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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]
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.
Every ring is an abelian group with respect to its addition operation. In a commutative ring the invertible elements, or units, form an abelian multiplicative group. In particular, the real numbers are an abelian group under addition, and the nonzero real numbers are an abelian group under multiplication.
Under addition, a ring is an abelian group, which means that addition is commutative and associative; it has an identity, called the additive identity, and denoted 0; and every element x has an inverse, called its additive inverse and denoted −x. Because of commutativity, the concepts of left and right inverses are meaningless since they do ...
This includes the existence of an additive inverse −a for all elements a and of a multiplicative inverse b −1 for every nonzero element b. This allows the definition of the so-called inverse operations, subtraction a − b and division a / b, as a − b = a + (−b) and a / b = a ⋅ b −1. Often the product a ⋅ b is represented by ...
The additive group of a ring is the underlying set equipped with only the operation of addition. Although the definition requires that the additive group be abelian, this can be inferred from the other ring axioms. [4] The proof makes use of the "1", and does not work in a rng.
The notation a < b < c stands for "a < b and b < c", from which, by the transitivity property above, it also follows that a < c. By the above laws, one can add or subtract the same number to all three terms, or multiply or divide all three terms by same nonzero number and reverse all inequalities if that number is negative.
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 .