Speed Mathematics Simplified (3 page)

If any one technique in this entire book is worth more than the price of admission, I would be tempted to put the left-to-right methods of working first on the list. There are other valuable techniques, but the left-to-right methods are utterly unique.

The value of this approach to your number sense can only develop as you learn the methods that make it possible. The point to be made here is simply this: work at it. It is, as you learn to use it, as black-and-white a difference as thinking of the number 462 or approaching it as 2, 6, 4.

Convert to Simpler Forms

Most of us convert some of our figuring problems to simpler forms, when we can and when we notice that we can, without thinking very much about it.

You wouldn't give a second thought to wondering how much you had in terms of dollars if you found three 25¢ pieces in your hand. We call 25¢ a quarter because that is just what it is—a quarter of a dollar. In fact, if you take one out of your pocket right now you will find that it doesn't even say anything about cents. The official designation is “quarter dollar.”

Whether anybody has ever called your attention to it or not, you are thinking now in terms of aliquots. An important chapter comes later on the short cuts that aliquots make possible. The whole concept, once you get used to it, is merely an extension and refinement of your instinctive understanding that 75¢ is the same as ¾ of a dollar.

This is conversion to a simpler form.

Perhaps, too, you have noticed that you can more easily multiply 692 by 99, by subtracting one 692 from a hundred 692's (69,200 – 692) than by setting up the whole problem with a pencil and paper and going through the classical form, which would look like this:

Which is quicker and easier? Yet in doing the first you were merely using a basic and helpful form of the technique we call “round off and adjust.” It can apply to many more numbers than 99.

This, too, is conversion to a simpler form.

Or perhaps, in quickly trying to come up with an appropriate tip for a meal check where 15% is standard, you noted that you could mentally take one-tenth of the check and then add one-half of that number to the one-tenth. A five-dollar check, for instance, would call for a 75¢ tip. One tenth of five dollars (50¢) plus one half of 50¢ (25¢), gives 75¢ quickly and easily.

It is obviously more convenient to arrive at 75¢ this way than to try (mentally or on the edge of the check) to multiply in the classic manner:

Yet in doing this little trick, you are merely practicing a fairly simple form of the short-cut method called “breakdown.”

There are other useful forms of conversion, such as factoring and proportionate change. The application of these methods to number sense will become plain as you learn and begin to apply them.

The Four Steps to Number Sense

Here, for quick review, are the four steppingstones to number sense:

Practice seeing relationships

How does 5 relate to 10? 3 to 9?

See only the answer

Read 4 + 3 as 7—not as four plus three.

Work from left to right

27 is 27—not 7, 2.

Convert to simpler forms

25¢ is both 25¢ and a quarter of a dollar.

99 is 100 minus 1.

15 is 10 plus ½ of 10.

(And more conversions to come.)

Before going on to the first real “working” chapter of this book, get in practice for using it as well as reading it by trying to see only the answers to the following multiplications. Remember, 6 x 7 is 42—not six times seven:

2

COMPLEMENT ADDITION

I
T 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.