Chapter 10 - Accuracy: The Quick Check

Producing a quick answer is not always the end of a problem in arithmetic. A wrong answer can be worse than none at all.

When you balance your checkbook, you care whether every stub was done perfectly because otherwise you face an unpleasant hour or so finding out why your balance does not agree with the bank statement. When you make out your income-tax return, you check and double-check every operation to make sure you are not paying too much, or else getting yourself into trouble with wrong arithmetic. And in business, where so many decisions are based on numbers, the wrong numbers can lead to wrong decisions.

There are two fast ways of checking your answers. The faster of the two is the subject of this chapter. A slightly more complex "back-up" check is discussed in the next chapter. Both of them are infinitely quicker than the standard technique of doing the problem over.

The standard way of checking an answer is effective, but very slow. It takes just as much time to tell whether an answer is right as it does to produce it in the first place. The standard way, of course, is to do the problem over again in the opposite way. If we got the answer by adding down, we check by adding up. This (to some extent) keeps us from repeating some habitual error that we might commit twice if we handled the figures in the same order both times. If we subtracted, we check by adding the answer and the smaller number to see if the total equals the larger number. If we multiplied 897 by 123, we check by multiplying 123 by 897. And if we divided, we check by multiplying the answer by the divider, adding the remainder, and seeing if the result equals the number divided.

There are two serious weaknesses to this "backward" method of checking.

First, it is slow and rather boring. Our object is speed and accuracy, with as little boredom as possible.

Second, it is not really a proof at all. If we get the same answer both times, we assume the first solution to be correct. Yet if we habitually think of 4 x 7 as 32 (and such habitual mistakes are not uncommon), then we might indeed get the same wrong answer twice. Even if our second try produces a different answer, we still do not know if one of them is right —or which one it is. We must do the problem still a third time.

The techniques of simplified mathematics you have learned are inherently more accurate (because they are simpler) than traditional methods, but it is still unwise to assume an answer is correct unless you know it is correct because you have checked it.

The two methods for checking answers you are about to learn are very similar. Neither one is new, although some of the short cuts in applying them are. The first method is known in mathematical circles as "casting out nines" or "the digit sum" method. The second is "casting out elevens." Both work on a system of check figures completely divorced from your calculations in solving the problem, so habitual errors are unlikely to be repeated, but the methods of deriving the check figures are quite different. Actually, each of them is a way of testing whether the remainders of nine or eleven remain properly constant through your calculations. This will be discussed at greater length in the next chapter. First, learn the technique of handling what is known as the digit sum.

The Digit Sum

The digit sum, as the phrase suggests, is simply the sum of all the digits in a number. This sum will be your "check figure" for each number.

Learn first how to find a digit sum. Then we will go on to the ways of using it. After you have found a few digit sums, you will be able to derive one almost as fast as you can read the number itself. It is really that quick.

If the digit sum is merely the sum of the digits in a number, then the digit sum of 23 should be 2 plus 3, or 5. Odd as this may seem at first, that is precisely right. The digit sum of 23 is 5.

The digit sum of 341 is 3 plus 4 plus 1. The digit sum of 341, then, is 8.

Just add the digits. The digit sum of 42 is—

Did you get 6?

Now, however, it becomes a little trickier. For quick utility, the check figure must always be a single digit. But the sum of the digits in longer numbers goes over ten.

In this case, we use the digit sum of the digit sum. This is the digit sum of the number itself.
This is how it works. The digit sum of 587, for instance, goes into two digits by the time we add 5 and 8, which make 13. When we add the final 7, we have a digit sum of 20.

You can reduce this to a single digit at the end, by adding 2 plus 0 and getting 2. Or you can reduce as you go along, like this: 5 plus 8 is 13. Reduce this by adding 1 plus 3 to get 4. 4 plus the final 7 is 11. Reduce this by adding 1 plus 1 and get 2.

This peculiarity of the digit sum is only a foretaste of those to come. Let us finish this thought before getting to that, however. Try one digit sum now. Add all the digits of the number 6934 and then add the digits of the answer until you come out with a single digit. Then reduce as you go along through the same number 6934 and see if you come out with the same final digit sum.

Done the first way, you add 6 plus 9 plus 3 plus 4 and get 22. Reduce this by adding 2 plus 2 to get 4. The digit sum of 6934 is 4.

Done the second way, you add 6 plus 9 to get 15. The digits of this total 6. 6 plus 3 is 9, plus 4 is 13. 1 plus 3 is 4. The digit sum is still 4.
 
There is, however, a third way. This third way is called casting out nines. The reason for the name is inherent in the digit sum, and is a fascinating byway in the mysteries of numbers.

The odd fact boils down to this: If you divide any number by nine, the remainder is the same as the sum of all the digits of that same number—reduced to one digit by continually adding the digits of the sum of the digits until you wind up with one digit.

In other words, the digit sum of any number divisible by nine will be nine. The algebraic proof of this is a little complicated for this book, but you can demonstrate it for yourself.

Take one of our examples of a minute ago. We found that the digit sum of 587 is 2. If you divide 587 by 9, you will get an answer of 65—and a remainder of 2.

The last example we tried was 6934. Our digit sum was 4. Try dividing 6934 by 9. The answer is 770—and a remainder of 4.

The digit sum is, in essence, the same as the "nines remainder"—the amount left over after an even division by nine. This is important not only to digit sums but in understanding how the entire check-figure system works, A more complete explanation comes in the next chapter.
The fact that the digit sum is the same as the remainder after dividing by nine brings up two more useful oddities. First, nine (for digit-sum purposes) becomes zero. Second, a digit 9 counts for nothing in the number itself.

This brings up a great short cut in deriving digit sums. As you add the digits, simply ignore any nines. They do not count.

Demonstrate this to yourself a few times. The thought takes a little getting used to.

Add the digits of 19. The total is 10. The digit sum of of this is 1. If you looked at 19 and ignored the 9, you would see 1 anyway.

Now try 29. 2 plus 9 is 11, which reduces (1 plus 1) to 2. Look at the same number, ignoring the 9, and you see 2.

See if you can find any combination of two digits, of which one is 9, which, when you add the digits and reduce, does not produce the digit which was not 9. This is an intriguing and frustrating search. 95 becomes 14, which reduces to 5. 89 becomes 17, which reduces to 8. 93 becomes 12, which reduces to 3.

Do not stop with two-digit numbers. Try any number you wish, that contains a quantity of 9's and any other digit. Convince yourself of this very peculiar truth by reducing these numbers to digit sums:

1999 9949 969
399 9299 99989

This is a strange phenomenon, but in addition to being strange it is highly useful. It means that in finding a digit sum your eye can simply skip over any 9's. They will not change the digit sum. The digit sum of 99999999999997 will be 7.

Perhaps your mind is already ranging ahead, wondering if digits in a number that add up to 9 behave in the same way. If the digit 9 does not change the digit sum, what about 3 and 6?

Try it and see. Find the digit sum of 361. Actually work it out. Now envision the 3 and the 6 as adding to 9, and therefore to be ignored. Cast both of them out, as you would cast out a 9. Your answer, of course, is 1—the 1 you see if you ignore the 3 and the 6 (because they add to 9) in 361.

The lesson is quite true. Since 9 will not affect the digit sum, you may ignore any 9's you see in the number—or any combination of digits that add to 9.

Try these:

145 727 463 273

254 381 574 186

In each case, you will find that adding all the digits and then reducing by adding together the digits of the sum (as many times as you need to) is precisely the same as the digit left after casting out digit combinations that would add to 9 —no matter where those digits appear in the number.

Zeros, too, obviously count for nothing. You would not add them as you added the digits anyway, so you can safely ignore any O's in any number as you derive its digit sum.

For digit-sum purposes, 9 and 0 are equal. This is only a device for this particular purpose, of course. But for simplicity in working, consider a final digit sum of 9 to be 0. It would come out to the same result in the end, and it can save a significant amount of time to wipe out the 9 to start with.

Before you learn how to apply digit sums in checking your results to problems, try deriving a few. Ignore any O's, 9's, or combinations of digits adding to 9 in the following numbers as you extract the digit sum of each:

16428  32,718,643

73619  84,600,372

24583  26,738,514

Notice that one of the digit sums above works out to 9. This, for digit-sum purposes only, can be treated as 0.

Running Adjustment

One more short cut is worth noting in developing digit sums. Since you know that 9 or any combination of digits adding to 9 (such as 324) can be ignored, you can also think of any pair of digits adding to ten as being worth 1, or any pair adding to 8 as subtracting 1 from the partial total already in your mind, and so on.
Glance back at the first example above. You can do it with extra speed by counting "1—(6 and 4 are complements, count as 1) 2—(2 and 8 are complements, count as 1) 3."

In the last example, you might start adding like this: "(2 and 6 are 8, or minus 1) from 7 is 6—and 3 is 9, or 0— (8 is minus 1) from 5 is 4—and 5 (group 1 and 4) is 9, or 0. Digit sum, 0."

I think you already see how you will soon be able to derive the digit sum of a number almost as fast as you can read the number itself. You simply add up to 9 and then start over, dropping each 9 in turn and not even recording it. In doing so, you use every trick of grouping you have learned.
 
These extra-speed tricks are helpful to very rapid work. They can become so fast and so easy you could, if you wish, make a parlor trick out of the idea. Glance at any figure and predict the total of its digits, totaled in turn until you get a single digit. You will have your result, if you play with these methods a bit, before your challenger has added the first three digits.

Try it once on this number:

869,325,008,462,118

Watch how quickly it goes: "8 from 6 is 5—skip 9— plus 5 (the 3 and 2) is 1 (ten reduced)—plus 5 is 6—skip the 0's—minus 1 is 5—plus 1 (the 4 and 6) is 6—plus 4 (the 2, 1, 1) is 1 (ten reduced)—less 1 (the final 8) is 0."

After a few more moments of practice, you will find yourself almost scanning a digit sum. You will ignore pairs adding to 9. You will add 1 for pairs that are complements, and subtract 1 for 8's or pairs adding to 8. Beyond this, you may begin to note pairs adding to 7 (or 7's themselves) as subtracting 2. You may even begin to skip around a little, "seeing" 485 in a long number as minus 1 because the 4 and 5 add to 9 and the 8 is minus 1.

This is such a joyous and useful byway of numbers that you will profit by making a game of finding digit sums as quickly as you can.

Checking Your Answers

The digit sum is not merely fascinating. Its utility is in the quick check.

The general rule for checking by digit sums is simply this: Do to the digit sums of the numbers in the problem whatever you did to the numbers themselves. The result must equal the digit sum of the answer—if the answer is correct.

If you add a column of numbers, then you simply add the digit sums of those same numbers. This result (reduced as always to a single digit) must equal the digit sum of the answer. If you multiply two numbers, then you multiply their digit sums. This, reduced, must equal the digit sum of the correct answer.

The reason why it works will be explored in the next chapter. For the moment, let us see how it works.

Follow this example in addition:

                                    Problem                 Check

                                    146                        2

                                    928                        1

                                    357                        6

Totals                           1431                      0

Digit sum of answer: 0

In this case, the sum of the digit sums is the same as the digit sum of the answer. Check.

Once you are in full training at digit-sum reduction, you will be able to check such a problem about as fast as you read it over. A peculiarity of checking problems in addition, especially, is that since you added the numbers you can merely add all the digit sums in one operation. That is, you can develop one digit sum for the columns of digits in one operation instead of getting a separate sum for each number. In the problem above, it would be equal to getting a digit sum for 146,928,357. If you try it, you will find that this digit sum is 0.

Now for a longer problem. Each digit sum appears on a separate line for clarity, but you do not need to do it this way. You can go through the three numbers one after the other until you have one final digit sum—which in this case will be 3:

                                                Problem                          Check
                                                68,352                            6
                                                97,834                            4
                                                35,876                            2
Totals                                       202,062                          3

Digit sum of answer: 3

Now try these problems and check your answers by using digit sums. Be sure to work at your new habits: work from left to right, use complements, and record tens:

Cover the answers and their check figures with your pad until you have finished.

Here are the totals, together with their digit sums:

212 (5) 277 (7) 17,122(4)

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:

                                                Problem                       Check

                                                615                              3

                                                -593                             8

                                                22                                8

from 3 won't work. But 8 from 12 (3 plus 9) is 4.

Digit sum of answer: 4

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:

7,382   1,123   586,493

- 6,987 - 1,099 - 465,906

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:

395 (8) 24 (6)   129,587 (5)

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:

                                                Problem                       Check
                                                421                              7
                                                x 17                             8

7,157   7 x 8 is 56. This reduces to 11, which reduces to 2.

Digit sum of answer: 2
 
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:

                                                Problem                       Check
                                                568                              1
                                                x 4                               4
                                                2,372                           4
Digit sum of answer: 5

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:

362      874
x 43     x 736

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

15,556 (5)        643,264 (7)

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:
 
                                                Problem                       Check
                                                3 8                               2
                                                x 9                               0
                                                3 4 2                            0

Digit sum of answer: 0

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:

Problem           Check

Check: 4 x 8 is 32, which reduces to 5. This is the digit sum of the number divided.

You can easily see the trouble you would have trying to divide the digit sum of the divider (8) into the digit sum of the number divided (5) and produce any rational whole-digit result. The reason for this lies in the special reduction of digit sums, which pretends (for digit-sum purposes) that is 1, that 14 is 5, and that 9 is 0. The system works perfectly if you multiply as outlined above, but cannot possibly work if you try to divide.

If there is a remainder in the answer to the division, add one more step. First multiply the digit sums of the divider and the answer, as before. Now, however, add the digit sum of the remainder. This total, reduced, should equal the digit sum of the number divided.

Here is how it works:

problem            Check

Check: 2 x 5 is 10, which reduces to 1. 1 plus 2 is 3, which is the digit sum of the number divided. Right.
Now try these problems, using your pad to cover the correct solutions as you always do. Caution: Any digit multiplied by 0 must give 0.

The illustrations below will, as always, be in shorthand division. Look at them after you have finished your practice.

Problem Check

Check: 1 x 8 is 8. Right.

Problem           Check

Check: 0 x 8 is 0. 0 plus 3 is 3. Right.

This is all there is to know about digit-sum checking. The back-up check in the next chapter works the same way, but the check figures will be quite different.

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