Chapter 5 - Building Speed In Subtraction

Certain parts of this book may seem repetitious.

This is intentional. Repeating the basic points is the easiest and most painless form of review. Doing one essential exercise over several times—but not over and over in succession—is the most effective way to build the automatic response that is the foundation of high-speed mathematics.

Read the following line as if it were a sentence of words. But instead of words, read the complements of these digits:

4 7 2 8 5 1 9 6 5

Before going on to some necessary practice in complement subtraction, reinforce your understanding of the principle at work by describing in words completely different from any used in this book precisely what a complement is.

Now explain to yourself, as if you had never heard of the idea before, how you can subtract 7 from 12 by adding 3 plus 2—and doing something else in the answer. It might be a good idea to set up, on your pad, the three expressions 12 - 7, 12 - 10 + 3, and 12 + 3 (cancel).

Your speed and ease with numbers will depend not only on how easily and automatically you "sense" these new techniques, but also on how easily and automatically you see only the answer to any digit combination. We will now go through the basic vocabulary of subtraction. It will not take very long, because the combinations are really the same ones you have already practiced for addition. They are all pairs you recognize at sight, but in this case one-half of each pair and the addition-answer are given, and you must respond with the missing number. 3 + 5 is a pair you should be starting to read at sight as "8" instead of "3 + 5 is 8"; the same pair will show up here as 8 - 5 (see 3) and 8 - 3 (see 5).

Work for speed with these combinations. School yourself to think not about the digits you see, but only the answer. Where the bottom digit is smaller than the top digit, see only the answer. Where the bottom digit is larger than the top, work at seeing the result of adding the complement of the bottom digit to the top digit, and mentally slash an imaginary digit to the left in the answer. Therefore you "see" 6 - 7 as "3 + 6 (9)—slash."

Just as in the practice tables in adding, every possible digit combination has been included in these sections. If you learn to read the answers to these without effort, you know you will never handle a single combination that you did not have a chance to practice.

See how quickly and automatically you can subtract these:

A certain amount of your speed at handling these must be pure habit, of course. There is no way to avoid developing the "automatic response" that only practice can bring. But the number sense at which you worked in Chapter II will be a substantial help here. The better you can visualize the relationships of numbers, the more quickly you will develop astonishing mastery of basic mathematical figuring.

Remember, too, that while this practice series shows 94 pairs (the 45 pairs, the 45 pairs upside down, plus the four hardest pairs repeated just to make it come out even), you need only be concerned with those twenty easiest combinations, plus the five complement pairs. Looked at this way, it should certainly be a reasonable task to master fully and automatically. What you are really doing, in effect, is learning to recognize those twenty easiest pairs whether they show up in simple smaller-from-larger form or disguised in complement applications.

Use your complements faithfully, and you will never deal with any combinations adding to more than ten—or subtract a larger digit from a smaller.

This finishes up all the possibilities:

 
That covers everything: 94 expressions of only twenty combinations, plus five complement pairs. Every problem you will ever face contains only these basic combinations, arranged in a different order. The only extra complication is your remembering to slash the answer-digit to the left whenever you use a complement. Even that is a far simpler system than trying to remember to "borrow."

Try this example. Be sure to use your pad:

A long subtraction, indeed. Yet you do it, step by step, in precisely the same way you would do your scales.

Use a clean page of your pad now and go through it once more. Compare the two answers. If they are not the same, you had better do it once again. Speed mathematics is useful only if it is also accurate.

Do these problems now, to help build your habits:

If you got an answer of any kind to that last one, take another look at it and bring your "number sense" to bear. There cannot be an answer, other than a minus one. It was put there to make sure you practice the reality, not an imitation.

Now do these:

Each of these examples illustrates some variation of the pattern your left-to-right subtraction will form. Some of them require carrying back a slash to one or more preceding digits in the answer. Others may momentarily surprise you because they do not require the use of complements at all, and you will find no slashes whatsoever in your answer.

Do another group now:

The system works just as well, naturally, with dollars and cents. You can slash across a decimal point without hesitation, because as you move left each digit becomes ten times as important whether or not a decimal point appears between two digits. All the decimal point does is break the number into a whole quantity and a fraction. The digits retain precisely the same relative value right across the decimal point: ten times in value for each place a digit moves to the left.

In order to make sure that a decimal does not slow you up in your handling of canceled tens, work through these with your pad:

If you are rewriting your answers with each slashed digit reduced in value by one, then you have already had some good practice at reading such answers directly, without bothering to rewrite them.

If you read through that like an expert, see if you can tell what is wrong with this answer:

I would urge you not to skip over this. Unless one glaring error caught your eye in that answer, you would do well to

Prove this to yourself by seeing if you can read the answer below as you would rewrite it, without pausing to figure out what each slashed digit should represent:review the last chapter on the subject of carried-back slashes. This is important—and will become even more important as you apply some of the elements learned so far to future sections of this book.

What Have You Learned

Unless you are unusually at home with numbers, or have a natural liking for them (which few of us do, although new mastery of any subject often brings enjoyment with it), now is a good time to pause and make sure everything covered so far is solidly entrenched.

You will profit most from this book if you take it in easy stages. Whenever a point seems a little difficult to understand on one reading, go back and reread it once or twice. That same point is very probably one that will crop up again as something you will be expected to know thoroughly in new applications for multiplying and dividing. Take a pencil and your pad and doodle with the obscure point for a bit. See if you can set up different expressions of it, as we did for addition and subtraction involving complements. The idea is to visualize it as clearly as you can. In this way, you will understand the why as well as the how.

If you truly understand the why, I promise that you will never forget the how. Even if you did, you could easily reconstruct it—because you know why it works.

The next chapter will take up another major area of basic mathematics: multiplying. It is a fascinating and quite new approach, but do not tackle it until you feel completely comfortable with the complement, left-to-right methods of adding and subtracting. Between them, they account for 75% of the arithmetic used in the average business.

A final re-check would be in order now, to make sure your base is really solid.

First, find your own words to describe exactly what a complement is and how it works in adding. If you have trouble putting the theory into words, then set up the three expressions for adding 6 plus 9 on your pad in the same way we did before. Then do the same thing for subtracting 7 from 13.
 
Once you have lived a little longer with the idea of complements, they will seem to be the most natural and useful devices in the world. They are basic to the structure of our ten-based counting system. Yet, oddly enough, nobody had ever formalized their use for arithmetic until the Japanese found how much they simplified calculation on the abacus.

Skip ahead now to that last secret of extra speed in adding called grouping. Practice the technique briefly once again by grouping the following pairs at a glance as if they were together in a column you were adding and you wished to handle each pair as a single digit:

You are well on your way to mastery if your only reaction to that third group was "nothing, fold."

One more reading of the complements is now indicated. See the complements to the following digits as quickly and automatically as you can:

3 6 1 5 8 4 9 5 2 7

This brush-up on your basic vocabulary is not casual. It provides one more opportunity to drive the new habits a little more deeply into your mind, as well as to refresh your understanding of the principles at work.

Add the following example from left to right, using either the finger or line method of recording tens. Make sure to group whenever you can. Do it on your pad:
 
It is entirely normal to hesitate a bit over some of the operations at this point. Do not worry if this happens to you. It takes quite a bit of living with any radically new methods before they become second nature. But if you have thoroughly understood each new idea and done each practice section conscientiously, you should have gone through each step without too much trouble.

Now re-check your left-to-right complement subtraction on the following problem. Use the slash method of canceling in the answer rather than borrowing in the larger number whenever you use a complement to

"subtract" a larger digit from a smaller:
 
If you tackled each of these examples with dispatch and confidence, then you are ready for the brand-new method of no-carry multiplication.

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