Speed Mathematics Simplified (24 page)

Try this part of the technique once more. Do not bother to complete these examples. Just practice “seeing” the first answer digit by dividing with the first digit only of the divider, raised by one:

Now we will combine everything we know about multiplication and subtraction, both of which are continuously involved, with this simplified digit-finding technique, to make the complete shorthand division method both easier and faster than the cumbersome method of long division.

Shorthand Division

We began the explanation of no-carry multiplication by taking apart a sample problem and seeing how the answer develops. Let us do the same thing with a sample division:

We have called the process of division “continuous approximation.” The first approximation you got in the above problem was really 60, not 6: 7 goes into 441 something more than 60 times. You know this because there is obviously another answer digit to come.

We get the second answer digit by finding out first how much of the 441 is left after subtracting from it exactly 60 7's. In long division, we multiply the answer digit by the divider and put this product under the portion of the number divided that produced the digit.

That product here is 420. We do not ordinarily bother with the 0, any more than we bothered with the 0 in 60, since careful placement of each digit takes them into account.

By subtracting 42 from 44 and then “bringing down” the remaining 1 in the number divided, we find that there is 21 left over. Actually, we really subtracted 420 from 441. The “bringing down” completes that process. It would be helpful to inspect two expressions of this situation:

Dividing now the 7 into the remainder, we find that it will go exactly 3 times. In long division, we verify this by multiplying 7 x 3 and subtracting it from the remainder, getting a final remainder of 0.

Again, try to feel the identity of these two expressions of the current situation:

Now we will accomplish the same result with a fraction of the pencil work involved in long division.

The two most tedious parts of long division are (1) multiplying the answer digit by the divider and writing it down as you go, and (2) subtracting this product from the part of the number divided involved, in order to establish the remainder so far.

The European system, you recall, involves doing these two steps in your head. You write down only the final result of each subtraction. But this involves handling several digits at once in your head—contrary to the best approach to speed mathematics.

Since you know how to multiply from left to right, digit by digit, and also how to subtract from left to right, digit by digit
—without carrying or borrowing
—you can combine the two and accomplish the European result
without
ever handling more than one digit at a time.

We will use that same problem as the first model:

The following process, remember, is multiplication and subtraction done in one-two order, one digit at a time:

One: 7 x 6 is in the 40's, and 4 from 4 is 0:

We do not bother to write the 0. As you become accustomed to this system, you will not even bother to make any mark at all for this result. The
lack
of a digit there shows you that the result was 0.

Two: 7 x 6 ends in 2, and 2 from 4 is 2:

Traditional long division would now require you to rewrite the next digit of the number divided—1—beside the 2. You do not need to do this. You can bring it down mentally and see that the remainder is now 21, by reading the problem like this:

The next digit of the answer is 3, and you know this is right merely by inspection. Just to get the technique thoroughly established, however, we will verify it as you would in a more complicated problem:

One: 7 x 3 is in the 20's, and 2 from 2 is 0.

Two: 7 x 3 ends in 1, and 1 from 1 is 0.

Compare the work you have now finished with the same problem spelled out in long division:

Get out your pad and pencil and actively follow each step in this demonstration:

Although we have not mentioned it before, you naturally divide by any single digit without raising it in value by 1. If this divider were 84, we would divide by 9 because 84 is somewhere between 8 and 9. But 8 is obviously nothing but 8.