Speed Mathematics Simplified (22 page)

Step three: 8 x 4 ends in 2:

The step two above is as complicated as no-carry multiplication can ever get. You have to remember the 8 while getting the 3. If you have learned to read answers, you would think only “8, 3, 1, record.” The same point in schoolbook multiplication would involve these thoughts: “Carry the 3 from 32. 4 x 7 is 28. Add the carried 3 to 8, which makes it 11. Put down 1 and carry 1 to the 20. Put down 3.”

This review is to encourage you to spend some of your practice time on the two-digit tables that follow. It would be impractical to include every possible combination (there are just under a thousand of them), but you will find a good spread of every type.

The first time you do this section, work slowly and evenly, disciplining yourself to think along the lines we have covered:

Read only the answer to each digit combination.

Work from left to right.

Think an initial zero if this is the left-hand digit of the first product.

Add the center digits of the answer with a complement if it goes over ten, and mentally record the ten by underlining the imaginary digit to the left in the answer.

Say aloud
the answers to these problems:

The most important thing you undoubtedly noticed is that your ease with these problems is based very directly on your ability to read automatically the left- and right-hand digits of the products of each combination. If they pop into your mind without thought—as they will after surprisingly little practice—then expanding your practice to two-digit examples is almost painless. But if you have to stop and think hard to get each digit, then you will find this section much harder and slower than it should be.

If you experienced trouble in “reading” the left- and right-hand digits to make these problems easy, go back and review your single-digit tables once or twice before going on. Each time you do them, the answer will come a little more automatically.

Now read from left to right the answers to these problems. Make sure you are building the right habits as you do so. Make it a point to use the proper technique:

The final practice table of this chapter follows. You have already practiced all the essentials. If you can handle two-digit tables with snap and decisiveness, then you can keep on doing step two through twenty-digit problems. You already know how to line up your columns for multipliers of two digits or more, and you know how to add more effectively and quickly than ever before. The final section asks you to say aloud, from left to right, the answers to a variety of multiplications with single-digit multipliers but differing numbers of digits in the numbers multiplied.

Just as you did with both the one-digit and two-digit tables, work slowly and carefully the first time over this varied practice group. Get the foundation of proper habits firmly established. Say your zero first digits where they are required, think an underline to the left when you use a complement, and do your very best to read only the answer to each combination—not the combination itself.

For longer problems you may wish to write down your answer. Just put your pad under the problem and jot down the answer from left to right, as it develops naturally in your mind.

Spend several minutes at this:

The mixing of problems with one, two, three, and more digits in this section was intentional. This is the way problems are presented to us in business. They do not ordinarily come neatly packaged in orderly rows of similar problems. Your mastery of each method and technique always has to become adaptable as well as proficient.

8

SHORT-HAND DIVISION

S
O FAR, we have covered three out of the four basic arithmetical computations.

In adding, we learned to use complements for the tougher combinations—those that would add over ten if we ever added over ten—and to record tens in such a way that the answer forms naturally in our mind, just as, on the modern abacus, the answer forms naturally on the board.

In subtracting, we learned never to subtract a larger digit from a smaller, and to avoid that crude and precarious method of “borrowing,” so that again the answer forms itself easily and naturally in the mind or on the paper.

In multiplying, we have torn apart the multiplication table so that we use only half of it at a time. This enables us to discard the idea of “carrying,” and furthermore produces the answer from left to right. When we have to record tens in preceding digits in our answer, we adopt a simple and effective method that—again—gives us a natural development of the answer.

Now, what about dividing?

There is no
single
secret for speed division comparable to the secrets of complements or no-carry multiplication. But by leaning on
both
complements and no-carry multiplication, we can build a streamlined technique for division that, in its total effect, can save you as much time and effort in this field as the single secrets can in theirs.

In order to get our ground firmly established, let us look at a sample problem in division and work it in the traditional long-division way:

This is a fairly simple problem. It has no remainder. Everything comes out even. Yet a great deal of pencil work was involved. It
looks
complicated.

Just for comparison, although the figures will be meaningless to you at the moment, let us show what the same problem would look like in the shorthand method you will learn in this chapter:

Certain elements of these numbers should be familiar to you—the underline and the slashes. The shorthand method will rely on your confident handling of complement subtraction and no-carry multiplication.

The two hardest parts of traditional long division, you will undoubtedly agree, are (first) determining at a glance the next digit of the answer and (second) going through the complex pencil work of verifying that digit and finding the remainder into which you divide in order to determine the next digit of the answer.

This chapter will offer a simpler way of doing each of these processes. Before we go into them, however, consider a few basic facts about the process of division.

Continuous Approximation

Long division, by which we mean division by a number of several digits, is really a progressive estimate that gets more accurate as we finish more of it.

In this sense, division is radically different from adding, subtracting, or multiplying. It is the only one of the four processes that we were taught to do from left to right, in the natural way. Since this is true, division is already self-estimating, just as the new methods for doing the other three processes are.

The familiar process we call long division, incidentally, seems to be a special crutch developed only in England and America, which, because every single step is spelled out (and written down in detail), no rational person in school can fail to learn to handle. It is certainly accurate and easy enough, but it is also infernally slow and cumbersome.

For another comparison, look at the division we just examined, next to the same problem solved in the European short-hand method:

If you have never before confronted this European (in England it is called “Continental”) method, you may feel some awe of European education when you learn that the difference is simply this: the multiplying and subtracting are done entirely in the head. They are never written down at all. The two lines of working figures you see under the problem are merely the
results
of each subtraction.