Speed Mathematics Simplified (4 page)

The appearance of this table is not random. It could be set up in slightly different shapes, but the order and pattern of this particular arrangement are especially instructive. You will find it worthwhile to examine the pattern with some care. Note, among other things, the heavy concentration of pairs adding up to totals around ten, and how the possibilities taper off toward high and low totals.

In the system about to be explained, here is how we will handle the forty-five different combinations:

We use the twenty combinations adding up to less than ten just as we do now. They are the easiest ones. We use the five combinations that add up to ten (1 + 9, 2 + 8, 3 + 7, 4 + 6, and 5 + 5) even more than we do now, so we learn them extra well. We
forget
those twenty hardest combinations that add up to more than ten and learn the technique of complement addition.

Add With Complements

The basic rule for the new technique is this:

To “add” over ten, subtract the complement of the larger digit from the smaller digit—and record a ten.

First we need to learn what complements are. Then we will take up how to record tens. Both are easy.

What is a complement? A complement is simply the digit that, added to the digit you have, adds up to ten. You might say that a complement is that digit needed (in addition to the one you have) to
complete
a ten.

For example: The complement of 9 is 1, because 9 + 1 is ten. The complement of 8 is 2, because 8 + 2 is ten. The complement of 7 is 3, and the complement of 6 is 4, because 7 + 3 is ten and 6 + 4 is ten. Even in your sleep you would answer that the complement of 5 is 5.

Those are all the complements you ever have to remember in adding the longest column of figures. There are only five of them: five pairs, you will note, that add up to ten in the table of possible combinations.

Before learning how to add with complements, make doubly sure that you have the idea by looking at the following digits and giving their complements. Try to “read” the complement of each as you are beginning to “read” the answer to a simple problem:

The way you add with complements takes a bit of getting used to. But it is one of the most fascinating and fruitful approaches known to short-cut arithmetic. You “add” two digits that total more than ten by subtracting the complement of the larger digit from the smaller digit and recording a ten.

In order to add 6 + 7, you subtract the complement of 7 (3) from 6, and record a ten. 6 – 3 gives 3. The recorded ten makes it 13.

Or to add 8 + 4, you subtract the complement of 8 (2) from 4 and record a ten. 4 – 2 gives 2. The recorded ten makes it 12.

It is useful to subtract the complement of the larger digit rather than the complement of the smaller. In this way you cut in half the number of complements you have to remember at this stage—though the other half of the complements are really only the same pairs of digits that add up to ten turned around. Just as 2 is the complement of 8, so is 8 the complement of 2.

Try it yourself, before going any further. Add 7 + 9 by subtracting the complement of the larger digit from the smaller digit. The complement of 9 is—. 7
^–
—is—. Remember to record a ten, in ways you will learn very soon. So the answer is 16. I hope that is what you arrived at through the new method, even the first time. If not, then it hasn't become clear yet. Another reading of the last few pages is indicated.

Now add 3 + 8. Would you subtract the complement of 8 from 3? What is the complement of 8? Don't forget to record a ten.

Strange and complex as this system undoubtedly seems at the moment, it is really far faster. This is because you are working with only the easier half of the forty-five digit combinations, the half that add up to less than ten. Even subtracting the complement will shortly become no problem, because you are always subtracting digits from pairs in the top part of the table. Look back at it again for a moment. In the complement system of addition, you
cannot possibly
get into that bottom part of the table—those twenty toughest (and slowest) combinations.

Give it one more try before going on. Each time you use it, the system will become a little easier and more natural.

Add 6 + 5. The complement of 6 is—. 5 minus—is—. Record a ten.

Recording Tens

You recall that we said the soroban, or modern Japanese abacus, is not really a calculating instrument at all. It is a recording instrument. By recording the results of each step in a calculation, it frees the operator to concentrate on skill and speed in taking each step.

We can record steps, too. Our methods of recording will enable us to concentrate on speed, just as the soroban does.

There are two good ways to record tens each time you use complements. The first way is simply to put a line at each place in a column of figures whenever you use a complement or add to ten. If you adopt this system, make it a habit so it becomes automatic. Then, when you write your final total, you just sweep your eye over the lines in that column and put down the total number of lines as your “tens” digit, one place to the left. Instead of remembering “37,” for instance, you have in your mind at that point only the
single digit seven
, but you will find three lines along the column.

We will go through one problem slowly and carefully, step by step. At first, the process will seem quite long and complicated because each step must be made clear. Actually, as you will find with use, it is far simpler and faster than the traditional method of addition.

Remember that we always work from
left to right:

First column

     
2 + 3 is 5.

     
5 – 1 (complement of 9) is 4. Record ten by putting a line under the 9.

     
4 – 3 (complement of 7) is 1. Record ten by putting a line under the 7.

     
Put down the remembered 1 under the column.

     
Count the lines. There are two. Put a 2 one place to the left of the column.

Second column

     
4 + 5 is 9.

     
9 – 1 (complement of 9) is 8. Record ten by putting a line under the 9.

     
8 – 1 (complement of 9) is 7. Record ten by putting a line under the 9.

     
Put down the remembered 7 under the column.

     
Count the lines. There are two. Put a 2 one place to the left of the column, under the 1 from the first column.

Now you simply add and get the answer, 237.

While this has taken some time to explain step by step, in practice you will find it infinitely faster than the old way. When you do it automatically, you will think only “5, 4 (line), 3 (line); 9, 8 (line), 7 (line); 237.”

One element about the problem may be a little confusing. We combine the next figure in the column with the figure in our mind from previous additions, not with the figure above it. For instance, in the first column of the problem above, we subtract the complement of 9 (1) from 5—the result of adding 2 and 3—not from the 3. It works just like regular addition in this respect. The use of complements does not change it.

Try the next example, in which we will go through the steps in a much more condensed way. See if you can follow each step, identify the complement being used in each case, and understand why we record a ten with a line each time we do so:

This example should have gone a little more easily. Take it slowly now, so you can build on a solid base of thorough understanding in later parts of the book.

Rather than go on with more practice at this point, let us get into the second method of recording tens. Of the two, this is quicker and more generally useful. But, in this case and in many alternate choices in the “short cuts” section later in the book, you should adopt the one that seems most natural to you and concentrate on it. Continuous use of one system will build the desirable habit pattern and accelerate your speed.

Record on Your Fingers

The second way to record tens is to use your fingers. We were taught not to count on our fingers, so the idea may come as something of a shock. Actually, however, the purpose here is vastly different. We were taught not to count on our fingers because using them for
counting
is leaning on a crutch that interferes with genuine mastery of the calculating skill itself. Using them for
recording
, as you will see, approaches the automatic-recording advantages of the soroban, and frees you to concentrate on adding the digits with extra speed.

Should you need any more encouragement, take note of the fact that top abacus operators become amazingly proficient at mental arithmetic by learning to close their eyes and visualize the soroban as they calculate—and they use their fingers for recording. So no matter how much distaste for using your fingers your school training may have left you, keep firmly in mind that this is recording rather than counting, and give it a try. Speed mathematics can and should make use of any device that simplifies and speeds up the solving of problems.

Here is how the system works. To record the first ten (when you first use a complement or add to ten), fold the little finger of your left hand into the palm. If you write with your left hand, there is no reason why you cannot record on the right. To record the second ten, fold the next finger alongside the little finger. This means two tens. If you use another complement or add to ten in the same column, fold the next finger. This records three tens. And so on, up to five tens.

If you have more than five tens in a long column, open the hand and start over with the little finger again. Perhaps you will feel happier about remembering to add five to the second running total of tens if you make a line in the column when you start over. Or use any other signal to yourself that makes sense.

This is not silly. Any mechanical aid that fits your habits and personality is a valid and useful device for freeing your mind to concentrate on the basic objective: speed and ease with fingers.

Whatever signal you adopt in a case like this, be consistent with it. Settle down to use this method for every single calculation you do, no matter how simple it is or where you do it. Habits are very important. Making a habit of consistently using the fastest techniques is what gives speed.

The use of fingers instead of lines to record tens does not change what you do at the end of each column, of course. First you put down the digit in your mind from the final addition. Then you put, one place to the left, the number of fingers you have folded—adding five if you had to start over again.

Here is how we solve a problem with this system. Work from left to right:

First column

     
5 + 1 is 6.

     
6 + 3 is 9.

     
7 – 1 (complement of 9) is 6. Fold a finger.

     
Put down the 6 in your mind.