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The stepped reckoner or Leibniz calculator was a ... which can hold 16 decimal ... in a procedure equivalent to the familiar long multiplication and long ...
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
The second most important decision is in the choice of the base of arithmetic, here ten. There are many considerations. The scratchpad variable d must be able to hold the result of a single-digit multiply plus the carry from the prior digit's multiply. In base ten, a sixteen-bit integer is certainly adequate as it allows up to 32767.
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 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 ...
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 ...
Using the multiplication tables embedded in the rods, multiplication can be reduced to addition operations and division to subtractions. Advanced use of the rods can extract square roots. Napier's bones are not the same as logarithms, with which Napier's name is also associated, but are based on dissected multiplication tables.
The arithmometer (French: arithmomètre) was the first digital mechanical calculator strong and reliable enough to be used daily in an office environment. This calculator could add and subtract two numbers directly and perform long multiplications and divisions effectively by using a movable accumulator for the result.