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A binary computer does exactly the same multiplication as decimal numbers do, but with binary numbers. In binary encoding each long number is multiplied by one digit (either 0 or 1), and that is much easier than in decimal, as the product by 0 or 1 is just 0 or the same number.
Booth's multiplication algorithm is a multiplication algorithm that multiplies two signed binary numbers in two's complement notation. The algorithm was invented by Andrew Donald Booth in 1950 while doing research on crystallography at Birkbeck College in Bloomsbury, London. [1] Booth's algorithm is of interest in the study of computer ...
It requires memorization of the multiplication table for single digits. This is the usual algorithm for multiplying larger numbers by hand in base 10. A person doing long multiplication on paper will write down all the products and then add them together; an abacus-user will sum the products as soon as each one is computed.
4 layer Wallace reduction of an 8x8 partial product matrix, using 15 half adders (two dots) and 38 full adders (three dots). The dots in each column are bits of equal weight. A Wallace multiplier is a hardware implementation of a binary multiplier, a digital circuit that multiplies two integers.
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.
4 layer Dadda reduction of an 8x8 partial product matrix, using 7 half adders (two dots) and 35 full adders (three dots). The dots in each column are bits of equal weight. Bits with lower weight are rightmost. The example in the adjacent image illustrates the reduction of an 8 × 8 multiplier, explained here.
The first step away from slide rules was the introduction of relatively inexpensive electronic desktop scientific calculators. These included the Wang Laboratories LOCI-2, [31] [32] introduced in 1965, which used logarithms for multiplication and division; and the Hewlett-Packard HP 9100A, introduced in 1968. [33]
At the end of a complete modular multiplication, the true binary result of the operation has to be evaluated and it is possible that an additional addition or subtraction of r will be needed as a result of the carries that are then discovered; but the cost of that extra step is small when amortized over the hundreds of shift-and-add steps that ...