ADDITION

SUBTRACTION

MECHANICAL MATH ON THE ABACUS

END

A story sequence similar to the one following might be used to guide the solution process for mechanical addition on the abacus. There is a king who owns all the land around his castle. He lets the village farmers grow crops on his land, but they have to pay him with copper and silver coins, they get from the village merchants.

MECHANICAL ADDITION ON THE PYTHAGOREAN ABACUS

The farmers bring the coins to the count. One farmer brings six silvers and eight copper coins, and another farmer brings five silvers and seven copper coins. The count collects the copper coins, and puts them in stacks.

6 8+ 5 7

The stacks of copper coins are represented on abacus A. The top number is shown by beads on the left. The bottom number is shown by beads on the right Once the coins are collected, the count counts the copper coins. If there are more than nine copper coins, the king gets ten, and the count gets the remaining coins.

6 8+ 5 7

To get the kings ten coins take a number equal the beads remaining in the leftmost column (two) from the seven pushed to the right to add to the eight beads pushed to the left.

6 8+ 5 7

+2

-2

Since the king has decreed that one silver coin equals ten copper coins, he will exchange the ten copper coins for one silver coin; therefore take one of the two previously taken from the right six and push it a little rightward to represent the one silver. This arrangement of beads will leaves five copper coins or beads in the right most array and one above it representing ten.

6 8+ 5 7

+2

-2

Now the count collects the silver coins. Beads from the left side again show the top number. Beads from the right columns added to the one above the rightmost five show the bottom number. If there are more than nine silver coins, the king gets ten, and the count gets the remaining coins.

6 8+ 5 7 5

+2

-2

1

To get the kings ten coins take a number equal the beads remaining in the leftmost column of the triangle (four) from the six pushed to the right to add to the six beads pushed to the left.

6 8+ 5 7 5

+2

-2

1

+4

-4

Since the king has decreed that one gold coin equals ten silver coins, he will exchange the ten silver coins for one gold coin; therefore take one of the four previously taken from the right six and push it a little rightward to represent the one gold.

6 8+ 5 7 2 5

+2

-2

+4

-4

This array correctly represents the solution to the addition problem.

6 8+ 5 71 2 5

+2

-2

+4

-4

When a queen is rescued from an angry dragon it sets the villagers' crops and hamlets a fire. A brave Prince, the King and his knights drive the dragon from the kingdom. In the dragons lair they find bags of treasure. The Queen decides to give the treasure to the villagers who lost their crops and hamlets. Each village received as many silver and copper coins it needed to buy seed and straw. Each silver piece was counted as was counted as ten copper pieces.  

SUBTRACTION ON THE ABACUS

The first village master asked for seventeen copper coins for his village. The Queen chose a bag of treasure with three silver pieces but only five copper pieces. Abacus A shows the number of copper pieces in the bag to the right. There were enough silver pieces in the bag but not enough copper pieces.

3 5- 1 7

She took one silver piece from the bag, leaving two sliver pieces in the bag, and exchanged it for ten silver pieces.

3 5- 1 7

12

The abacus shows ten silver pieces pushed to the left. The queen now had enough to give the village master the seven copper pieces he needed. These seven pieces are taken from the ten.

3 5- 1 7

10+2

Now the remainder from the ten, three, are added to the right most array. Adding the remainder from the ten always gives the correct number of ones.

3 5- 1 7 5

10+2

3+

Now two beads are pushed over to show the silver pieces in the bag, representing the remaining tens.

3 5- 1 7 8

10+2

3 5- 1 7 8

10+2

After giving the village master the one silver piece he needed, the queen had eight copper pieces and one silver piece remaining in her bag,

3 5- 1 7 1 8

10+2

And now the solution is correctly represented.