Addition
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 x and subtracting x are inverse functions.

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.

Multiplication can be thought of as repeated addition. If a single term x appears in a sum n times, then the sum is the product of n and x. If n is not a natural number, the product may still make sense; for example, multiplication by −1 yields the additive inverse of a number.

In the real and complex numbers, addition and multiplication can be interchanged by the exponential function:

This identity allows multiplication to be carried out by consulting a table of logarithms 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 groups, where it relates multiplication of infinitesimal group elements with addition of vectors in the associated Lie algebra.

There are even more generalizations of multiplication than addition. In general, multiplication operations always distribute over addition; this requirement is formalized in the definition of a ring. In some contexts, such as the integers, distributivity over addition and the existence of a multiplicative identity is enough to uniquely determine the multiplication operation. The distributive property also provides information about addition; by expanding the product (1 + 1)(a + b) in both ways, one concludes that addition is forced to be commutative. For this reason, ring addition is commutative in general.

Division is an arithmetic operation remotely related to addition. Since a / b = a (b), division is right distributive over addition: (a + b) / c = a / c + b / c. However, division is not left distributive over addition; 1 / (2 + 2) is not the same as 1/2 + 1/2.
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