Search results
Results From The WOW.Com Content Network
Modulo operations might be implemented such that a division with a remainder is calculated each time. For special cases, on some hardware, faster alternatives exist. For example, the modulo of powers of 2 can alternatively be expressed as a bitwise AND operation (assuming x is a positive integer, or using a non-truncating definition):
The logical operator XOR sums 2 bits, modulo 2. The use of long division to turn a fraction into a repeating decimal in any base b is equivalent to modular multiplication of b modulo the denominator. For example, for decimal, b = 10.
They do not behave like the integers 0 and 1, for which 1 + 1 = 2, but may be identified with the elements of the two-element field GF(2), that is, integer arithmetic modulo 2, for which 1 + 1 = 0. Addition and multiplication then play the Boolean roles of XOR (exclusive-or) and AND (conjunction), respectively, with disjunction x ∨ y ...
Modulo is a mathematical jargon that was introduced into mathematics in the book Disquisitiones Arithmeticae by Carl Friedrich Gauss in 1801. [3] Given the integers a, b and n, the expression "a ≡ b (mod n)", pronounced "a is congruent to b modulo n", means that a − b is an integer multiple of n, or equivalently, a and b both share the same remainder when divided by n.
This table does not describe the logic operations necessary to implement this operation, rather it simply specifies the function of inputs to output values. With respect to the result, this example may be arithmetically viewed as modulo 2 binary addition, and as logically equivalent to the exclusive-or (exclusive disjunction) binary logic ...
[1] [2] [3] An example is the ring of integers modulo 2. Every Boolean ring gives rise to a Boolean algebra , with ring multiplication corresponding to conjunction or meet ∧ , and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨ , [ 4 ] which would constitute a semiring ).
GF(2) can be identified with the field of the integers modulo 2, that is, the quotient ring of the ring of integers Z by the ideal 2Z of all even numbers: GF(2) = Z/2Z. Notations Z 2 and Z 2 {\displaystyle \mathbb {Z} _{2}} may be encountered although they can be confused with the notation of 2 -adic integers .
Implements the mathematical modulo operator. The returned result is always of the same sign as the modulus or nul, and its absolute value is lower than the absolute value of the modulus . However, this template returns 0 if the modulus is nul (this template should never return a division by zero error).