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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]
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.
The multiplicative identity 1 and its additive inverse −1 are always units. More generally, any root of unity in a ring R is a unit: if r n = 1, then r n−1 is a multiplicative inverse of r. In a nonzero ring, the element 0 is not a unit, so R × is not closed under addition.
Inverse elements of vector addition: For every v ∈ V, there exists an element −v ∈ V, called the additive inverse of v, such that v + (−v) = 0. Compatibility of scalar multiplication with field multiplication: a(bv) = (ab)v [nb 3] Identity element of scalar multiplication: 1v = v, where 1 denotes the multiplicative identity in F.
The additive inverse of a number is unique, as is shown by the following proof. 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.
The nimber square of a Fermat 2-power x is equal to 3x/2 as evaluated under the ordinary multiplication of natural numbers. The smallest algebraically closed field of nimbers is the set of nimbers less than the ordinal ω ω ω , where ω is the smallest infinite ordinal.
Informally, a field is a set, along with two operations defined on that set: an addition operation written as a + b, and a multiplication operation written as a ⋅ b, both of which behave similarly as they behave for rational numbers and real numbers, including the existence of an additive inverse −a for all elements a, and of a multiplicative inverse b −1 for every nonzero element b.
This process can be reversed if 2 is an invertible element of R: [b] if b is an involution, then 2 −1 (1 − b) and 2 −1 (1 + b) are orthogonal idempotents, corresponding to a and 1 − a. Thus for a ring in which 2 is invertible, the idempotent elements correspond to involutions in a one-to-one manner.