Speed Mathematics Simplified (8 page)

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

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:

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.

4

COMPLEMENT SUBTRACTION

S
UBTRACTION is merely the other side of the coin of addition.

For most of us, however, it causes far more trouble. There are probably two reasons for this. While many of us learned our “addition tables” by heart in school, few of us really mastered the conversion of these into “subtraction tables” with anything approaching the same thoroughness. More important, however, the traditional process of “borrowing” is a tricky concept. Many of us find ourselves forgetting to borrow, or borrowing twice, because it is basically unnatural.

This chapter will eliminate both these handicaps. It brings to your work in subtracting three important aids to speed and accuracy.

First, complement subtraction will enable you to work from left to right. This is quite impossible in any other method of speed mathematics, but, surprisingly, the left-to-right procedure works
best
with complements. You should begin to have some feeling at this point of how much left-to-right working helps preserve and build your number sense.

Second, you will use a new technique that does away with “borrowing” entirely. The same necessary step will develop naturally and easily in your answer, just as it does on the abacus.

Third, you will apply to subtraction the same complement technique you have just learned for addition. This means that never again will you have to subtract a larger digit from a smaller—the process that causes so much confusion and error. Just as you now do in adding, you will work entirely with the twenty easiest combinations and the five pairs that “complete” tens—and forget the twenty hardest combinations altogether.

Before getting into the complement portion of subtraction, it will be helpful to get used to handling subtraction from left to right on a few problems in which you can work from left to right with standard methods. Such problems are those in which each digit in the smaller number is smaller—or the same size as, but never larger—than its corresponding digit in the larger number. In other words, in any problem that does not involve “borrowing” you can as easily work from left to right as from right to left:

Take your pad and pencil and subtract the above problem from
left to right.
It will feel strange the first time, but your answer will come out right. If you feel at all uneasy about it, reassure yourself by doing it over in the way you are accustomed to working and note that the answer is the same.