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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 fraction a / b is b / a. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one ...
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.
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 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 ...
The multiplicative inverse (reciprocal) of the square root of two is a widely used constant, with the decimal value: [20] 0.70710 67811 86547 52440 08443 62104 84903 92848 35937 68847... It is often encountered in geometry and trigonometry because the unit vector, which makes a 45° angle with the axes in a plane, has the coordinates
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.). A formal power series is a special kind of formal series, of the form.
Reciprocal polynomials have several connections with their original polynomials, including: is not 0. p(x) = xnp∗(x−1). [2] α is a root of a polynomial p if and only if α−1 is a root of p∗. [4] If p(x) ≠ x then p is irreducible if and only if p∗ is irreducible. [5] p is primitive if and only if p∗ is primitive.
The mean reciprocal rank is a statistic measure for evaluating any process that produces a list of possible responses to a sample of queries, ordered by probability of correctness. The reciprocal rank of a query response is the multiplicative inverse of the rank of the first correct answer: 1 for first place, 1 ⁄ 2 for second place, 1 ⁄ 3 ...