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In modular arithmetic, the modular multiplicative inverse of a is also defined: it is the number x such that ax ≡ 1 (mod n). This multiplicative inverse exists if and only if a and n are coprime. For example, the inverse of 3 modulo 11 is 4 because 4 ⋅ 3 ≡ 1 (mod 11). The extended Euclidean algorithm may be used to compute it.
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 ...
Every power of one equals: 1 n = 1. ... 3, 5 Power functions for n = 2, 4, 6. ... is defined only if x has a multiplicative inverse. ...
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.
[3] [4] Quaternions are ... multiplication, and multiplicative inverse are ... The fourth power of the norm of a quaternion is the determinant of the corresponding ...
The set {3,19} generates the group, which means that every element of (/) is of the form 3 a × 19 b (where a is 0, 1, 2, or 3, because the element 3 has order 4, and similarly b is 0 or 1, because the element 19 has order 2).
7.3 Power series. 7.4 Noncommutative ... r is the multiplicative inverse of q ... possibly raised to a nonnegative power. As usual, exponents equal to one and factors ...
4.3 Exponents. 4.4 Other ... are inverse operations—much like the same way multiplication and division are inverse operations, and addition and subtraction are ...