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The reciprocal function: y = 1/x.For every x except 0, y represents its multiplicative inverse. The graph forms a rectangular hyperbola.. In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x −1, is a number which when multiplied by x yields the multiplicative identity, 1.
A modular multiplicative inverse of a modulo m can be found by using the extended Euclidean algorithm. The Euclidean algorithm determines the greatest common divisor (gcd) of two integers, say a and m. If a has a multiplicative inverse modulo m, this gcd must be 1. The last of several equations produced by the algorithm may be solved for this gcd.
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.
A unit is an element a having a multiplicative inverse; in this case the inverse is unique, and is denoted by a –1. The set of units of a ring is a group under ring multiplication; this group is denoted by R × or R* or U(R).
If this is the case, then the matrix B is uniquely determined by A, and is called the (multiplicative) inverse of A, denoted by A −1. Matrix inversion is the process of finding the matrix which when multiplied by the original matrix gives the identity matrix. [2] Over a field, a square matrix that is not invertible is called singular or ...
Once we have defined multiplication for formal power series, we can define multiplicative inverses as follows. The multiplicative inverse of a formal power series A is a formal power series C such that AC = 1, provided that such a formal power series exists. It turns out that if A has a multiplicative inverse, it is unique, and we denote it by ...
The product and the multiplicative inverse of two roots of unity are also roots of unity. In fact, if x m = 1 and y n = 1, then (x −1) m = 1, and (xy) k = 1, where k is the least common multiple of m and n. Therefore, the roots of unity form an abelian group under multiplication. This group is the torsion subgroup of the circle group.
The multiplicative inverse of an element may be computed by using the extended Euclidean algorithm (see Extended Euclidean algorithm § Modular integers). Let F {\displaystyle F} be a finite field. For any element x {\displaystyle x} in F {\displaystyle F} and any integer n {\displaystyle n} , denote by n ⋅ x {\displaystyle n\cdot x} the sum ...