Speed Mathematics Simplified (26 page)

Now let us pick up the final technique from multiplying that enables you to handle shorthand division with dividers of any length.

No-Carry in Division

So far, you have been dividing by only one digit. In such divisions, you would ordinarily do most or all of these steps entirely in your head anyway and not worry about putting down the remainders as you went along. It is really short division, and we have started with this only to get the general technique firmly established.

When you divide by numbers of two or more digits—by 653, for instance—you will add to your simultaneous left-to-right multiplication and subtraction the efficient and handy no-carry system.

This is the point where the European shorthand method becomes really difficult for most of us. When dividing by a number of two or more digits, the European system requires you to multiply (including carrying) and subtract (including borrowing), all in your head. This can involve juggling as many as six digits all at once in your mind. With our left-to-right methods, however, we can do everything digit by digit.

If you understand thoroughly both the no-carry multiplication method and the division method covered so far, then you could probably work out the entire method yourself without further help. Since it is using no-carry multiplication within a new framework, however, we will go into the entire process step by step.

Recall, as we get into this, the technique of dividing by only the first digit of the divider—raised by one:

Our first answer digit we “see” by dividing 4 into 13, and putting 3 over the 5. Remember that the first answer digit starts as many places over the number divided as there are digits in the divider—plus one if you start dividing into two digits instead of one.

Now we develop the remainder:

One: 3 x 3 is in the zeros, and 0 from 1 is 1:

Two: 3 x 3 ends in 9. 8 x 3 is in the 20's. 9 + 2 is (complement of 9 from 2) 1, and record. Now you have
two
things to do: subtract 1 from 3, and record the ten. Since we are subtracting while we multiply, this ten obviously gets
subtracted.
How? Just by canceling; slash left:

Three: 8 x 3 ends in 4, and 4 from 5 is 1:

There was a curve hidden in point two of that example, but it seemed best to slide it in quietly. It is a new application of recording by slashing, because the only digits we jot down are the results of the subtraction and the final effect of a recorded ten from the multiplication is obviously to
reduce
the preceding digit in the answer to the subtraction by 1.

So one of the side-rules of speed division grows from this: when your no-carry multiplication involves a complement (and therefore a recorded ten), slash the digit to the left in your working figures.

Our remainder so far is 2,128. 38 will go into 13,528 300 times, with 2,128 left over. If the size of these figures jars you, inspect the work so far and think back to the identity-expressions at the start of this section.

Note that we have two digits in the answer to the subtraction. We still bring down the next digit (mentally), so the next division is 38 into 212. We “see” it as 4 into 21, and put down 5. Now we develop the remainder:

One: 3 x 5 is in the 10's, and 1 from 2 is 1:

Two: 3 x 5 ends in 5. 8 x 5 is in the 40's. 5 plus 4 is 9, and 9 from 1 is—larger from smaller. Add the complement of 9 to 1, and put down 2. Cancel by slashing to the left:

This brings up an interesting point. You have
two
occasions to slash to the left when doing shorthand division: when a complement is used in no-carry multiplication, and when a complement is used in subtraction. Both involve the use of a complement, and both result in a slash to the left. One special result of this will develop later.

Three: 8 x 5 ends in 0, and 0 from 2 is 2:

The remainder at this point is 228. This is the excess after subtracting 38 x 350 from 13528.

Before going on to the final digit of this answer, and determining the remainder (if any), recall our earlier comments about revised digits. Dividing by only the first digit of the divider is the quickest and easiest way to produce the next digit of the answer, but sometimes it will need revising. This is a price paid happily by operators of the high-speed abacus, because it saves more time in producing each digit than it costs in revision.

The final digit of this answer will demonstrate such a case.

See the next digit of the answer as 4 into 22—5. Now let us work out the remainder (if any) and see what happens.

One: 3 x 5 is in the 10's, and 1 from 2 is 1:

Two: 3 x 5 ends in 5. 8 x 5 is in the 40's. 5 plus 4 is 9, and 9 from 2 is—complement of 9 plus 2, and slash:

Three: 8 x 5 ends in 0, and 0 from 8 is 8: