Speed Mathematics Simplified (34 page)

Locating Errors

Before going on to the ways of using digit sums in other types of problems, face one imperfection in the system—and learn a special advantage in return.

Digit sums do not invariably catch every type of error. The errors they miss are so unlikely that for all practical purposes you can almost forget them, but you should know about the possibility.

Since for digit-sum purposes 9 is the same as 0, you can easily see that this method of checking will not catch an error in which one digit in your answer is 9 when it should be 0, or 0 when it should be 9. If you have two correct digits, but have them reversed (36 instead of 63) it will not catch this either. Or if by any odd chance your error consisted of a digit or combination of digits that was exactly 9 more or less than it should be, the digit-sum check would not ferret this out either.

Actually, years of experience have shown that the errors not caught by the digit sum are exceedingly rare. For most needs, it is perfectly adequate—far more accurate than doing the problem over, in fact.

In return for these shortcomings, however, the digit-sum check offers a substantial bonus.

The digit sum will not only tell you if your answer is wrong; it will also tell you
how much it is wrong.
If the digit sum of your answer is 4, and you find that it should be 7, then you know that one digit of your answer is too low by exactly 3. You do not know which digit it is, but the fact that one digit is precisely 3 less than it should be is helpful in locating the error quickly.

Checking Subtraction

Our general rule is that you do to the digit sums of the numbers whatever you did to the numbers themselves. This result, reduced, must equal the digit sum of the correct answer.

In subtraction, it is important to recall that for digit-sum purposes we can consider 9 to be 0. This is because you will sometimes have to subtract a larger digit sum from a smaller. The way to do it is to add 9 to the digit sum that is otherwise too small to be subtracted from.

Here is an example of this situation:

You do not always have to add 9 to one digit sum before you can subtract the other. About half the time, the digit sum of the larger number will be as large as or larger than the digit sum of the smaller number. In this case, of course, you do not tamper with either digit sum; you simply subtract.

Another way to tackle the check when the situation is as above is not to subtract at all. You will get exactly the same result by
adding
the digit sum of the answer to the digit sum of the smaller number. This, if the answer is correct, must equal the digit sum of the larger number. Try it on the example above: The digit sum of the answer (4) plus the digit sum of the smaller number (8) is 12, which reduces to 3. This is the digit sum of the larger number. Check.

Try these subtractions and check them with digit sums. Remember to work from left to right, use complements, and cancel in the answer:

Because finding digit sums themselves is and should be entirely a mental process, you may not have used your pad recently. Locate it now and actually do the above problems and their digit-sum checks before uncovering the answers below.

Now compare your results with these:

If you have any difficulty in determining how the digit sums of the numbers in each problem worked to produce the digit sum of each answer, go back over the last two or three pages. You cannot subtract 3 from 2—but you can subtract 3 from 11, or add 3 and 8 to get 11, which reduces to 2.

Checking Multiplication

In checking multiplication, you follow the same general rule that applies to all digit-sum proving: since you multiplied two numbers, you multiply their digit sums. This result, reduced to a single digit, must equal the digit sum of the correct answer.

Here is an example:

Odd as it may seem to multiply digit sums together, that is just what you do in order to prove multiplication. As you can see, it works.

Suppose, though, that you set out to check a multiplication and found this result:

Something is wrong. The product of the digit sums does
not
equal the digit sum of the product.

The key here is that the digit sum of the answer is 1 higher than it should be—if the digit sums of the individual answers are correct. If the digit sum of the answer is 1 higher than it should be, then one digit of the answer is 1 higher than it should be, too.

Does this help you locate the error more quickly than you otherwise would? Try it and see. One digit of the answer is exactly 1 higher than it should be.

Try the two following examples on your pad, covering the answers below with the pad until you are finished. Work from left to right with the no-carry method, and check your answers with digit sums:

In order to check your answers to these problems, of course, you multiply the digit sums and reduce. Here are the results:

Don't forget that when any digit is multiplied by 0, the result is 0. So if the digit sum of
either
of the multiplied numbers is 0 (or 9) the digit sum of the answer must be 0. For instance:

In a case like this, keep in mind that despite the apparent extra dangers of multiplying by 0 (which would seem to permit any digit sum at all for the other number without changing the final check figure), the answer to the problem must also have a digit sum of 0 in order to check out. So it is as accurate as any other digit-sum proof.

Checking Division

When we come to checking division with digit sums, we have to use a special application of the general rule. Instead of trying to divide the digit sum of the divider into the digit sum of the number divided, work the process in reverse. Multiply the digit sum of the divider by the digit sum of the answer. This, reduced, should equal the digit sum of the number divided.

The reason for this special handling is illustrated by the following example: