Chapter 11 - Accuracy: The Back-Up Check

The digit-sum, or "casting out nines," method is the quickest and easiest way to check any problem. Once you become fully accustomed to it, you will find yourself checking a problem about as quickly as you could read it over.

It is not, however, completely foolproof. The last chapter explained the types of errors to which it is quite blind. As someone once pointed out, the digit-sum method will tell you that a problem is wrong, but it will not tell you for sure that it is right.

This chapter explains how to "cast out elevens." This is a little slower but inherently more accurate than casting out nines. In cases of critical accuracy, some experts advise using both methods. You can easily do one right after the other in much less time than it would take to check by conventional methods, and if both your digit-sum and your "elevens" results check out, you can be quite sure you have a perfect answer.

Casting out elevens, or simply "elevens" as we will call it, works on precisely the same check-figure method as does casting out nines. In fact, adding up the digits is really only casting out nines because the proof of a number's divisibility by nine is the addition of its digits. If the sum is nine (or 0), the number is exactly divisible by nine. Any other result is the remainder you will have after dividing by nine.

Both casting out nines and casting out elevens are merely special (and convenient) applications of a general rule. You could check a problem by "casting out" any number at all. You could find the remainder of each number after dividing it by four, say, and use these remainders as check figures. Nines and elevens are merely the easiest numbers to cast out that also depend for their divisibility on every digit in the number.

This use of a division-remainder is not as odd as it might sound at first. If you add a series of numbers exactly divisible by four, then their total must obviously be divisible by four. If one of those numbers has a remainder of two after a division by four, then the answer must also have a remainder of two after a division by four. If you multiply two numbers each of which is exactly divisible by seven, then their product must also be exactly divisible by seven.

When the numbers are not exactly divisible by whatever number you use for your check figure, then the remainders of each number get carried along through the arithmetic too, and once you do to these remainders whatever you did to the numbers themselves, they must come out in exactly the same relationship to the remainder of the answer.

In order to get a clearer understanding of what is behind this general method of checking, try "casting out" the fives in the following example. That is, use as a check figure the remainder of each number after dividing it by five:

                                                Problem                       Check
                                                7                                  2
                                                X 8                              3

56        2 x 3 is 6. Reduce by dividing by 5 and showing only the remainder: 1

Five-remainder of answer: 1. Right.

Note that in none of these check figures do we count the answer to any division by the "base" of our check figure. It is only the remainder we watch—because the remainders must stay in order through the calculations. If the remainders do not check out, we know the answer is wrong.

As a general exercise in number sense, try "casting out" the sevens in the next example. Your check figure in each case is now the remainder after dividing by seven, and you use the check figures just as you would use digit sums:

5 8,792
-49,296

Cover up the explanation below with your pad while you do this problem (from left to right, canceling in the answer) and then check your results by dividing each number by seven and using only the remainder as your check figure. Handle the check figures just as you would digit sums.

Here is the working:

Problem           Check

Check figure of answer: 4

The one weak point of casting out any single-digit number for checking purposes is that any one digit in your answer that happens to wrong by the exact size of the digit you are casting out will not be caught. "Casting out" (or dividing by) a two-digit number is by nature more accurate. The easiest two-digit number to cast out—which also depends on every digit in the number when casting it out, unlike ten for example—is 11. There are three different ways to test divisibility by 11, or to determine the remainder after a division by 11 to use as a check figure. None of them is quite as simple as adding up the digits (which casts out nines), but with a little practice it goes quite fast.

Dividing By Eleven

In your work with numbers in the past, you may have learned to recognize numbers exactly divisible by eleven because of the pattern they form.
 
All two-digit numbers divisible by eleven, for instance, are paired digits—from 11 through 99.

For two-digit numbers, then, you can quickly get the elevens-remainder by subtracting from the number (mentally) the next lower number with paired digits. Here are some examples:

Note with special care that next-to-last example. When you cast out elevens, nine is no longer "0." Nine is "0" only for digit-sum purposes. Both nine and ten are check figures you will use when casting out elevens. When you cast out elevens, eleven becomes 0. Since you are using remainders as check figures, within the check system the number you cast out becomes 0.

The check figure of 88, when you cast out 11's, is 0. The check figure of 98 is ten. The check figure of 97 is nine. Don't forget and call it 0.

Numbers from 100 to 999 also form a particular pattern when exactly divisible by 11. The two "outside" digits of any three-digit number will (when added) equal the "middle" digit or else exceed it by 11—if the number is divisible by 11. In other words, a three-digit number is exactly divisible by 11 if the sum of the first and third digits equals the middle digit or else exceeds it by 11.

Here are some examples:

At this point, the pattern becomes more of a figuring job and less an obvious shape you can "scan" as you glance at the number. The above examples, particularly if you test them out by dividing with 11 and watching why the patterns form as they do, is an excellent exercise in number sense. Just as important, however, they lead to two general rules for determining 11 's remainders.

Numbers divisible by 11 continue to form patterns, but more complicated ones, as the number of digits goes above three. The patterns, however, are the reasons why the rules work. Try the first rule on the above numbers to gain some feeling of why it works.

Odd and Even Digits

A quick way to extract a check figure based on division by 11 is to subtract the total of all the digits in even places (starting from the right) from the total of all the digits in odd places.

In the first example above, the only even-placed digit is 9. (Even, of course, means divisible by two.) The first and third, or odd, digits (starting from the right) are 1 and 8. These total 9. 9 from 9 is 0. The 11's remainder is 0.

In deciding "odd" and "even" places, you always start from the right. This is the only place in the entire book where you are permitted to read a number from right to left, but you have to for this purpose.

In the last example above, the only even-placed digit is 0. The total of the two odd-placed digits (5 and 6) is 11. Perhaps you can guess that, since 11 is 0 for 1 l's-remainder purposes, you are in effect subtracting 0 from 0—or if the middle (even) digit were 2, you would be subtracting 0 from 2.

If you have any trouble remembering whether "even" or "odd" comes first—is to be subtracted from the other—just recall that E (for even) appears in the alphabet before 0 (for odd). In professional memory-expert circles this is called a mnemonic key. After a few days' disuse, such a key can be very useful.

Here is how this technique works with a few numbers you already can "feel":

23        46        308      154      429

In order, here is the working:
 
The even-placed digit (counting from the right) in 23 is 2. 2 from 3 is 1. This is the 11 's remainder.

In the number 46, you subtract 4 from 6 and find the check figure 2. Test this against dividing 46 by 11 and finding the remainder.
For 308, the even-placed digit is 0. Subtract this from the sum of the odd-placed digits (3 plus 8) or 11. The result is 11. For 1 l's-remainder purposes, this is 0.

Do the last two on your own.

Now one complication creeps in. Sometimes, you will find that the total of the even-placed digits is greater than the total of the odd-placed digits—and not always by an exact 11, which we consider to be 0. Consider:

6 9 1

The only even-placed digit is 9. The total of the odd-placed digits is 7. You cannot subtract.

But, as you might suspect in this system, you can add 11 to that 7 and then subtract. 7 plus 11 is 18. 18 minus 9 is 9.

The rule is this: When the total of your even-placed digits is smaller than the total of your odd-placed digits, add 11 to the total of the odd-placed digits and then subtract.

This method works on numbers of any length. In general, it is most useful for numbers of three, four, and five digits. Above that, another method will become more useful. First, however, reinforce your understanding of the even-from-odd method by trying it on the following numbers:

791      2,648   540      8,623

The 11 's remainders of these four numbers are, in order, 10, 8, 1, and 10.

The even-from-odd technique is useful primarily for numbers in which you can spot the even numbers and hold their total in your mind while adding the odd numbers, then (after adding 11 if necessary) subtract. The optimum size for rapid "scanning" (after some practice) is four or five digits. For longer numbers, still a third alternative becomes most useful.
 
For any number, no matter how many digits it contains, there is a technique for finding the 11 's remainder in one continuous process from left to right. It is not (alas) quite as much of a snap as adding up digit sums, but it is as simple as we can make it. Once you really learn the technique, you will find it amazingly swift.

The method is to subtract the first digit from the second, this result from the third, this result from the fourth, and so on through the very end of the number. If any succeeding digit is too small to be subtracted from, add 11 and then subtract.
Notice how it works on a simple example:

1 3 4

Start by subtracting 1 from 3. Answer, 2. Now subtract this answer from the next digit: 2 from 4. Answer, 2. Test the correctness of this 11's check figure by finding the remainder by the even-from-odd method: 3 from the sum of 1 and 4 is also 2.

Try the continuous subtraction technique on this number:

13 5 7 9

Working from the left, the process goes: 1 from 3 is 2, from 5 is 3, from 7 is 4, from 9 is 5. 11's remainder, 5. Verify it, if you wish, by subtracting the total of the even-placed digits from the total of the odd-placed digits: 7 plus 3 is ten. 9 plus 5 plus 1 is 15. 10 from 15 is 5.

So far, continuous subtraction seems almost as easy as digit sums. Now, however, try it on the same number reversed:

9 7 5 3 1

To start with, you cannot subtract 9 from 7. First you must add 11 to the 7, then subtract: 9 from 18 is 9. This 9, in turn, cannot be subtracted from 5. It can, however, be subtracted from 5 plus 11:9 from 16 is 7. Once more, you have to add 11 to the 3 before you can subtract: 7 from 14 is 7.
 
Adding 11 to the final 1, you find that 7 from 12 is 5. The ll's remainder is 5.

This number is an extreme. On the average, you have to adjust with an extra 11 in about half of the digits, not all of them. A more typical process would go like this:

4 6 17 9 8

Here is how it goes: 4 from 6 is 2, from 12 (1 plus 11) is 10, from 18 (7 plus 11) is 8, from 9 is 1, from 8 is 7. ll's remainder, 7.

Take special care to go through any zeros at the end of the number. Zeros after a decimal point do not count (unless followed by another digit), but zeros before a decimal must be included in your calculation. For instance:

1 0 0

You can "feel" what the ll's remainder of this is by mentally subtracting the next-lower two-digit number with paired digits: 99 from 100 is 1. Continuous subtraction, for demonstration, would go like this: 1 from 11 (0 plus 11) is 10, from 11 (0 plus 11) is 1.

Use of Complements

If you have learned your complements thoroughly, you will find that they can speed up this process. You subtract, of course, by adding the complement of the number to be subtracted to the number from which you are subtracting—if the number to be subtracted is larger than the other.

You can make a routine of this for continuous subtraction, with the extra little kicker that you add one extra 1 each time you use a complement. This gives the same result as adding 11.

Try this technique on this number:

8 4 2 5 3

Complement-kicker subtraction goes like this: Complement of 8 (2) plus 4 plus 7 is 7; complement (3) plus 2 plus 1 is 6; complement (4) plus 5 plus 1 is 10; (no complement) plus 3 plus 1 is 4. 11 's remainder, 4.

Checking Addition

Except that you extract your check figures in a different fashion, proving your answers with 11 's works precisely the same way as checking with digit sums. Find your 11 's remainders, do to them whatever you did to the numbers, and the result must equal the 11 's remainder of the correct answer.

When adding, you add the check figures, reduce if need be by casting out the 11 's of your total (you can no longer reduce by adding the digits, remember; that is for digit sums only) until you have a final check figure of 10 or less. This is equal to the 11 's remainder of the answer.

Follow the checking of this problem step by step:

Problem Check

11 's remainder of answer: 2 from 4 is 2 from 8 is 6 from 11 (0 plus 11) is 5; or 10 (8 plus 2) from 15 (4 plus 0 plus 11 to adjust) is 5.

Try this one on your pad:

Work out the answer and check it with 11 's before comparing your results with this explanation:

The check figure of 638 is 0; of 147 is 4; of 269 is 5.
 
The total of these is 9. The correct answer is 1054, which has a check figure of 9: 1 from 11 (for the 0) is 10, from 16 is 6, from 15 is 9. Or the even-placed digits 5 and 1 total 6, from 4 plus 0 plus 11 (to adjust) is 6 from 15, or 9.

Checking Subtraction

In subtraction, just as in using digit sums, you subtract your check figures to see if the result equals the check figure of your answer. If the check figure of the larger number is smaller than the check figure of the smaller number, add 11 to it before subtracting. If you prefer, add the check figures of the answer and smaller number; this must equal the check figure of the larger number.

Problem Check

Check figure of answer: 9.

Try this one on your pad before looking at the answer and its proof:

Remember to work from left to right and cancel in the answer.

The 11 's remainder of the larger number is 5, of the smaller number is 3. 3 from 5 is 2. The check figure of the correct answer, 108352, is 2. Right.

Checking Multiplication

You prove your multiplication answer by multiplying the check figures of the numbers you multiplied to see if the result—reduced by casting out ll's—equals the check figure of your answer.

Problem Check

Check figure of answer: 3. Try it yourself. Now carry one through on your own:

Cover the answer and its proof with your pad until you have finished.
The ll's remainder of 735 is 9. The check figure of 48 is 4. 9 x 4 is 36, which reduces (3 from 6) to 3. The correct answer is 35280, and has a check figure of 3.

Checking Division

You recall that in checking division with digit sums, you could not divide the digit sums even though you had divided the numbers. This is inherent in all check figures because (with the two remainders we use as check figures) either 9 or 11 is "0."

Just as in checking with digit sums, you check with ll's by multiplying the check figure of the answer by the check figure of the divider—adding the check figure of the remainder, if any—and seeing if this equals the check figure of the number divided.
 
Here is an example:

Problem Check

The check figures work like this: 11 's remainder of answer (0) times remainder of divider (3) is 0, plus check figure of remainder (2) is 2. The 11 's remainder of the number divided is 2. Everything checks out.

Try this one, working out the solution in shorthand division and checking it by casting out ll's:

Cover the explanation with your pad until you have finished.

The answer is 248, which has an ll's check figure of 6. There is no remainder. The check figure of the divider is 4. 6 x 4 is 24, which reduces (2 from 4) to 2. The check figure of the number divided is also 2. Perfect.

Duplicate Proofs

Several times, we have mentioned the advisability in critically important cases of double-checking. Unlike the traditional double-check of doing the problem over twice in opposite directions, the use of both 9's and ll's gives an absolute, unquestioned proof of accuracy—completely divorced from the human possibility of multiplying 4x8 and getting 28 three times in a row.

Here is one final example of division, the trickiest both to solve and to check, worked out and proved in both ways:

Problem 9's Check ll's Check

Proof with 9's: 3 x 1 is 3, plus 1 is 4. Check figure of number divided is 4.

Proof with ll's: 9 x 2 is 18, which reduces (1 from 8) to 7. 7 plus 10 is 17, which reduces (1 from 7) to 6. Check figure of number divided is 6.

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