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In a vector space, the additive inverse −v (often called the opposite vector of v) has the same magnitude as v and but the opposite direction. [11] In modular arithmetic, the modular additive inverse of x is the number a such that a + x ≡ 0 (mod n) and always exists. For example, the inverse of 3 modulo 11 is 8, as 3 + 8 ≡ 0 (mod 11). [12]
For example, in the 2×2 matrices over the integers the additive identity is = [] In the quaternions, 0 is the additive identity. In the ring of functions from , the function mapping every number to 0 is the additive identity. In the additive group of vectors in , the origin or zero vector is the additive identity.
The axioms of modules imply that (−1)x = −x, where the first minus denotes the additive inverse in the ring and the second minus the additive inverse in the module. Using this and denoting repeated addition by a multiplication by a positive integer allows identifying abelian groups with modules over the ring of integers.
[1] [2] For example, 0 is an identity element of the addition of real numbers. This concept is used in algebraic structures such as groups and rings . The term identity element is often shortened to identity (as in the case of additive identity and multiplicative identity) [ 3 ] when there is no possibility of confusion, but the identity ...
The negations or additive inverses of the positive natural numbers are referred to as negative integers. [2] The set of all integers is often denoted by the boldface Z or blackboard bold Z {\displaystyle \mathbb {Z} } .
Modular addition, defined in this way for the integers from to , forms a group, denoted as or (/, +) , with as the identity element and as the inverse element of . A familiar example is addition of hours on the face of a clock , where 12 rather than 0 is chosen as the representative of the identity.
In mathematics, −1 (negative one or minus one) is the additive inverse of 1, that is, the number that when added to 1 gives the additive identity element, 0. It is the negative integer greater than negative two (−2) and less than 0 .
For example, addition is a total associative operation on nonnegative integers, which has 0 as additive identity, and 0 is the only element that has an additive inverse. This lack of inverses is the main motivation for extending the natural numbers into the integers.