Chapter 2 - Complement Addition

It has been estimated by experts that, for the average business, the total time spent in arithmetical computations breaks down to 70% addition, 5% subtraction, 20% multiplication, and 5% division.

These exact proportions may or may not hold in your particular business or profession. But chances are that they are not far wrong if you include all the number work you do.

So the obvious first job of becoming better at figures is to simplify by a very substantial margin that 70% of the time spent adding. What is simpler is, by nature, faster. Since adding is the single most-often-used process, it is worth spending a little extra effort at the beginning to learn a new approach that is guaranteed to make your work both easier and much, much speedier.

The approach you are about to learn is quite different from the one taught in any school. In fact, it has never even appeared in any of the books on the subject and is practically unknown in this country.

There is a reason for this. The reason is that the basis of this system is not part of our western civilization at all. The basis comes from Japan.

Back in 1946, an amusing story appeared in many American newspapers. The story said, incredibly, that in a contest in the Ernie Pyle Theatre in Tokyo the most expert electric calculator operator of General MacArthur's headquarters had been roundly defeated in a public match by—of all things—an abacus!

In a long series of problems, ranging from addition and subtraction of as many as fifty numbers with three to six digits each, through division and multiplication problems with up to twelve digits each, the electric calculator had gone down to resounding defeat. The winner was a "primitive" instrument of beads on rods.

An abacus is really nothing more than a recording, not a calculating, device. It is basically so simple and useful a machine that different forms of it were used in Rome, India, China, Japan, and many other countries. The varieties used have been very different indeed, some of them about as clumsy as they were useful, but in Japan the highest mathematical thinking was brought to bear on the problem. An entirely new, "streamlined" version called the soroban was developed within the last few decades.

The soroban still consists of beads on rods. This is basic to anything that can be called an abacus. But it has fewer beads on each rod than any other variety. Where some contemporary Chinese models still have as many as fifteen beads on each rod, the soroban has exactly nine.

The number nine rings a bell. It is the highest of all single-digit numbers . . the basis of our decimal (tens) counting system.

The Japanese mathematicians saw this fact. After thousands of years of using the device in their calculating, they sat down and realized that it was silly to record ten or more on any one rod, because that ten could be recorded on another rod with just one bead in precisely the same way that we record a ten on paper—with a one moved over one place to the left.

Actually, of course, the electric calculator in that Tokyo contest was not defeated by the abacus at all. The operator of the calculator was defeated by the operator of the abacus —a man trained in the Japanese system of soroban arithmetic, which is so much simpler and faster than ours that he could solve and record each step of a problem faster than the electric calculator operator could punch them into his keyboard.

The soroban operator was no number genius, incidentally. He was a champion operator, but (as he himself stated) no better than many other first-class operators. After all, the soroban is still the basic tool of Japanese arithmetic, which today is building an industrial complex producing the most sophisticated binoculars and cameras and advanced radios.

If today you want a number job in Japan, don't bother to learn how to operate an adding machine. Learn the soroban.

Soroban Theory

The soroban, or modern Japanese abacus, is useful to us here because it is a valuable tool for calculating in its own right and because in order to use it with such incredible efficiency and speed the Japanese had to develop the theory.

Three parts of this theory are especially useful and applicable to our technique of streamlined arithmetic:

1. Do each step one at a time, recording the results in the quickest and easiest way.

2. Work from left to right.

3. Never calculate over ten.

That last one is a surprise. It surprised me some years ago when I was researching the whole field of short-cut mathematics for a program I was editing and, remembering that story about the Tokyo contest, I did some research on modern soroban theory.

Never add over ten? The whole idea violates everything we learned in school and everything we think we know about numbers. At first sight, the method for doing so will look more complicated. We are tempted to dismiss the idea and go on to something else.

But it does make sense. It makes enough sense for a soroban operator to beat the pants off an electric calculator operator.

Never add over ten. It takes time to get used to this idea.
 
If you react as I did when I first read the theory and method, then applied it to streamlined math and found how well it worked, you will need several days to adjust to the concept. But use it anyway. Force yourself. At first it will take longer than the way you now do arithmetic, because you will be breaking old habits and building new ones: new ones you'll prize for the rest of your life. Soon, if you keep working at it, you will find that you can do problems far more quickly and accurately than you have ever done them before.

Never add over ten! What about 5 + 6? 8 + 3? 9 + 7? We will get to that very shortly. Before going into it, though, you should understand thoroughly why this system is so fast.

Even though you have already memorized the addition tables up to 9 + 9 or even more, you will gain tremendously if from now on you concentrate on just about half of them— the easier half, at that. Soon you will naturally, almost unavoidably, become almost twice as fast on the easier half you really use.

Combine this with an automatic-recording system for taking care of the tens, such as the soroban provides or the two techniques developed especially for this system, and your speed accelerates still further.

Look at the following table of all possible combinations of two digits. You will find that there are forty-five of them in all, from 1 + 1 to 9 + 9. Now notice that of the forty-five combinations, twenty add up to less than ten. Five add up to ten. Twenty add up to more than ten.
The twenty combinations that add up to more than ten, incidentally, are also the twenty hardest to remember quickly and the ones on which most of us stumble most often.

The table, incidentally, shows each pair only once. That is, 2 + 5 is shown in the "two" column but 5 + 2 is not shown at all; it is merely the same pair backwards.
 
The appearance of this table is not random. It could be set up in slightly different shapes, but the order and pattern of this particular arrangement are especially instructive. You will find it worthwhile to examine the pattern with some care. Note, among other things, the heavy concentration of pairs adding up to totals around ten, and how the possibilities taper off toward high and low totals.

In the system about to be explained, here is how we will handle the forty-five different combinations:

We use the twenty combinations adding up to less than ten just as we do now. They are the easiest ones. We use the five combinations that add up to ten (1 + 9, 2 + 8, 3 + 7, 4 + 6, and 5 + 5) even more than we do now, so we learn them extra well. We forget those twenty hardest combinations that add up to more than ten and learn the technique of complement addition.

Add With Complements

The basic rule for the new technique is this:
To "add" over ten, subtract the complement of the larger digit from the smaller digit—and record a ten.

First we need to learn what complements are. Then we will take up how to record tens. Both are easy.

What is a complement? A complement is simply the digit that, added to the digit you have, adds up to ten. You might say that a complement is that digit needed (in addition to the one you have) to complete a ten.

For example: The complement of 9 is 1, because 9 + 1 is ten. The complement of 8 is 2, because 8 + 2 is ten. The complement of 7 is 3, and the complement of 6 is 4, because 7 + 3 is ten and 6 + 4 is ten. Even in your sleep you would answer that the complement of 5 is 5.

Those are all the complements you ever have to remember in adding the longest column of figures. There are only five of them: five pairs, you will note, that add up to ten in the table of possible combinations.

Before learning how to add with complements, make doubly sure that you have the idea by looking at the following digits and giving their complements. Try to "read" the complement of each as you are beginning to "read" the answer to a simple problem:

7 6 9 5 8

The way you add with complements takes a bit of getting used to. But it is one of the most fascinating and fruitful approaches known to short-cut arithmetic. You "add" two digits that total more than ten by subtracting the complement of the larger digit from the smaller digit and recording a ten.

In order to add 6 + 7, you subtract the complement of 7(3) from 6, and record a ten. 6 — 3 gives 3. The recorded ten makes it 13.

Or to add 8 + 4, you subtract the complement of 8 (2) from 4 and record a ten. 4 — 2 gives 2. The recorded ten makes it 12.
 
It is useful to subtract the complement of the larger digit rather than the complement of the smaller. In this way you cut in half the number of complements you have to remember at this stage—though the other half of the complements are really only the same pairs of digits that add up to ten turned around. Just as 2 is the complement of 8, so is 8 the complement of 2.

Try it yourself, before going any further. Add 7 4- 9 by subtracting the complement of the larger digit from the smaller digit. The complement of 9 is — 7ˉ — is — Remember to record a ten, in ways you will learn very soon. So the answer is 16. I hope that is what you arrived at through the new method, even the first time. If not, then it hasn't become clear yet. Another reading of the last few pages is indicated.

Now add 3 + 8. Would you subtract the complement of 8 from 3? What is the complement of 8? Don't forget to record a ten.

Strange and complex as this system undoubtedly seems at the moment, it is really far faster. This is because you are working with only the easier half of the forty-five digit combinations, the half that add up to less than ten. Even subtracting the complement will shortly become no problem, because you are always subtracting digits from pairs in the top part of the table. Look back at it again for a moment. In the complement system of addition, you cannot possibly get into that bottom part of the table—those twenty toughest (and slowest) combinations.

Give it one more try before going on. Each time you use it, the system will become a little easier and more natural.

Add 6 + 5. The complement of 6 is — 5 minus — is — Record a ten.

Recording Tens

You recall that we said the soroban, or modern Japanese abacus, is not really a calculating instrument at all. It is a recording instrument. By recording the results of each step in a calculation, it frees the operator to concentrate on skill and speed in taking each step.
 
We can record steps, too. Our methods of recording will enable us to concentrate on speed, just as the soroban does.

There are two good ways to record tens each time you use complements. The first way is simply to put a line at each place in a column of figures whenever you use a complement or add to ten. If you adopt this system, make it a habit so it becomes automatic. Then, when you write your final total, you just sweep your eye over the lines in that column and put down the total number of lines as your "tens" digit, one place to the left. Instead of remembering "37," for instance, you have in your mind at that point only the single digit seven, but you will find three lines along the column.

We will go through one problem slowly and carefully, step by step. At first, the process will seem quite long and complicated because each step must be made clear. Actually, as you will find with use, it is far simpler and faster than the traditional method of addition.

Remember that we always work from left to right:
 
Now you simply add and get the answer, 237.

While this has taken some time to explain step by step, in practice you will find it infinitely faster than the old way. When you do it automatically, you will think only "5, 4 (line), 3 (line); 9, 8 (line), 7 (line); 237."

One element about the problem may be a little confusing. We combine the next figure in the column with the figure in our mind from previous additions, not with the figure above it. For instance, in the first column of the problem above, we subtract the complement of 9 (1) from 5—the result of adding 2 and 3—not from the 3. It works just like regular addition in this respect. The use of complements does not change it.

Try the next example, in which we will go through the steps in a much more condensed way. See if you can follow each step, identify the complement being used in each case, and understand why we record a ten with a line each time we do so:

This example should have gone a little more easily. Take it slowly now, so you can build on a solid base of thorough understanding in later parts of the book.

Rather than go on with more practice at this point, let us get into the second method of recording tens. Of the two, this is quicker and more generally useful. But, in this case and in many alternate choices in the "short cuts" section later in the book, you should adopt the one that seems most natural to you and concentrate on it. Continuous use of one system will build the desirable habit pattern and accelerate your speed.

Record on Your Fingers

The second way to record tens is to use your fingers. We were taught not to count on our fingers, so the idea may come as something of a shock. Actually, however, the purpose here is vastly different. We were taught not to count on our fingers because using them for counting is leaning on a crutch that interferes with genuine mastery of the calculating skill itself. Using them for recording, as you will see, approaches the automatic-recording advantages of the soroban, and frees you to concentrate on adding the digits with extra speed.

Should you need any more encouragement, take note of the fact that top abacus operators become amazingly proficient at mental arithmetic by learning to close their eyes and visualize the soroban as they calculate—and they use their fingers for recording. So no matter how much distaste for using your fingers your school training may have left you, keep firmly in mind that this is recording rather than counting, and give it a try. Speed mathematics can and should make use of any device that simplifies and speeds up the solving of problems.

Here is how the system works. To record the first ten (when you first use a complement or add to ten), fold the little finger of your left hand into the palm. If you write with your left hand, there is no reason why you cannot record on the right. To record the second ten, fold the next finger alongside the little finger. This means two tens. If you use another complement or add to ten in the same column, fold the next finger. This records three tens. And so on, up to five tens.

If you have more than five tens in a long column, open the hand and start over with the little finger again. Perhaps you will feel happier about remembering to add five to the second running total of tens if you make a line in the column when you start over. Or use any other signal to yourself that makes sense.

This is not silly. Any mechanical aid that fits your habits and personality is a valid and useful device for freeing your mind to concentrate on the basic objective: speed and ease with fingers.

Whatever signal you adopt in a case like this, be consistent with it. Settle down to use this method for every single calculation you do, no matter how simple it is or where you do it. Habits are very important. Making a habit of consistently using the fastest techniques is what gives speed.

The use of fingers instead of lines to record tens does not change what you do at the end of each column, of course. First you put down the digit in your mind from the final addition. Then you put, one place to the left, the number of fingers you have folded—adding five if you had to start over again.

Here is how we solve a problem with this system. Work from left to right:

First column

5          + 1 is 6.
6          + 3 is 9.
7          — 1 (complement of 9) is 6. Fold a finger.

Put down the 6 in your mind.
One finger folded. Put down 1 one place to the left.

Second column

                                                574
                                                198
                                                365
                                                744

                                                1661
                                                  22

                                                1881

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 modern 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:

Can you feel the identity of all three processes? We chose 9 as the demonstration example because it is so obviously 1 less than ten. Just as surely as 9 is 1 less than ten, 8 is 2 less than ten, 7 is 3 less than ten, and so on. The principle does not change one bit when we use these other combinations.

As one further example, let us show all three ways of expressing another "identity":

Take your pad at this point and work out the addition of 5 and 8 in all three ways. The closer you can come to "feeling" the identity of all three pictures of the same process, the more confidently you will handle complements.

One analogy that has proved helpful to some people is to visualize the process of adding as climbing a series of ladders, each with ten rungs, from level to level. At any point, you know your position on a ladder and you know on which ladder you stand. For instance, you are now standing on the sixth rung of the third ladder—an analogy of the number 36. You are told that you can advance eight more rungs, and wish the quickest and easiest way of projecting where you will be standing after eight rungs.
 
First, you know that you will be on the next higher ladder (in the 40's), because there are not eight more rungs above you on the third ladder. Adding 6 and 8, let us say, is something you have never been taught to do. You do know that if you could advance a full ten rungs you would be on the corresponding rung of the fourth ladder—46. But since 8 fails by 2 to complete a ten, you will be 2 rungs lower—44.

So, in any addition that crosses the next ten-point, you will fail to reach the corresponding number across that ten-point by precisely the amount that the number you add fails to reach ten. That is its complement.

Before going on to the next chapter, work for a few moments at making the use of complements a habit by using them conscientiously in adding the following problems. Use either lines or fingers as you prefer, but standardize now on one system or the other.

Do not add these pairs. In each case, subtract the complement of the larger digit from the smaller and record a ten. Just "see" the answer; don't write it down:
 
As you "read" these examples (and you should be trying to "read" rather than "solve" them) it may help to channel your thoughts in the right direction if you lip-read them the first few times. This is not good permanent practice, but it will help break your old habit patterns. You would lip-read the first problem, for example, as "5 - 3 is 2; finger," to help you avoid slipping back into the thought pattern of "5 + 7 is 12." Ultimately, you will try to "see" it as merely "2, finger."

The first key to speed in this system is obviously knowing your complements at sight, without pausing to think for a second. Review them quickly. Try to "read" the complement of each digit as you see it, without stopping to ponder:

8 7 9 6

These are the only digits for which you have to remember complements at this point. Five is the complement of five, but you never use it that way in adding because when faced with a five and a five you simply react "0, finger." When faced with a five and a larger digit, you use the complement of the larger digit.

What is the complement of 7?

If you had to pause for even a flicker, build your base for rapid progress later in the book by reading the above digits again. React without thought with the complements to these digits:

6 9 8 7
The sheer repetition here is not overdone. It is essential to mastering the new system. One of the two major approaches to teaching machines uses precisely this principle.

Go through this brief check-up to make sure you are ready for the next chapter, which will begin to build your speed and confidence in complement addition.

What is a complement?

What is the complement of 7?

When you add two digits that would go over ten do you add or subtract the complement of one of them?

Is it quicker to use the complement of the larger digit, or the smaller one? Why?

What is the complement of 6?

When adding a column, do you keep all the tens in your mind, or do you record them?

What is the complement of 8?

How do you record a ten?

What is the complement of 9?

In adding a column, do you combine each new digit ith the digit above it, or with the digit in your mind from above additions?

Could you explain to a friend why complements work as they do? Pretend he has just asked you, and see if you can.

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