Speed Mathematics Simplified (7 page)

Before finishing the random series of all digit combinations, take a breather by hearing the famous (and possibly apochryphal) end to the story of that Tokyo contest between the abacus and the calculating machine. The electric calculator, according to the story, was made by International Business Machines, whose company-wide motto is THINK.

After the American machine-operator was roundly defeated by the soroban-operator, he is reported to have said: “Maybe his way is faster. But all I have to do is punch buttons. He has to think.”

Now we will finish up our speed practice in basic digit combinations. Remember to use complements where the addition would go over ten, and fold a finger or think “line”:

That's all. Those are all the possible digit combinations. You will never, in all your life, face any combination of digits that you haven't just practiced.

Some of the addition we do in our jobs or at home consists of single pairs, such as the examples you have just done. Much of it, however, does not. We frequently have to add three or more digits in each column of a particular addition, whether it is sales in seven different territories or prices of twelve lots from the real-estate developer.

Handling more than two digits using the complement system is something you already understand but might profitably use a little practice on. This involves handling complements when one of the digits to be combined is in your mind (from adding the previous digits in the column) and the other is the next digit in the column, rather than with two digits set up just for you to practice with.

Consider this addition:

When you add the first two digits, you subtract 3 (the complement of 7) to get 4, and record a ten—14. The only digit you carry in your mind, however, is 4. The ten you record with a line or a folded finger, and promptly ignore for greater speed and accuracy.

Now you glance at the last 7. You combine it, of course, not with the 7 above it but with the 4 in your mind. 4 – 3 (complement of 7) is 1, with another recorded ten. You have recorded two tens and are remembering 1, so your answer is 21

This answer “formed itself naturally” in your mind, just as it forms itself naturally on the board of the soroban.

While you know all this, you will handle the process more easily and quickly if you spend a few minutes consciously practicing the use of it. Run through the next column with the complement technique. Then see if your handling agrees with the description below it.

The complement system, assuming you use fingers (if you use lines, read “lines” for “fingers”), would go like this: “7 (finger), 3 (finger), 1 (finger), 0 (finger), 5, 2 (finger). 5 fingers plus 2—52.”

Note especially that between the 5 and the last 7 there is no finger. Why?

Now read through these examples, using complements in each case and seeing if the total of your recorded tens plus the number in your mind comes out the same as the answer. If not, do them again:

The last one was put in there on purpose, just to remind everyone that we don't
always
use complements. They only apply when addition goes over ten.

Compound Your Speed by Grouping

You have learned, and begun to practice, two basic elements of real addition speed: recording tens, and using complements instead of adding over ten.

There is one other major contributor to high-speed adding. It is a standard “short-cut” method. But it is easier than ever to use with complement addition, because you will get to know the twenty-five combinations to which it most easily applies by first name, instead of scattering your memory over all forty-five possible combinations.

Your full mastery of those twenty-five easiest combinations can speed up your addition still further if you stretch it to include the technique called grouping. In grouping, you “see” any pair of digits adding to less than ten as one digit, and any complementary pair as leaving the number in your mind unchanged but worth another recorded ten.

Just as you look at the two letters
i
and
t
and see—not
i
and
t
—but “it,” so you will learn to look at “3” and “4” and (if you are adding) see only “7.” It works like this:

An expert will handle this as the addition of 7, 8, (record), and 8. He will “see” the 3 and 4 as 7, and so on. Simply think “5 (finger), (finger), 3 (finger)—33.” It's fast—and surprisingly easy.

Now try grouping on these examples:

In any future addition examples, make a special effort to group digits that add up to less than ten as well as to ten exactly. Steady work with complements will help flag 3 plus 7 as worth exactly one folded finger (or one line), without changing the number in your mind from previous additions.

All your adding practice so far has been single-column work. Some of the adding we do in our jobs or at home is of this nature, but it is more than likely that a large part of it includes several columns.

Now is the time to refresh your memory on working from left to right. The abacus is always used this way. That Japanese operator who so thoroughly beat the calculator operator would not dream of working from right to left. It just would not be natural.

Remember that when we add several columns, we put down under each column the last digit that developed naturally in our mind, and one place to the left of it we put the number of recorded tens. Under the first column we can place our recorded tens immediately to the left, but under later columns they have to go down one line because of the totals of those columns. Follow, using all your new techniques, this example and see if your answer agrees. Work from left to right:

This example shows one or two special points. Note that in the next-to-last column, there are no tens recorded and therefore there is no digit placed to the left of that column. Note also that in adding the two sub-totals, you carry one “ten” back from the next-to-last column, through the column before that, to the column before that one. When you come to adding your sub-total lines, you will sometimes have to do this. Since you never add more than two lines of sub-totals, a glance ahead will show when you need to “carry back” a ten. If this proves difficult, simply
underline
a digit to which you find you have to carry back a ten. The underline raises the value of the underlined digit by one—a technique you will learn to use automatically when we get to multiplication.