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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 second formula is then written as =. Many specific examples are given in the article on multiplicative functions. The theorem follows because ∗ is (commutative and) associative, and 1 ∗ μ = ε, where ε is the identity function for the Dirichlet convolution, taking values ε(1) = 1, ε(n) = 0 for all n > 1. Thus
In three-dimensional Euclidean space, these three planes represent solutions to linear equations, and their intersection represents the set of common solutions: in this case, a unique point. The blue line is the common solution to two of these equations. Linear algebra is the branch of mathematics concerning linear equations such as:
cos x−1 = cos(x)−1 = −(1−cos(x)) = −ver(x) or negative versine of x, the additive inverse (or negation) of an old trigonometric function; cos −1 y = cos −1 (y), sometimes interpreted as arccos(y) or arccosine of y, the compositional inverse of the trigonometric function cosine (see below for ambiguity)
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.
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 ...
An involution is a function f : X → X that, when applied twice, brings one back to the starting point. In mathematics, an involution, involutory function, or self-inverse function [1] is a function f that is its own inverse, f(f(x)) = x. for all x in the domain of f. [2] Equivalently, applying f twice produces the original value.
For example, the sine of angle θ is defined as being the length of the opposite side divided by the length of the hypotenuse. The six trigonometric functions are defined for every real number , except, for some of them, for angles that differ from 0 by a multiple of the right angle (90°).