domingo, 3 de dezembro de 2017


DOING ARITHMETIC USING AN ABACUS. Fredric M. Menger, Emory University, Atlanta, GA. USA.

The abacus is an ingenious device for doing arithmetic that has been used for thousands of years in China where it was invented. Shown below is a Japanese-style abacus. It is separated into columns representing, from right to left, ones, tens, hundreds, thousands etc. The single buttons above the horizontal partition represent, from right to left, 5, 50, 500, 5000 etc. The four buttons below the horizontal are each the ones, tens, hundreds, thousands etc. One obtains a desired value by sliding the lower buttons up, and the upper button down, until the partition has been reached. Below the abacus picture is a schematic of the number 654,321 with the dotted line being the partition.

To show how an addition is carried out, let us use a 3-column abacus to add 233 to 548. The first step is to set the abacus to 548:

To add 233 to 548, begin with the left-hand column (the hundreds place) where one wants 700 (namely 2OO + 5OO). Thus, two “1OO’s” are elevated from the bottom with the single “500” remaining where it is. Then on the middle column (the tens place) one must create a 7O (namely 3O + 4O) by moving the “50” and two of the “10’s” down. The two maneuvers on the left-hand and middle columns have up till this point formed the arrangement shown below.

The next step in adding 233 to 548 is to accommodate the 3 + 8 = 11 addition in the right-hand column. Since an 11 is bigger than the maximum of 9 capacity of the column, one must first elevate a single 1O button in the middle column. It remains only to convert the right-hand column into the needed additional 1. The result of all these manipulations is shown below:

The abacus has correctly calculated that 233 + 548 = 781. Subtraction, multiplication, and division can also be handled by an abacus. (Please go on-line if this interests you). Note that fractions are also possible. The same procedure is used for fractions, such as 123.4, except that the right-hand column now refers to the 0.1 - 0.9 numbers, whereas it is now the second column to the right that gives the 1-9 numbers.

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