Editing Addition (section)

=== 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>.