Speed Mathematics Simplified (33 page)

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:

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:

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 0'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 0's, 9's, or combinations of digits adding to 9 in the following numbers as you extract the digit sum of each:

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:

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:

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:

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: