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Some chips implement long multiplication, in hardware or in microcode, for various integer and floating-point word sizes. In arbitrary-precision arithmetic , it is common to use long multiplication with the base set to 2 w , where w is the number of bits in a word, for multiplying relatively small numbers.
The product is obtained by multiplying together all factors obtained by substituting the multiplication index for an integer between the lower and the upper values (the bounds included) in the expression that follows the product operator. More generally, the notation is defined as
The Schönhage–Strassen algorithm is based on the fast Fourier transform (FFT) method of integer multiplication. This figure demonstrates multiplying 1234 × 5678 = 7006652 using the simple FFT method. Base 10 is used in place of base 2 w for illustrative purposes.
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 ≡ ...
Karatsuba multiplication of az+b and cz+d (boxed), and 1234 and 567 with z=100. Magenta arrows denote multiplication, amber denotes addition, silver denotes subtraction and cyan denotes left shift. (A), (B) and (C) show recursion with z=10 to obtain intermediate values. The Karatsuba algorithm is a fast multiplication algorithm.
The multiplicative order of a number a modulo n is the order of a in the multiplicative group whose elements are the residues modulo n of the numbers coprime to n, and whose group operation is multiplication modulo n. This is the group of units of the ring Z n; it has φ(n) elements, φ being Euler's totient function, and is denoted as U(n) or ...
Graphs of functions commonly used in the analysis of algorithms, showing the number of operations versus input size for each function. The following tables list the computational complexity of various algorithms for common mathematical operations.
However, not every integer has a multiplicative inverse (as is the case of the number 2), which means that under multiplication is not a group. All the rules from the above property table (except for the last), when taken together, say that Z {\displaystyle \mathbb {Z} } together with addition and multiplication is a commutative ring with unity .
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