Speed Mathematics Simplified (23 page)

Difficult? To us, yes. To a French or German schoolboy it is something he is expected to learn; and learn it he does, or flunks out and spends the rest of his life hoeing potatoes. But to those of us trained in the “dot-every-
i
, put-down-every-digit” methods of American arithmetic, it is rather difficult to learn late in life.

What you will discover before this chapter is over, however, is that applying your new mastery of simplified left-to-right multiplication and subtraction will make it not only possible to divide in a way similar to the European method, but actually easier than it was in the standard long-division way.

Before we get into this subject, we will first explore a method for rapid answer-producing that removes the first major stumbling block to quick and easy dividing.

Automatic Division

The point at which most of us hesitate longest in working our way through any long division is deciding on the next digit of the answer.

Consider this example:

The first step is to divide 87 into 42 or, since this “won't go,” into 426.

Almost all of us, no matter how good our number sense is otherwise, lack any sort of genuine feel for such an answer. We are not dividing by 8, but by 87. Think back, and you will probably find that you often try two or three “trial answers” in your mind before deciding on one to put down.

Here is a simple trick to overcome this difficulty—a trick that automatically delivers to you the next digit of your answer no matter how complicated the divider is. It makes dividing by 34,968 as simple (at this point) as dividing by 4.

The trick is this: Do not divide by the divider. Divide only by its first digit,
raised by one.
Do not divide into the number divided. Divide only into its first digit (if that digit is larger than the divider digit) or into its first two digits (if the first digit is smaller than the divider digit).

This technique, by the way, is also adapted with minor variations from modern soroban theory. It is considered as basic to speed and ease on the abacus as is the use of complements for adding and subtracting.

In the example above, you do not try dividing 87 into 4263. Instead, divide 9 into 42. This is much, much easier. You should “see” the answer 4 at a glance.

The reason this works is that 87 is somewhere between 80 and 90, but for simplicity we consider it to be 90. A little over half the time, this first digit will be correct. Less than half the time, it will need revision—but the revision will be automatic and quick, just as it is on the abacus.

Try this technique on these examples:

On these three problems, our automatic division works like this:

5 (instead of 47) into 26 (instead of 268) is 5. This is the correct first digit of the answer.

7 (instead of 65) into 51 (instead of 513) is 7. Right.

3 (instead of 28) into 13 (instead of 136) is 4. This also checks out.

Caution
: Note with special care that using this trick to “see” each successive digit of your answer does
not
alter the position of each answer digit. In the first example, you put the answer digit 5 over the 8, not over the 6. You still follow your classical rule for placing your answer: start as many digits over in the number divided as there are digits in your divider—plus one if you start by dividing into two digits instead of one:

In the first case, we “see” 8 into 9 and put down the answer digit 1 two places to the right because there are two digits in the divider. In the second case, we “see” 8 into 17 and put down the answer digit 2 three places to the right because there are two digits in the divider
and
we started the division into two digits of the number divided.

Now get the idea firmly in hand by trying these:

Remember that we are not yet finishing these divisions. At the moment, we are concerned only with developing this rapid and foolproof way to produce automatically each digit of the answer without hesitation.

Check your reactions to the above three examples. Did you see the first as 6 into 21, and put down 3 over the 6? Did the second become 10 instead of 9—indicating that the answer digit 1 goes over the 8? When you got to the third, did you “see” 4 into 8 as 2, and put it over the 3? If any of your answer digits got misplaced, review the general rule once more:

If your first division is into a single digit (4 into 9), the first answer digit appears as many places to the right over the number divided as there are digits in the divider.

If your first division is into two digits (4 into 23), the first answer digit moves one more place to the right.

The principle of finding each digit of the answer by dividing with only the first digit of the divider, raised by one, works with problems of any length. Experiment only on the examples provided, however, until we come to automatic revision.

Go through the following problem on your pad. Find the two digits of the answer by dividing with the first digit only (plus one) of the divider:

Work this out completely in your traditional handling of long division, applying to it at the moment only the new automatic digit-finder.

The final answer is 65. The first digit is produced by dividing 9 (not 876) into 57 (not 5714) and putting the resulting 6 over the 4. When you multiply out and subtract, you divide into the remainder 4580 for the second digit. 876 might make you hesitate between 5 and 6 for the second answer digit, but 9 into 45 can only be 5. We have produced two digits of the answer by simple inspection. For now, we will ignore the fractional remainder.

Now we will go on to the special aid that makes this technique useful on any problem at all, not merely on carefully selected examples.

Automatic Revision

Consider this case:

Start with the trick of dividing 9 into 43, instead of 876 into 4380. 9 will go into 43 no more than 4 times. Yet if you multiply out the divider by 4 and subtract, you find a remainder of 876. This is the divider itself. The answer to this problem is 5, not 4. Then it comes out even.

What is wrong? Nothing, really. We said earlier that division is really a continuous approximation from left to right. The digit of the answer we first put down is an approximation that may need revising before we finish.

On the abacus, each trial digit is produced by dividing with the first digit of the divider but without raising it first. Revision is frequently necessary, just as it is in this system. But revision on the abacus is always to
reduce
the trial digit by one (sometimes two), adding in this revision factor to the trial remainder. In our system of using a digit that is one higher than the first digit of the divider, the only way we ever have to revise is
upward.
As you will see when the technique develops fully, this is easier and more automatic with pencil and paper.

Since you are in effect dividing by a number larger than your real divider, you could not possibly try too large an answer digit. It is child's play to revise your answer upward in our system, but it would be quite difficult to revise it downward.

You have learned in no-carry multiplication how to increase the value of a digit by one without rewriting it. You simply underline it. The answer to the example above, when finished, would look like
4
. You read it as 5.

All of this will be drawn together as we assemble the various parts of the complete division system. For the moment, remember only that you speed up your division by “seeing” the answer to 9
rather than trying to work out an answer to 876