Speed Mathematics Simplified (36 page)

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 11'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”:

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 11'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:

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:

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.

Continuous Subtraction

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:

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:

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:

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 11'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: