Speed Mathematics Simplified (6 page)

Can you
feel
the identity of all three processes?

We chose 9 as the demonstration example because it is so obviously 1 less than ten. Just as surely as 9 is 1 less than ten, 8 is 2 less than ten, 7 is 3 less than ten, and so on. The principle does not change one bit when we use these other combinations.

As one further example, let us show all three ways of expressing another “identity”:

Take your pad at this point and work out the addition of 5 and 8 in all three ways. The closer you can come to “feeling” the identity of all three pictures of the same process, the more confidently you will handle complements.

One analogy that has proved helpful to some people is to visualize the process of adding as climbing a series of ladders, each with ten rungs, from level to level. At any point, you know your position on a ladder and you know on which ladder you stand. For instance, you are now standing on the sixth rung of the third ladder—an analogy of the number 36. You are told that you can advance eight more rungs, and wish the quickest and easiest way of projecting where you will be standing after eight rungs.

First, you know that you will be on the next higher ladder (in the 40's), because there are not eight more rungs above you on the third ladder. Adding 6 and 8, let us say, is something you have never been taught to do. You do know that if you could advance a full ten rungs you would be on the corresponding rung of the fourth ladder—46. But since 8 fails by 2 to complete a ten, you will be 2 rungs lower—44.

So, in any addition that crosses the next ten-point, you will fail to reach the corresponding number across that ten-point by precisely the amount that the number you add fails to reach ten. That is its complement.

Before going on to the next chapter, work for a few moments at making the use of complements a habit by using them conscientiously in adding the following problems. Use either lines or fingers as you prefer, but standardize now on one system or the other.

Do not add these pairs. In each case, subtract the complement of the larger digit from the smaller
and
record a ten. Just “see” the answer; don't write it down:

As you “read” these examples (and you should be trying to “read” rather than “solve” them) it may help to channel your thoughts in the right direction if you lip-read them the first few times. This is not good permanent practice, but it will help break your old habit patterns. You would lip-read the first problem, for example, as “5 – 3 is 2; finger,” to help you avoid slipping back into the thought pattern of “5 + 7 is 12.” Ultimately, you will try to “see” it as merely “2, finger.”

The first key to speed in this system is obviously knowing your complements at sight, without pausing to think for a second. Review them quickly. Try to “read” the complement of each digit as you see it, without stopping to ponder:

These are the only digits for which you have to remember complements at this point. Five is the complement of five, but you never use it that way in adding because when faced with a five and a five you simply react “0, finger.” When faced with a five and a larger digit, you use the complement of the larger digit.

What is the complement of 7?

If you had to pause for even a flicker, build your base for rapid progress later in the book by reading the above digits again. React without thought with the complements to these digits:

The sheer repetition here is not overdone. It is essential to mastering the new system. One of the two major approaches to teaching machines uses precisely this principle.

Go through this brief check-up to make sure you are ready for the next chapter, which will begin to build your speed and confidence in complement addition.

     
What is a complement?

     
What is the complement of 7?

     
When you add two digits that would go over ten, do you add or subtract the complement of one of them?

     
Is it quicker to use the complement of the larger digit, or the smaller one? Why?

     
What is the complement of 6?

     
When adding a column, do you keep all the tens in your mind, or do you record them?

     
What is the complement of 8?

     
How do you record a ten?

     
What is the complement of 9?

     
In adding a column, do you combine each new digit with the digit above it, or with the digit in your mind from above additions?

     
Could you explain to a friend why complements work as they do? Pretend he has just asked you, and see if you can.

3

BUILDING SPEED IN ADDITION

I
N 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.