Speed Mathematics Simplified (11 page)

There is one more important element in this high-speed method of subtraction. This element is handling a slashed zero—
.

A slashed zero is, like any other digit, reduced in value by one. Since it must have a digit to the left of it in the answer (or you could not subtract), then obviously the zero must become 9—and the digit to the left of it must also be slashed, to reduce it in value by one (since you “borrowed” from it in order to get 09 from 10).

Consider this:

You would read the answer as 9. If the example were
you would read it as 19. In practice, particularly in a long problem, it is important to slash both digits. In reading that last
you would read
as 1, and
as 9.

This may sound formidable, but it is really not as complicated as borrowing continuously to the left as you sometimes have to do in ordinary subtraction. Go through the steps in this example, and note where we start canceling tens:

Follow each step carefully: Nothing from 1 is 1. Put down 1. 5 from 5 is 0. Put down 0. 3 from 3, and so on, gives you zeros until you come to the final column.

In the final column, 9 is larger than 8. Do not subtract. Add the complement of 9 (1) to 8. Put down 9, and cancel a ten by slashing to the left.

The digit to the left is 0. Slash it. Whenever you slash a 0, you must go back and slash the digit to the left of it too. That next digit is also a 0, so you have to keep on slashing until you slash a digit that is not a zero.

This may still sound a little strange. If you have any lingering doubts, do the problem above in the old-fashioned, schoolbook fashion. You will find that you have to do precisely the same thing, but in the more complex, error-prone method of borrowing over and over for
each
subtraction.

Try two longer problems now. Remember, as always, to practice the new technique as you do them. Work from left to right. Subtract a smaller digit from a larger digit just as you do now. But do not subtract a larger digit from a smaller. Instead, add the complement of the larger digit to the smaller digit and slash left in the answer. If you slash a zero, remember to go back a step and slash the digit to the left of the zero too.

The next chapter will carry you on to developing speed and accuracy at complement subtraction. Before you turn to it, however, let's cover another major advantage of adding and subtracting from left to right instead of from right to left.

Automatic Estimating

Any left-to-right method of doing arithmetic is self-estimating. Since you develop your answer from the left, the important end, you can always carry it exactly as far as you need for the accuracy you require and stop there.

Many of us have often tried to do this in the old-fashioned method of working when under pressure, but that is a backwards method and very difficult. Complement addition and subtraction does it automatically.

Suppose, for instance, you are production manager of a company making brass buttons. Your inventory as of the moment is 37,852 buttons. Today's orders total 16,965. The salesman selling to a large chain of stores calls to see how many buttons you could ship tonight on an emergency order. You must know, while he waits on the phone, about how many buttons you have.

Quick now: 1 from 3, 2. 6 from 7, 1. You have about 21,0 buttons. You have done merely the first two steps of your regular process in complement subtraction, instead of changing your method for estimating needs.

Suppose you need the next figure, too. 9 from 8. Add the complement of 9 (1) to 8: 9. Slash the digit to the left:
. 20,900 buttons.

See how quickly and accurately you can give a three-digit estimate of the following subtractions:

For estimating—as well as for many rounded-off computations—you simply ignore the relatively unimportant numbers to the right, and carry your subtraction just as far as you wish.

The automatic estimating feature applies just as much to complement addition as it does to subtraction. The only thing to beware of in adding is that whenever you stop, the next column could make a substantial change in your stopped-at digit. In subtracting, the next column can never affect your stopped-at digit by more than a reduction in value of one.

These two examples illustrate this point: