Chapter 4 - Complement Subtraction

Subtraction 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.

Because working from left to right is a much harder adjustment to make in subtraction than it is in addition, do a few more examples in this way before going on to the complement techniques:
 
Just to make sure that you really have the idea, do them over again to see if your answers agree.

When we come to problems in which any digit of the smaller number is larger than the corresponding digit of the larger number, we face the situation handled in traditional methods by "borrowing." The relationship is really the reverse of the similar situation in adding two digits that go over ten, which traditionally calls for "carrying" but which we now handle by "recording." Just as we have substituted recording for carrying, we will now in subtraction throw out the concept of borrowing and substitute for it a new technique we call canceling.

Here is a situation in which you must borrow or cancel:

Schoolbook thinking would approach this problem, from right to left, in this fashion: "7 from 14 (borrow the 1 from the 3) is 7. 2 from 3—no, we borrowed a 1 so it is now 2—2 from 2 is 0. Answer: 7."

Working from left to right in complement subtraction, our thinking is quite different. First, we glance at the first column and "see" 3 — 2 as 1. We put it down. There is a reason for this, so bear with the obvious wrongness of that 1 for a moment—you will see why. Then we glance at the second column and "see" 4 — 7 as 4 plus the complement of 7—and cancel a ten.

The complement of 7 is 3. 4 plus 3 is 7. Put it down under the second column.

Keeping in mind that subtraction is just the reverse of addition, it should make sense that when subtracting you add a complement, just as when adding you subtract it. A full explanation comes later, but for the moment just remember that you are (in effect) doing addition in reverse and so your complements are added rather than subtracted.

Now we have used a complement, and when we use a complement in subtraction we must cancel a ten—just as when we use one in addition we must record a ten.

The method that makes possible our left-to-right working is that we cancel that ten in the answer—rather than "borrowing" it in the larger number. The technique for this is quite simple. We merely slash the 1 we put down under the first column:

A slashed digit in the answer to a subtraction is a digit from which a ten has been canceled. In this particular case there is only one ten there—the ten of 17—so the answer is 7.

The general rule goes like this: To cancel a ten, slash the digit to the left in the answer. That digit is then reduced in value by one.

If there seems to be any confusion over the apparent interchangeability of the words "ten" and "one" here, reflect on the fact that each digit increases in importance by a factor of ten as it moves one place to the left.

Note the similarity of these answers to the last one, and follow the left-to-right process by which each was produced:

Now, however, keep in mind that a slashed digit is reduced in value by one—it is not wiped out entirely—and go through the development of these answers:

At this point the necessity for putting down that first digit at all, then slashing it and reading it as "one less" than it was before it was slashed, may be obscure. Its value and utility in working from left to right will become apparent when we get into longer problems with many columns, so make sure you understand the process thoroughly.

Why the Process Works

After visualizing the way complements function in adding, you have perhaps already seen the reason why the reverse should be true in subtracting. Let's go through a similar group of comparisons, however, to drive the point home.

Remember that group of ten-rung ladders. You are now standing on the third rung of the fourth ladder. Your instructions are to step down exactly eight rungs. Where will you be standing then?

Obviously, you must drop down to the next ladder because you are only on the third rung of this one and you are to go down eight. If you descended a full ten rungs, you would then stand on the corresponding rung of that next-down ladder, or at the number 33. But you are to go down a number of rungs that fails by two (the complement of eight) to reach the corresponding rung—so you will be two rungs higher. You add the two, by which your eight-move fails to make ten, to the corresponding rung (three) and know that you will be on the fifth rung of the third ladder.

In simpler terms, 43 — 8 is 35. But you have arrived at this fact without ever subtracting 8 from (borrow) 3. Instead, you added the complement of 8 (2) to 3 to get the 5, and canceled a ten to reduce 4 to 3.

First, compare these two expressions:

Now see if you can feel the identity of these two expressions with the third, which describes our method of complement subtraction:

Using complements instead of subtracting a larger digit from a smaller digit gives you not just one, but two major advantages in speed and accuracy. First, most of us find the process of adding easier than subtracting. Second, your thinking is restricted to the twenty easiest digit combinations and five complement pairs; you never deal at all in the pair 8 + 5, for example, which is the digit-pair called for in our first expression 43 — 8. Instead, your thinking is converted to the simpler pair 3 + 2 by the use of a complement.

You also have a simple and highly automatic signal for the proper time to use a complement. In adding, it is when the two digits would add up to more than ten. In subtracting it is even easier. You use a complement whenever you would otherwise have to subtract a larger digit from a smaller.

Just remember, always, that subtraction is the reverse of addition. In adding, you subtract a complement. In subtracting, you add the complement—and always the complement of the digit being subtracted.

When adding, you record a ten every time you resort to a complement. When subtracting, you cancel a ten every time you use a complement.

Put the theory to use now by doing these four simple problems in the left-to-right method, using complements:
 
Easy as these are, they are designed to start you off with confidence in complement subtraction. Be sure to do them carefully and properly with the new technique.

The first example should develop like this: Nothing from 1 is 1. Put down 1. 4 is larger than 2, so do not subtract. Add complement of 4 (6) to 2. Put down 8, and immediately (before you forget) slash the 1 to cancel a ten. The answer is 1 8, or 8

Second: 1 from 2 is 1. Put down 1. 6 is larger than 5, so do not subtract. Add the complements of 6 (4) to 5 and put down 9. At once slash the 1 to cancel a ten. Answer, 19. or 9.

Third: 2 from 3 is 1. Put down 1. 8 is larger than 7, so do not subtract. Add the complement of 8 (2) to 7. Put down 9. Immediately cancel a ten by slashing the 1. Answer, 1 9, or 9.

The last example: 3 from 4 is 1. Put down 1. 6 is the same as 6. Nothing, or 0. No complement, no cancel. The answer is 10.
 
Perhaps the last one caught you. It was designed to. Complements only apply when we subtract a larger digit from a smaller. You will still subtract, about half the time, a smaller digit from a larger one or from one of the same value.

Why We Slash Digits

In the examples so far, it has really been a little childish to bother slashing digits in order to cancel tens. A fourth-grade schoolboy knows that 4 from 12 is 8. But you are exploring a new technique, a technique that applies not merely to 4 from 12 but also to 8,344,897 from 9,432,752. Learning to go through the proper steps is as important as learning to play the scales before tackling Chopin.

Play a few scales right now. First, make your complement-reaction just a little faster by "reading" the complements to these digits:

5 9 2 4 8 6 3 5 1 7

You will notice that in subtraction we now use both halves of each complement pair. We find it faster to use only the larger of each pair in addition, but you have to use all of them in subtraction. This is no problem, because there are still only five pairs. If you pause to wonder why we can pick which half of each complement pair we wish to use in adding, but have no choice in subtracting, notice that you can add 7 + 9 or 9 + 7 as you choose, but have no choice of complements in each of the two corresponding subtractions: 16-9 or 16 ― 7.

Subtract the following examples from left to right. Put down on your pad or card every digit as you go along, even if it seems silly. This habit is important to your successful handling of longer and more complicated problems. Whenever you come to a larger digit from a smaller, add the complement of the digit to be subtracted to the digit you are subtracting from, and cancel a ten by slashing the digit to the left in the answer:

One more point. A slashed 5 ($) is read as a 4, because the slash "borrows" or more properly "cancels" in the answer. But until this too becomes second nature, you may wish to rewrite answers before considering them finished. Remember that a slashed digit is reduced in value by one; then a subtraction answer that looks like this would be rewritten or would read like this 67623085.

After you have used this technique steadily for a few days, you will probably not bother to rewrite answers in this fashion. But until you have fully mastered the art of reading a slashed digit as one less than it was before the slash, you will profit by making sure you interpret such answers without error by rewriting them.

Take your pad now. Use it to cover the rewritten version of the following subtraction answer as you copy it in final form. Every slashed digit becomes the next digit smaller:

After you have rewritten this answer, compare your version with the one that follows. If you got any of the digits wrong, it would be worth while to do it again.

Here is how your copied answer should read:

4 2 7 6 1 0 9 5 4 2

Now try these examples. Remember to work from left to right, use complements where indicated, and "borrow" by slashing the preceding digit in the answer:

By this time you should be finding it a little easier to work rom left to right, and canceling tens in the answer—rather than "borrowing" in the larger number—should be beginning to feel natural. Once you become fully used to it you will find it far more natural and infinitely more foolproof than the older system.

It has been estimated that 80% of all mistakes in subtraction come from forgetting to borrow, or borrowing too much. Since we eliminate borrowing altogether, this method is by nature more accurate as well as faster.

Carrying Back Slashes

There is one more important element in this high-speed method of subtraction. This element is handling a slashed zero—0.

A slashed zero is, like any other digit, reduced in value by one. Since it must have a digit to the left of it in the answer (or you could not subtract), then obviously the zero must become 9 — and the digit to the left of it must also be slashed, to reduce it in value by one (since you "borrowed" from it in order to get 09 from 10).

Consider this:
 
You would read the answer as 9. If the example were £0, you would read it as 19. In practice, particularly in a long problem, it is important to slash both digits. In reading that last 20, you would read 2 as 1, and 0 as 9.

This may sound formidable, but it is really not as complicated as borrowing continuously to the left as you sometimes have to do in ordinary subtraction. Go through the steps in this example, and note where we start canceling tens:

Follow each step carefully: Nothing from 1 is 1. Put down 1. 5 from 5 is 0. Put down 0. 3 from 3, and so on, gives you zeros until you come to the final column.

In the final column, 9 is larger than 8. Do not subtract.

Add the complement of 9 (1) to 8. Put down 9, and cancel a ten by slashing to the left.

The digit to the left is 0. Slash it. Whenever you slash a 0, you must go back and slash the digit to the left of it too. That next digit is also a 0, so you have to keep on slashing until you slash a digit that is not a zero.

This may still sound a little strange. If you have any lingering doubts, do the problem above in the old-fashioned, schoolbook fashion. You will find that you have to do precisely the same thing, but in the more complex, error-prone method of borrowing over and over for each subtraction.

Try two longer problems now. Remember, as always, to practice the new technique as you do them. Work from left to right. Subtract a smaller digit from a larger digit just as you do now. But do not subtract a larger digit from a smaller. Instead, add the complement of the larger digit to the smaller digit and slash left in the answer. If you slash a zero, remember to go back a step and slash the digit to the left of the zero too.

The next chapter will carry you on to developing speed and accuracy at complement subtraction. Before you turn to it, however, let's cover another major advantage of adding and subtracting from left to right instead of from right to left.

Automatic Estimating

Any left-to-right method of doing arithmetic is self-estimating. Since you develop your answer from the left, the important end, you can always carry it exactly as far as you need for the accuracy you require and stop there.

Many of us have often tried to do this in the old-fashioned method of working when under pressure, but that is a backwards method and very difficult. Complement addition and subtraction does it automatically.

Suppose, for instance, you are production manager of a company making brass buttons. Your inventory as of the moment is 37,852 buttons. Today's orders total 16,965. The salesman selling to a large chain of stores calls to see how many buttons you could ship tonight on an emergency order. You must know, while he waits on the phone, about how many buttons you have.

Quick now: 1 from 3, 2. 6 from 7, 1. You have about 21,000 buttons. You have done merely the first two steps of your regular process in complement subtraction, instead of changing your method for estimating needs.

Suppose you need the next figure, too. 9 from 8. Add the complement of 9 (1) to 8: 9. Slash the digit to the left: 1. 20,900 buttons.

See how quickly and accurately you can give a three-digit estimate of the following subtractions:

For estimating—as well as for many rounded-off computations—you simply ignore the relatively unimportant numbers to the right, and carry your subtraction just as far as you wish.

The automatic estimating feature applies just as much to complement addition as it does to subtraction. The only thing to beware of in adding is that whenever you stop, the next column could make a substantial change in your stopped-at digit. In subtracting, the next column can never affect your stopped-at digit by more than a reduction in value of one.

These two examples illustrate this point:

The illustrations are admittedly extreme. A first-column-only estimate of the addition would give a rough total of 50, while actually the real total is 95. A first-column-only approximation of the subtraction would be 60, while the real answer is 51.
 
The reason why the first digit of the addition can be changed in this case by 4, and the first digit of the subtraction is changed only by 1, is that you might be adding any quantity of numbers and any two of them can add up to more than ten—carrying back as much as ten for each two numbers. In subtraction, you never deal with more than two numbers and the maximum amount that can be canceled is one ten.

In subtraction, the safe approach is to work out your subtraction to one more digit than you really need, and round off. In adding, carry your addition at least one more place than you really need and assume that the final digit is raised by one for each two numbers you have added, then round off; or else carry it two digits beyond the accuracy required and round off.

Try one estimate in addition at this point. Give a rounded-off three-digit approximation of the following problem (The section immediately following takes up rounding off, in case you are not acquainted with the technique.):

If you worked this out to four digits and assumed the last digit would be raised by 3 (since you added six numbers), your working figures would be 2874 plus 3, or 2877. This you would round off to 1,880,000,000. If you went to five digits, they would be 28770. You would still round off to 2,880,000,000.
A properly rounded-off three-digit estimate can never at worst be more than one per cent wrong, incidentally, and more usually is restricted to no more than one-half of one per cent. The maximum error would be in an estimate of 100 when the accurate answer is 101. An estimate of 999, if properly rounded off, cannot be wrong by more than one-tenth of one per cent. Numbers in between have a maximum possible error that increases as the first digit decreases, from 9 to 1, but it cannot go over one per cent. This, once again, is because each digit becomes just one-tenth as important as it moves one place to the right.

How to Round Off

If anyone doesn't know how to round off, he has missed one of the greatest time- and energy-savers in modern business. Traditional accountants kicked and dragged their heels until they had worked with it a bit, then became its most enthusiastic supporters.

Rounding off simply means expressing any quantity to the nearest standard unit. The standard unit may be whatever you say it is. In the three-digit estimates you just did, we in effect determined that the standard unit would be one in which there could not be an error greater than one per cent.

The standard unit in a U. S. personal income-tax report is one dollar. $3.99 is rounded off to $4.00. $3.01 becomes $3.00. To become a little subtler, $3.51 becomes $4.00 and $3.49 becomes $3.00. The usual rule is to give away an even half, and call $3.50 an even $4.00.

Any other standard unit that makes sense for a particular situation can be adopted. The operating and financial statements of many companies are rounded off to even thousands. $357,800 is expressed on the statement as 358—with a note at the top of the report, of course, that all figures are in thousands of dollars. Smaller companies may round off to tens or hundreds of dollars. Very large corporations may even round off to hundreds of thousands or, for certain purposes, to the nearest million!

At the other extreme, there is an almost forgotten currency value in this country of one mil—a tenth of a cent. It was used primarily in state sales taxes, before sales taxes went up to much higher rates. Naturally, people working with quantities of mils soon learned to round off their reports—to the nearest cent!

The most accurate way to estimate in adding or subtracting, as we have said, is to work out your figures to one place more than the accuracy needed, and round off. If the extra (not needed) digit is 5 or more, raise the preceding digit by one before reporting the estimate. If the extra digit is 4 or less, leave the final significant digit alone.

The theory is that roundings-off tend to cancel each other out in practice. You will add half or less to just about as many numbers as those from which you subtract less than half. To the surprise of many old-line accountants and bookkeepers, several test-runs of complicated reports and statements proved this to be completely true. The error is hardly ever likely to be larger than a single rounding-off.

Review quickly now the three secrets of speed in subtraction, before going on to practice that speed. The three major secrets—in addition to the over-all speed-math secret of "seeing only the answer"—are

1. Subtract from left to right.

2. Never subtract a larger digit from a smalle instead, add the complement of the larger digit to the smaller digit and—

3. Cancel tens in the answer by slashing, rather than "borrowing" in the larger number.

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