Speed Mathematics Simplified (18 page)

Now try doing a problem that involves this situation on your own. Write your answer, left to right, before going on. Then check your steps against the explanation that follows:

Here is the way the no-carry method works on this problem:

Step one: 7 x 8 is in the 50's:

Step two: 7 x 8 ends in 6. 8 x 8 is in the 60's. Complement of 6 (4) from 6 is 2, and record the ten:

Step three: 8 x 8 ends in 4:

Rewrite this answer as 624, and you are done.

Sometimes, too, your recorded ten will affect a first-place zero. Here is such a case:

Work this one out yourself, being sure to put down a zero if the left-hand, or tens, digit of any of the products is in the zeros, and see if this changes as you go through the full answer.

Work it out now on your pad.

If you did each step correctly, your answer should look like
0
52, which you rewrite as 152.

Do the next two problems on your pad. Be sure to go through exactly the steps we have been demonstrating, and check your answers to make sure they are right. If you hesitate too much, or come up with a wrong answer on two or three tries, then a review of the last few pages is in order.

How Complements Help

By this time, you have surely noticed a surprising and delightful fact: complement addition is a tremendous aid to no-carry multiplication. You need have no worry about remembering when to record a ten. The occasion is signaled to you automatically. You record a ten every time you use a complement or add to ten, and that is all. One goes with the other.

Whenever you use a complement, of course, this also gives you the digit to enter in the answer more quickly and accurately than would trying to add (say) 9 plus 9 and getting 18—of which you would have to put down the 8 and carry the ten. Working the new way, you think simply “8, record.”

It is almost foolproof, once it becomes a habit. No-carry multiplication is the closest possible approach to the secret of the soroban: to make as much of the operation as possible mechanical, so you can give your attention to the digit-by-digit sums and products and divisions without worrying about carrying and holding numbers in your mind. Since you are released from much of the mental labor of ordinary mathematics, you can concentrate on the single most important skill needed to handle this quickly and easily: your ability to “see” 6 x 7 as “40's” and “ends in 2” without hesitation or effort. The next chapter will go more fully into this.

Two-Digit Multipliers

No part of no-carry multiplication changes when we approach multipliers of two or more digits. It would be theoretically possible to produce the answer to two- or three-digit multipliers in one operation, but this means juggling four digits in your mind at once. This, for most of us, is impractical. Some “short-cut” mathematics books do urge this method, but it is really going in precisely the wrong direction for true speed. Unless years of practice go into them, the methods for producing the answer to 59 times 38 in one operation, on one line, are more apt to get mixed up and give a wrong answer than to speed up your results.

You now know how to produce a left-to-right answer to any multiplication by one digit with greater speed and accuracy, as well as ease. Let us stick to this head start, and do longer problems in the simplest and fastest way. We will use a different line for each digit in the multiplier. We will arrange these lines, however, in the reverse of the classical system. Start your top line with the
left
digit of the multiplier, and put each following line one place to the right.

This method is both easier and faster once you get the hang of it. It keeps you working from left to right—which is more natural, and also makes the system self-estimating.

Watch this demonstration:

Step one: 2 x 9 is in the 10's:

Step two: 2 x 9 ends in 8. 6 x 9 is in the 50's. Complement of 8 (2) from 5, and record:

Step three: 6 x 9 ends in 4:

Now, for the second line. This we get by multiplying the 8 by each digit in turn of the number multiplied, and we place the answer one place to the
right
(we work always from left to
right
):

Step one: 2 x 8 is in the 10's: