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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]
A closely related idea in geometry is that of "inverting" a point. In the plane, the inverse of a point P with respect to a reference circle (Ø) with center O and radius r is a point P ', lying on the ray from O through P such that ′ =. This is called circle inversion or plane inversion.
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 manipulations of the Rubik's Cube form the Rubik's Cube group.. In mathematics, a group is a set with an operation that satisfies the following constraints: the operation is associative, it has an identity element, and every element of the set has an inverse element.
An alternative definition of the Minkowski difference is sometimes used for computing intersection of convex shapes. [3] This is not equivalent to the previous definition, and is not an inverse of the sum operation. Instead it replaces the vector addition of the Minkowski sum with a vector subtraction. If the two convex shapes intersect, the ...
Replacing i by −i, j by −j, and k by −k sends a vector to its additive inverse, so the additive inverse of a vector is the same as its conjugate as a quaternion. For this reason, conjugation is sometimes called the spatial inverse.
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 ...
V is the symmetry group of this cross: flipping it horizontally (a) or vertically (b) or both (ab) leaves it unchanged.A quarter-turn changes it. In two dimensions, the Klein four-group is the symmetry group of a rhombus and of rectangles that are not squares, the four elements being the identity, the vertical reflection, the horizontal reflection, and a 180° rotation.
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