Speed Mathematics Simplified (14 page)

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

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.

6

NO-CARRY MULTIPLICATION

M
ULTIPLYING, according to the same estimates we mentioned before, averages about 20% of the figuring done in normal business.

But while it is used less than addition, multiplication is dreaded by more people, and done poorly or inaccurately by more people, than either addition or subtraction.

Perhaps this is because multiplication, particularly of one long number by another long number, becomes so fearfully complicated in comparison to the simpler process of adding or subtracting. Most of us have little trouble visualizing that adding ten lines of numbers may involve more work, but is really no more complex, than adding two lines of digits. But multiplying 2,958 by 165 is something that few of us can really “see” as a whole. We tackle it step by step, by pure rote, in inefficient and inherently slow traditional methods.

Our new way to multiply solves a large part of this. It involves three secrets. Two of them you already know from adding and subtracting. The third is brand-new.

The first secret is the one we use throughout this book: work from left to right. Tackle 165 as 165, not as 5, 6, 1. The less we have to use methods that violate the plain common sense of the way we normally read numbers, the better off we are. Our number sense becomes sharper instead of becoming dulled by backward absurdities.

Working from left to right also makes our new method of multiplying a self-estimating system, just as our left-to-right addition or subtraction is.

There is no simple way to work from left to right in the classical method of multiplying. But our no-carry method works just as easily from left to right as from right to left, so you will find it natural to work in the proper direction.

The second secret, again, is the same as its equivalent secret in adding or subtracting: “see” only the answer, combining digits at a glance. This is simply a matter of practice, but chances are you have already had more practice at this than you had for adding. Most schools spend far more time on drill in multiplication tables than they do on drill in addition and subtraction tables. So you are probably closer to mastering this step in multiplication than you were for the two earlier processes.

If you were taught in the standard way, however, you would do well to begin practicing the deletion of the slowdown steps taught in school. Instead of reading the example

as “4 times 7 is 28,” make a conscious effort to look at it and think only “28.” You do not look at “me” and think “m and
e
is ‘me.'”

In fact, the entirely new way to multiply involved in the third step will bring up quite a different way of looking at 4 x 7. You will never, oddly enough, think the whole product at all, but only half of it at a time.

The third secret is the new method. It is radically different from the traditional way to multiply because you never have to “carry.” The greatest trouble with standard multiplication, and the greatest source of errors, is carrying. It is very much like the difficulty in “borrowing” in subtraction. Either you forget to carry, or carry twice, or carry the wrong figure—and wind up hating numbers.

The no-carry method of multiplying works without remembering to carry at all. It may look a little strange at first, but once you try it a few times you will get the idea.

The easiest way to approach this method is to take apart a sample multiplication and see what makes it tick. Make sure you fully understand every step of this, because once you understand why the system works as it does you will find it very easy to use. If you simply try to learn the technique by rote, however, it will always seem complicated.

Let's take this multiplication apart:

Look at the answer, digit by digit, and see how it really develops.

The first digit, 3, is simply the
left-hand
(tens) digit of 4 times 8—
3
2.

Look back at the example and note this. The first digit of this answer is merely the tens, or left-hand, digit produced by multiplying the first digit of the number multiplied by the multiplier.

The second digit is a little more complicated. This 7 is the sum of two other digits. It is the sum of the
right-hand
(units) digit of the multiplication we just examined—4 times 8—and the
left-hand
(tens) digit of 7 times 8. The right-hand digit of 4 times 8 is 3
2
, or 2. The left-hand digit of 7 times 8 is
5
6, or 5. 2 plus 5 is 7—the middle digit in our answer.

Look back at the example again to make sure this is completely clear. Read the above explanation again if you need to.

If you remember our earlier comments about left-to-right working, in which we pointed out that each digit increases in value by a factor of ten as it moves one place to the left, then you can see why the middle digit in this answer is the sum of the
unit
part of the 4 times 8, and the
tens
part of the 7 times 8. It is because the 4 in 47 is really ten times 4 because of its position—or 40.

The last digit in this answer is 6. This is simply the
right-hand
(units) digit of 7 times 8—5
6
.

This is a new way of looking at multiplication for most people. Get it clear now, and everything that follows will fall into place naturally and easily.

Now let us try multiplying those same numbers left to right in the new method, using the understanding above of how the answer really develops. If the method seems unclear at any point, re-check the explanation above.

Step one: Look at 4 x 8 to see only what the left-hand (tens) digit of the product will be. In other words, is 4 x 8 in the teens, twenties, thirties, forties, or what?

4 x 8 is in the 30's. The tens digit of this pair is 3. For the moment, you do not care what the right-hand, or units, digit is. All you care about is the 3.

For the first digit of your answer, put down that 3: