases.{{sfnp|Mikhalkin|2006|p=2}} The approximation can be made exact by extracting a constant <math> h </math>, named by analogy with the [[Planck constant]] from [[quantum mechanics]],{{sfnp|Litvinov|Maslov|Sobolevskii|1999|p=3}} and taking the "[[classical limit]]" as <math> h </math> tends to zero:
<math display="block">\max(a,b) = \lim_{h\to 0}h\log(e^{a/h}+e^{b/h}).</math>
In this sense, the maximum operation is a ''dequantized'' version of addition.{{sfnp|Viro|2001|p=4}}

=== Other ways to add ===
[[Convolution]] is used to add two independent [[random variable]]s defined by [[probability distribution|distribution functions]]. Its usual definition combines integration, subtraction, and multiplication.{{sfnp|Gbur|2011|p=300}} In general, convolution is useful as a kind of domain-side addition; by contrast, vector addition is a kind of range-side addition.