Speed Mathematics Simplified (9 page)

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.