Speed Mathematics Simplified (19 page)

Step two: 2 x 8 ends in 6. 6 x 8 is in the 40's. 6 and 4 are complements. Zero, record:

Step three: 6 x 8 ends in 8:

Now you simply add these lines from left to right. The easiest way to handle the recorded tens is to add each underline as one, rather than rewrite the two lines before adding:

This should be clear through every step. If not, recheck your understanding of the preceding pages.

There is one special point to watch carefully. You recall the stress we put on putting down a first digit for each line, even if that first digit happens to be a zero. If you forget to do this, your lines for each digit in the multiplier will get out of order and your answer will be wildly wrong. Notice how it works in this example:

If you had not put down the zero in front of the 7 in that first line, you might very possibly have lined up the two partial-products like this:

Not only is the answer absurdly wrong, but it is the type of error that is infernally hard to catch. You might do the problem over several times, get every line right, and still get the wrong answer. So watch, very carefully, your first-place zeros.

Go through the process once yourself. Doing is the secret of remembering:

Cover the solution below with your pad while you work out this example.

If you remembered that you always put down a left-hand digit even if it is a zero, then your answer looked like this:

Note how we handled the two recorded tens in adding the lines of partial answers. You are already learning to read underlined digits as increased in value by one. Occasionally, in very complex multiplications with three, four, or more digits in the multiplier you will need to record more than one ten for the same place. That is, sometimes you will need to add two or three to a digit already put down as you work from left to right.

In any multiplication likely to become this involved, you may wish to handle the addition of partial answers just as if it were a major addition problem, and record tens on your fingers to be noted beneath each digit to the left. But multiplication seldom becomes this complex. In most cases, you will find it quicker and easier to use two or even three underlines when you need to.

Now it is time to handle a couple of two-digit multiplications entirely on your own. Cover up the answers below with your pad while you use it to work out the answers. Use your new techniques exclusively.

After you have finished, compare your working with these answers:

Larger Multipliers

When you get into three- and four-digit multipliers, you simply follow the steps you already know. They are no different in kind, only in degree. Just add one more line for each new digit in the multiplier, stepping one place to the right for each line, and remember to put down a left-hand digit even if it is a zero in order to keep the lines in proper order.

There is no magic to getting the right answer for a long multiplication problem. There are two ways of getting very rapid, quite accurate estimates—the slide rule, and the self-estimating feature of no-carry multiplication—but for a full, complete answer you simply have to go patiently through all the steps. Those steps are made more natural and easier as well as quicker by this method.

Watch the step-by-step development of the answer to the following example. You should be able to understand why each new digit appears without trouble. If anything does not seem clear, review the last few pages and try again. It might be a good idea to follow with pencil and pad, too.

Now add, from left to right of course: