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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.
The group (/) is cyclic if and only if n is 1, 2, 4, p k or 2p k, where p is an odd prime and k > 0. For all other values of n the group is not cyclic. [1] [2] [3] This was first proved by Gauss. [4] This means that for these n:
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 inverse x ≡ a −1 (mod m) may be efficiently computed by solving Bézout's equation a x + m y = 1 for x, y, by using the Extended Euclidean algorithm. In particular, if p is a prime number, then a is coprime with p for every a such that 0 < a < p; thus a multiplicative inverse exists for all a that is not congruent to zero ...
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 ...
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 ...
The multiplicative inverse for an element a of a finite field can be calculated a number of different ways: By multiplying a by every number in the field until the product is one. This is a brute-force search. Since the nonzero elements of GF(p n) form a finite group with respect to multiplication, a p n −1 = 1 (for a ≠ 0), thus the inverse ...
While the notation f −1 (x) might be misunderstood, [1] (f(x)) −1 certainly denotes the multiplicative inverse of f(x) and has nothing to do with the inverse function of f. [6] The notation might be used for the inverse function to avoid ambiguity with the multiplicative inverse. [7]