Speed Mathematics Simplified (10 page)

The first example should develop like this: Nothing from 1 is 1. Put down 1. 4 is larger than 2, so do not subtract. Add complement of 4 (6) to 2. Put down 8, and immediately (before you forget) slash the 1 to cancel a ten. The answer is
8, or 8.

Second: 1 from 2 is 1. Put down 1. 6 is larger than 5, so do not subtract. Add the complements of 6 (4) to 5 and put down 9. At once slash the 1 to cancel a ten. Answer,
or 9.

Third: 2 from 3 is 1. Put down 1. 8 is larger than 7, so do not subtract. Add the complement of 8 (2) to 7. Put down 9. Immediately cancel a ten by slashing the 1. Answer,
9, or 9.

The last example: 3 from 4 is 1. Put down 1. 6 is the same as 6. Nothing, or 0. No complement, no cancel. The answer is 10.

Perhaps the last one caught you. It was designed to. Complements only apply when we subtract a larger digit from a smaller. You will still subtract, about half the time, a smaller digit from a larger one or from one of the same value.

Why We Slash Digits

In the examples so far, it has really been a little childish to bother slashing digits in order to cancel tens. A fourth-grade schoolboy knows that 4 from 12 is 8. But you are exploring a new technique, a technique that applies not merely to 4 from 12 but also to 8,344,897 from 9,432,752. Learning to go through the proper steps is as important as learning to play the scales before tackling Chopin.

Play a few scales right now. First, make your complement-reaction just a little faster by “reading” the complements to these digits:

You will notice that in subtraction we now use
both
halves of each complement pair. We find it faster to use only the larger of each pair in addition, but you have to use all of them in subtraction. This is no problem, because there are still only five pairs. If you pause to wonder why we can pick which half of each complement pair we wish to use in adding, but have no choice in subtracting, notice that you can add 7 + 9 or 9 + 7 as you choose, but have no choice of complements in each of the two corresponding subtractions: 16 – 9 or 16 – 7.

Subtract the following examples from left to right. Put down on your pad or card every digit as you go along, even if it seems silly. This habit is important to your successful handling of longer and more complicated problems. Whenever you come to a larger digit from a smaller, add the complement of the digit to be subtracted to the digit you are subtracting from, and cancel a ten by slashing the digit to the left in the answer:

One more point. A slashed 5 ($) is read as a 4, because the slash “borrows” or more properly “cancels” in the answer. But until this too becomes second nature, you may wish to rewrite answers before considering them finished. Remember that a slashed digit is reduced in value by one; then a subtraction answer that looks like this

would be rewritten or would read like this

After you have used this technique steadily for a few days, you will probably not bother to rewrite answers in this fashion. But until you have fully mastered the art of reading a slashed digit as one less than it was before the slash, you will profit by making sure you interpret such answers without error by rewriting them.

Take your pad now. Use it to cover the rewritten version of the following subtraction answer as you copy it in final form. Every slashed digit becomes the next digit smaller:

After you have rewritten this answer, compare your version with the one that follows. If you got any of the digits wrong, it would be worth while to do it again.

Here is how your copied answer should read:

Now try these examples. Remember to work from left to right, use complements where indicated, and “borrow” by slashing the preceding digit in the answer:

By this time you should be finding it a little easier to work from left to right, and canceling tens in the answer—rather than “borrowing” in the larger number—should be beginning to feel natural. Once you become fully used to it you will find it far more natural and infinitely more foolproof than the older system.

It has been estimated that 80% of all mistakes in subtraction come from forgetting to borrow, or borrowing too much. Since we eliminate borrowing altogether, this method is by nature more accurate as well as faster.

Carrying Back Slashes