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In mathematics, the additive inverse of an element x, denoted -x, [1] is the element that when added to x, yields the additive identity, 0 (zero). [2] In the most familiar cases, this is the number 0, but it can also refer to a more generalized zero element .
In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element x in the set, yields x.One of the most familiar additive identities is the number 0 from elementary mathematics, but additive identities occur in other mathematical structures where addition is defined, such as in groups and rings.
The inverse or multiplicative inverse (for avoiding confusion with additive inverses) of a unit x is denoted , or, when the multiplication is commutative, . The additive identity 0 is never a unit, except when the ring is the zero ring, which has 0 as its unique element.
An identity with respect to addition is called an additive identity (often denoted as 0) and an identity with respect to multiplication is called a multiplicative identity (often denoted as 1). [3] These need not be ordinary addition and multiplication—as the underlying operation could be rather arbitrary.
Additive and multiplicative identity: there exist two distinct elements 0 and 1 in F such that a + 0 = a and a ⋅ 1 = a. Additive inverses: for every a in F, there exists an element in F, denoted −a, called the additive inverse of a, such that a + (−a) = 0.
The additive identity is unique. The additive inverse of each element is unique. The multiplicative identity is unique. For any element x in a ring R, one has x0 = 0 = 0x (zero is an absorbing element with respect to multiplication) and (–1)x = –x.
The group axioms for identity and inverses may be "weakened" to assert only the existence of a left identity and left inverses. From these one-sided axioms, one can prove that the left identity is also a right identity and a left inverse is also a right inverse for the same element. Since they define exactly the same structures as groups ...
The base case b = 0 follows immediately from the identity element property (0 is an additive identity), which has been proved above: a + 0 = a = 0 + a. Next we will prove the base case b = 1, that 1 commutes with everything, i.e. for all natural numbers a, we have a + 1 = 1 + a.