MULTIPLICATION

DIVISION

MECHANICAL MULTIPLICATION AND DIVISION ON THE

PYTHABACUS

The people of the kingdom of Clotheah are weavers of the finest cloth in many lands. With the fine cloth they make beautiful cloaks. The king appointed his daughter, Princess Clotheah, mistress of the sewing circles. One of her duties is to attend the great bazaar and buy thread to make the beautiful cloaks.

MECHANICAL MULTIPLICATION

At the bazaar, thread is sold on single spools and in wraps of ten spools. Each cloak requires two wraps of ten spools plus one single spool of thread or twenty-one spools altogether. Princess Clotheah must buy thread for one village sewing circle and three farm sewing circles. For the village sewing circle she must buy enough thread to make ten cloaks and for each of the farm circles enough for one cloak Therefore the princess must buy enough thread for thirteen cloaks. So she will need twenty-one times thirteen spools of thread.

The digits of twenty-one are represented by the base beads of a triangle pushed to the left on abacus B.

The digits of twenty-one or represented by the base beads of a triangle pushed to the left on abacus B.

2

2

To represent thirteen, on abacus C, one, then three base beads are pushed to the left.

2 1

To represent thirteen, on abacus C, one, then three base beads are pushed to the left.

X 1

1

Then all four are pushed back to the right on abacus D

1 3

After a few simple manipulations, the columns of beads in the middle will represent the product of 21 times 3 and the product of 21 times 10. First push the 1 of 21 to the right.

1

When you touch the bottom bead of the adjacent column stop. As the arrow shows push the bead you touched on the rod above to the right.

When you touch the bottom bead of the next column stop. As the arrow again shows push the bead you touched on the rod above to the right

Return to the 1 bead of 21 and push it back to its place.

The answer is revealed to be 273. From left to right you have two hundreds, then six plus one tens and three ones. Note how the six and one are automatically position such that the bottom bead of the six is directly above the one to indicate the same place value.

2 7 3

Now that we know how to array the beads we can follow the mechanical process. Princess Clotheah first purchased thread for the three farm circles. She bought one times three single spools and two time three wraps. On abacus E one column of the right columns is pushed to the right leaving two columns. These columns represent sixty-three spools of thread.

1 X 3

The distributed columns represent 21 times 3.Now distributing the remaining column as indicated above will represent 21 times 1.

2 1 X 3

So next Princess Clotheah purchased thread for the village circle. She bought one times ten single spools and two times ten wraps. On abacus F one base bead from the left column is pushed to the right leaving two beads. These beads represent 21 times 10 spools of thread. Complete the mechanical process by adding the place value columns.

2 7 3

MECHANICAL DIVISION

To demonstrate mechanical division I will show how to divide the product of the previous multiplication example, 273, by the example’s multiplicand, 21, so that the example’s multiplier, 13, will be the quotient here. The divisor 21 is represented by a triangle with three beads in its base pushed to the left. Therefore the dividend will be an array of columns to the right of the divisor oriented down right as indicated by the down rightward pointing arrow.

So to configure this array beads are counted down right beginning with the leftmost bead.Count out two beads from the first column of three and push them to the left to represent the two in the hundreds place of the dividend. One bead will remain in the column and is carried rightward to be part of the seven in the tens place. To complete representing the seven six more bead, two from each subsequent column, are counted out and pushed to the left. The bottom bead of the six positions directly above the one from the first column to make seven in the tens placed.

Three beads will remain, one from each of the three columns of which two were taken to make the array of six. These three beads represent the the three in the one’s place of the dividend.

The 27 tens are first divided by 21. Since the 2 hundreds and 1 ten are represented by a single downright column 27 divided by 21 equals 1. This array also reveals that 1 times 21, the divisor, is 21. The difference between 27 and 21 is given by the 6 tens above the 1 ten. Now bringing your 3 remaining beads over to represent the ones leaves 63 to be divided by 21. The 63 is represented by a partition of 3 downright columns, so 63 divided by 21 equals 3. Since there are no beads positioned above the 3 ones 3 times 21 must equal 63.