Speed Mathematics Simplified (15 page)

Step two: Now look at 4 x 8 to see what the right-hand, or units, digit of this pair is…what the full product of 4 x 8 “ends in.” The units digit is 2, the 2 of 3
2
. Remember that 2 for just an instant while you look at 7 x 8 to see what its left-hand, or tens, digit will be. 7 x 8 is in the 50's. Add this 5 to the remembered 2 and put down the total as the second digit of your answer. 2 plus 5 is 7, so your answer now looks like this:

Step three: Look at 7 x 8 again to see only what digit the product ends in. The right-hand, or units, digit of 7 x 8 is 6—the 6 of 5
6
. Put it down as the last digit in your answer:

Pause here for a moment to let this sink in. It is just as shocking an idea in its own way as is the idea of complements for adding and subtracting, and just as useful. But, as with complements, you need a bit of time to adjust to the thought.

There is one point in step two when you must remember one digit while “seeing” another one to add to it. Check the traditional process taught in school, however, and you will find that you had to juggle three digits at this point. You had to carry the 5 from 56 while noting the 32, then remember the 3 from 32 while adding the 2 and 5 and putting down 7. After that, you had to remember to put down the 3 from 32. The new no-carry method is at least one-third simpler—and produces the answer from left to right as well.

Here is another run-through to reinforce your grasp of this method:

Step one: 8 x 9 is in the 70's. Write down 7 as the first digit of your answer:

Step two: 8 x 9 ends in 2. Remember 2. 3 x 9 is in the 20's. Add the remembered 2 and the 2 from the 20's and put down 4:

Step three: 3 x 9 ends in 7. Put down 7 as the last digit in your answer:

If any element along the way does not seem to make sense, go through the three steps again with pencil and pad. This is really an incredibly simple idea, but it is vastly different from the way we were taught to work with numbers.

Now we will try one more, adding another digit. This means simply that we shall do step two twice. More properly, steps “one” and “three” are special steps for the extreme left and right digits of the number multiplied. Step “two” is the step done for every pair of digits across the number multiplied; once for a two-digit number, twice for three digits, and so on.

Here is how it works with a three-digit number:

One: 5 x 7 is in the 30's. Put down 3:

Two (1): 5 x 7 ends in 5. Remember 5. 3 x 7 is in the 20's. Add 5 and 2. Put down 7:

Two (2): 3 x 7 ends in 1. Remember 1. 2 x 7 is in the 10's (teens). Add 1 and 1. Put down 2:

Three: 2 x 7 ends in 4. Put down 4:

That is the basic system. It is that simple, and that revolutionary. If there had been twenty digits in the number multiplied, you would simply have repeated step two until you got to the end.

Get out your pad, open to a clean page, and go through the steps exactly as described for the following example. Do not try it on other random problems yet, however, because there are two special techniques for special cases yet to be revealed.