Chapter 3 - Building Speed In Addition

In the last chapter you have had a taste of one of the newest and most exciting developments in the whole field of speed mathematics. Its sheer beauty and rapidity will grow on you as you begin to make it a habit.

Part of making it a habit is plain old-fashioned practice. There is simply no way of learning high-speed arithmetic without a pretty fair dose of practice. You cannot begin to master the systems without using them enough times to feel at ease with them.

It is always a temptation to skip the practice in a book of this kind. You are interested in the "meat," in the theories, in what comes next. There is a great deal coming next. But to skip the practice in its proper place would be unfair to yourself. The best theory, the finest technique in the world, is useless unless you can use it. You cannot use it simply by knowing the theory. The difference between knowing how something is done and knowing how to do it is skill. Only practice can build skill.

We will vary the practice, break it up into modest doses, to keep it as inviting as we can. But—don't skip it!

In order to encourage you to do the practice page by page, I have hidden right in the middle of it one more big step for even greater speed in addition.

Start now by reading at sight the answers to the following additions. Don't think or lip-read or even "see" the problem itself if possible; see only the answer. Remember your complements for groups that would go over ten:
 
Pause and ask yourself some questions here. Did you manage to see only the answer, not the two digits to be added? Did you begin to find yourself glancing at each group that would add over ten and automatically subtracting the complement of the larger digit from the smaller digit—and folding a finger?

If not, go back over them and make the special effort to use complements in these cases. Such combinations are mixed in with "under ten" combinations on purpose. The two are always mixed in the figure work we meet in our lives.

Now let us go on to another easy dose of practice. These numbers are not simply random, by the way. Every possible combination of digits has been recorded and appears in the practice tables. By the time you finish this chapter you will have practiced every single possibility.

See only the answers to these, using complements where appropriate:

That's enough for a moment. Arithmetic, even the streamlined variety, takes concentration. At the start, the new techniques take even more concentration than the old ones, because you have to stop and think about doing things in the new way.

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.
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 file 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.

Using this method, the final answer to the example above would look like this:

2 8 2 1 8 1 0 0 8

You underline the 1 because you have looked at the next column before putting it down and seen nothing to carry back. But when you add that next column (the 9 with nothing under it), you see that you will have to add a ten from the next-to-last column—the 9 plus 11—and this will change the 9 to a 0, with a ten carried back to the 1 you have already put down. It would be awkward to change the 1 by this time, so you simply underline it. In reading or copying the final answer, read the 1 as 2.

If this seems hard or slow, note that the same thing often happens when you add or multiply on the abacus; and it is considered more than worthwhile to carry back a ten in this fashion rather than pay the far greater price of working from right to left.

The obvious job remaining is to practice a bit more; practice so that the techniques become second nature, so that you begin to "see" only the answer, so that you group digits adding to ten or less without having to think about it.

Try reading right through the following problems, using all your newly learned techniques and noting your answers on your pad or cards for later reference:

At this point your practice is beginning to combine all the separate elements you have learned. Some columns involve complements and recorded tens; some do not. Some columns require you to carry tens back to a previous column in the final answer; some do not. Some columns contain digits you can combine at a glance; some do not. This is the variety of which our daily arithmetic is composed. It never comes in neat parcels designed especially to illustrate some special point.

Now go back, with a fresh page of your pad, and do the examples over again.

Compare the answers you got the. two different times. Are they the same, or different? If you have two different answers in any case, do it still once again—and find out where you went wrong.

Now go on to these:
 
Note your answers as you did before. These examples have fewer columns but more digits in each column. The variety is planned, in order to show examples of different applications of the techniques and to keep the

practice from becoming too monotonous.

Now turn your pad or card over and do the above problems again. Compare your answers to the ones you got the first time around. If they are the same, good. If not, learn from your mistakes by doing any problems to which you got different answers once more, and seeing which one is really right.

Because it is so important to everything you will do for the rest of your life in mathematics, review right now the twenty combinations of digits under ten. Other than complements, they are the only ones you have to handle from now on. Combine these pairs at a glance:

This table includes every possible digit combination in adding other than complement pairs. The complementary pairs, too, should be starting to feel as natural as breathing. Look at the following digits and, in a flash, see only the complement:

7 6 9 8

As a finale to this chapter, try your hand at one really big problem—the sort most of us approach with some reluctance when we have to solve it, yet which combines in just one practice session everything you have learned so far. Approach it with these rules in mind: first, work from left to right; second, add "over" ten by using complements and recording the ten; third, record the tens as you go; fourth, combine digit-pairs adding to ten or less at a glance and handle them as a single digit or recorded ten.

Work for speed on this one. Note down your answer, and come back from time to time to see if on another try you still get the same answer. Vary your practice by adding down one time, adding up the next:
 
Do this at least once before going on. It embodies, in one example, every possible technique from the last two chapters.

Are You Ready To Move Onto The Next Lesson? Click Here...