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Addition or subtraction is performed in a single step, with a turn of the crank. Multiplication and division are performed digit by digit on the multiplier or divisor digits, in a procedure equivalent to the familiar long multiplication and long division procedures taught in school.
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
First step of solving 825 × 913. To multiply by a multi-digit number, multiple rows are reviewed. For this example, the rows for 9, 1, and 3 have been removed from the board for clarity. Second step of solving 825 × 913. Each row is evaluated individually and each diagonal column is added as explained in the previous examples.
Add half of its neighbor to the right (dropping decimals, if any). The neighbor of the units position is 0. If the base-digit is even add 0 otherwise add 5. Add in any carryover from the previous step. Example: 693 × 7 = 4,851 Working from right to left: (3×2) + 0 + 5 + 0 = 11 = carryover 1, result 1. (9×2) + 1 + 5 + 1 = 25 = carryover 2 ...
The lattice technique can also be used to multiply decimal fractions. For example, to multiply 5.8 by 2.13, the process is the same as to multiply 58 by 213 as described in the preceding section. To find the position of the decimal point in the final answer, one can draw a vertical line from the decimal point in 5.8, and a horizontal line from ...
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.
A common technique for multiplication with larger numbers is called long multiplication. This method starts by writing the multiplier above the multiplicand. The calculation begins by multiplying the multiplier only with the rightmost digit of the multiplicand and writing the result below, starting in the rightmost column.
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 ]