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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.
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. [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.
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.
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 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} } .
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 .
As mentioned above, an additive inverse of a number is defined as a value which when added to the number yields zero. Let x be a number and let y be its additive inverse. Suppose y′ is another additive inverse of x. By definition, + ′ =, + = And so, x + y′ = x + y.