Search results
Results from the WOW.Com Content Network
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.
Integer multiplication respects the congruence classes, that is, a ≡ a' and b ≡ b' (mod n) implies ab ≡ a'b' (mod n). This implies that the multiplication is associative, commutative, and that the class of 1 is the unique multiplicative identity. Finally, given a, the multiplicative inverse of a modulo n is an integer x satisfying ax ≡ ...
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 Inverse Symbolic Calculator is an online number checker established July 18, 1995 by Peter Benjamin Borwein, Jonathan Michael Borwein and Simon Plouffe of the Canadian Centre for Experimental and Constructive Mathematics (Burnaby, Canada).
In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. The picture at the right illustrates the multiplication of (+) (+) = +. Because the real and imaginary part of 5 + 5i are equal, the argument of that number is 45 degrees, or π/4 (in radian).
This is a consequence of the fact that, because gcd(R, N) = 1, multiplication by R is an isomorphism on the additive group Z/NZ. For example, (7 + 15) mod 17 = 5, which in Montgomery form becomes (3 + 4) mod 17 = 7. Multiplication in Montgomery form, however, is seemingly more complicated.
Any involution is a bijection.. The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation (x ↦ −x), reciprocation (x ↦ 1/x), and complex conjugation (z ↦ z) in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the ...
The complexity of an elementary function is equivalent to that of its inverse, since all elementary functions are analytic and hence invertible by means of Newton's method. In particular, if either exp {\displaystyle \exp } or log {\displaystyle \log } in the complex domain can be computed with some complexity, then that complexity is ...