Speed Mathematics Simplified (5 page)

     
One finger folded. Put down 1 one place to the left.

Second column

     
7 – 1 (complement of 9) is 6. Fold a finger.

     
6 – 4 (complement of 6) is 2. Fold a finger.

     
2 + 4 is 6.

     
Put down the 6 in your mind.

     
Two fingers folded. Put down 2 one place to the left, under the 6 from the first column.

Third column

     
4 – 2 (complement of 8) is 2. Fold a finger.

     
2 + 5 is 7.

     
4 – 3 (complement of 7) is 1. Fold a finger.

     
Put down the 1 in your mind.

     
Two fingers folded. Put down 2 one place to the left, under the 6 from the second column.

Note especially that, because it is a faster habit to use the complement of the larger of the two digits to be added at any point (one being in your mind from the last addition, the other being the next digit in the column), sometimes you use the complement of the digit in your mind, and sometimes the complement of the next digit in the column. It makes no difference.

Now we will go through another example with a condensed explanation of the process:?

First column

     
1 (finger), 0 (finger), 3, 8. 8 under the column, 2 one place to the left.

Second column

     
0 (finger), 8, 4 (finger), 1 (finger). 1 under the

     
column, 3 one place to the left (below the 8).

Third column

     
8, 2 (finger), 3, 2 (finger). 2 under the column, 2 one place to the left (below the 1).

This example demonstrates one new fact. In developing the final answer you sometimes have to raise a digit you have already put down. In the problem above, the 2 in the very left column becomes 3 in the final answer of 3132. Since you are adding just two lines at this point, it should not be a problem. When we get into multiplication, where it can be a little harder, you will learn a special recording technique that makes it possible to work from left to right with quite complex problems in this way. But in adding you never have to add more than two lines, and no digit in the final answer ever needs to be raised in value by more than one. You should be able to work from left to right by merely glancing at the next column as you put down each digit to see if the total of the next column will be ten or more. If it will be (you don't care how much more than ten it will be at this point), just add one to the digit you are about to put down.

In the problem above, you glance at the second column and note that 8 + 3 will be more than ten. So instead of putting down 2 as the first digit, you put down 3. In a sense you are pre-recording a ten from the complement you will use when you get to the second column. For the second digit of the final answer, you subtract the complement of 8 (2) from 3 and put down 1. The ten has already been recorded by raising the first 2 to 3.

Why Complements Work

The use of complements is at the very heart and center of modem abacus theory in Japan, where today the soroban rather than the adding machine stands on the average bookkeeper's desk.

You don't have to understand the theory of complement addition to use it, but understanding always helps mastery. Learning simply by rote leads to a shaky mastery at best—to what W. W. Sawyer calls “imitation” instead of substance. So let us take apart the theory of complements and see why they work the way they do.

Since our counting base is ten, any addition is really a process of going up to ten and then
starting over again
—recording a ten by remembering “xxteen,” “twenty-xx,” and so on; or, with our new system, by using a line or a folded finger.

When we add two digits that would go over ten in complement addition, we really do just what a soroban operator does when he has to add some beads to a rod and finds that there are not enough beads on the rod. The streamlined abacus, or soroban, has only five beads on a rod: one representing a value of five, and four each representing a value of one. Altogether, they can record no more than nine.

Suppose the operator has recorded eight on one rod. Beads are moved toward the center divider in order to record, and a total of eight on one rod would look like this:

The five-bead is the one above the separator. It is moved to the center in order to record a five. Three one-beads have been moved toward the separator. This rod is recording the number eight—five plus three.

Now suppose the operator has to add nine to this number. He can't. There is only one bead not recording (the one on the bottom) and that would add only one. How can he add nine?

This is where modern abacus theory took over in Japan. Mathematicians developed the approach that the operator should never
try
to add more beads than he can find on the rod—even in his head, which was the way it had been done before. Instead, he should
subtract
the
complement
of the new digit, and record a ten on the rod to the left.

So, in order to add nine to the eight recorded above, the operator—knowing his complements cold, as he must—merely flicks one bead
away
from the separator and immediately flicks one bead on the rod to the left toward the separator to record the ten.

After he subtracts one (complement of nine) from this rod and adds one (ten) on the rod to the left, the answer looks like this:

Simple? Yes. But very subtle, and very revolutionary to our ways of doing arithmetic. The answer on these two rods is 17; one ten plus one five plus two ones. But it was produced
without
ever adding eight plus nine. It was produced by subtracting the complement of nine (one) from eight and recording a ten.

Soroban teaching calls this “letting the answer form naturally on the board.” What we are learning to do, in our mental adaptation of soroban theory, is let the answer form naturally in our mind.

Let us go a little more deeply into the theory of complements, in order to reinforce still further your “number sense” in using them.

Remember that each time we add beyond ten we start over again with one—11, 21, etc. Since using an addition table reaching beyond the next ten only compounds the number of possible combinations we must memorize and handle with ease, the use of complements enables us to deal only in combinations of ten or less and yet run through the entire counting system.

Take an example for which we would not normally use the complement system. You can add ten and nine in either of two ways:

This is very easy to understand at sight. 9 is 1 less than ten, so we can just as well add ten and subtract 1 as add 9. This is true no matter to what other digit we add it:

This works because, as you already know, 1 is the complement of 9. Working out each step of the theory, the complement approach may appear more complicated. Working out the addition of ten is what makes it appear to be so; we never bother to add ten as such, because we can simply record it.

Done in this fashion, the two above examples now look like this: