Speed Mathematics Simplified (13 page)

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.

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:

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 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 litle 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 easly 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.”