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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 multiplicative inverse of a ...
The product logarithm Lambert W function plotted in the complex plane from −2 − 2i to 2 + 2i The graph of y = W(x) for real x < 6 and y > −4.The upper branch (blue) with y ≥ −1 is the graph of the function W 0 (principal branch), the lower branch (magenta) with y ≤ −1 is the graph of the function W −1.
In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo of an odd prime number p: its value at a (nonzero) quadratic residue mod p is 1 and at a non-quadratic residue (non-residue) is −1. Its value at zero is 0.
Also, programs can be written that pull information from the worksheet, perform some calculations, and report the results back to the worksheet. In the figure, the name sq is user-assigned, and the function sq is introduced using the Visual Basic editor supplied with Excel. Name Manager displays the spreadsheet definitions of named variables x & y.
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 Dirichlet convolution of two multiplicative functions is again multiplicative, and every not constantly zero multiplicative function has a Dirichlet inverse which is also multiplicative. In other words, multiplicative functions form a subgroup of the group of invertible elements of the Dirichlet ring.
The reciprocal function f(x) = x −1 where for every x except 0, f(x) represents its multiplicative inverse. Exponentiation of a non‐zero real number can be extended to negative integers, where raising a number to the power −1 has the same effect as taking its multiplicative inverse: x −1 = 1 / x .
The Möbius function () is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated Moebius) in 1832. [ i ] [ ii ] [ 2 ] It is ubiquitous in elementary and analytic number theory and most often appears as part of its namesake the Möbius inversion formula .