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The stepped reckoner or Leibniz calculator was a mechanical calculator invented by the German mathematician Gottfried Wilhelm Leibniz (started in 1673, when he presented a wooden model to the Royal Society of London [2] and completed in 1694). [1]
5 is halved (2.5) and 6 is doubled (12). The fractional portion is discarded (2.5 becomes 2). The figure in the left column (2) is even, so the figure in the right column (12) is discarded. 2 is halved (1) and 12 is doubled (24). All not-scratched-out values are summed: 3 + 6 + 24 = 33. The method works because multiplication is distributive, so:
The basic algorithm, as best exemplified in a 2 • 2 multiplication problem, is as follows (where t represent a tens digit, u represent a units digit, and T symbolizes the tens digit multiplied by its respective power): Step 1 = (t 1 - u 1) • u 2 = A Step 2 = u 1 • t 2 = B Step 3 = (u 1 • u 2) + t (u 1 • u 2) = C
Take half of the neighbor, then, if the current digit is odd, add 5. Example: 42×5=210 Half of 2's neighbor, the trailing zero, is 0. Half of 4's neighbor is 1. Half of the leading zero's neighbor is 2. 43×5 = 215 Half of 3's neighbor is 0, plus 5 because 3 is odd, is 5. Half of 4's neighbor is 1. Half of the leading zero's neighbor is 2. 93 ...
The Wallace tree is a variant of long multiplication.The first step is to multiply each digit (each bit) of one factor by each digit of the other. Each of these partial products has weight equal to the product of its factors.
In elementary arithmetic, a standard algorithm or method is a specific method of computation which is conventionally taught for solving particular mathematical problems. . These methods vary somewhat by nation and time, but generally include exchanging, regrouping, long division, and long multiplication using a standard notation, and standard formulas for average, area, and vol
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 the decimal point in 2.13. (See picture for Step 4.)
The 2 on the top scale is placed over the 5.5 on the bottom scale. The resulting quotient, 2.75 , can then be read below the top scale's 1 : There is more than one method for doing division, and the method presented here has the advantage that the final result cannot be off-scale, because one has a choice of using the 1 at either end.