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© Copyright 1999, Jim Loy
When we do arithmetic, we should normally check our work. There are several ways to do this. Our first step should be to ask, "Is the answer reasonable?" Sometimes the answer is way too big, or way too small, or has the wrong number of decimal places. If it looks reasonable, then we check the work. One way is to re-do the arithmetic. This doesn't always catch our errors, as we may make the same mistake, in the same place. Humans are prone to that kind of mistake. A better way is to re-do the arithmetic, in a different order. Add a column of numbers, from bottom to top. Or, have someone else do the arithmetic.
One method, of checking our work, is called "Casting out nines." We convert each number into its casting-out-nines equivalent, and then redo the arithmetic. The casting-out-nines equivalent of this answer should be the casting-out-nines equivalent of the original answer. We get the casting-out-nines equivalent of a number by adding up its digits, and then adding up those digits, until you get a one digit number. If our answer is 9, then that becomes 0. As a short cut, we don't have to add in any of the 9's in our work, as these are the equivalent of 0. We can just "cast out" those 9's. For example, 19 becomes 1, without even adding 1 and 9 and getting 10, and then adding 1 and 0 and getting 1. As a further short cut, we can group numbers together which add up to 9, and replace them with 0. 2974 becomes 4, because we can cast out the 9 and the 2+7 (which is also 9 or 0). Well, let's try an arithmetic problem:
137892 3 + 92743 + 7 ------ -- 230635 1
We converted the first number to 3, by casting out nines (We threw out the 9 and the 7+2 and the 8+1, leaving the 3). The next number became a 7, for similar reasons. The answer became a 1. And, to check our answer, we add the 3 and 7, getting 10, which is 1. It checks out. We get the same answer in both directions.
We can use the same method for subtraction, multiplication, and division (check the division backwards, by doing the multiplication). It doesn't always catch our error. Many numbers reduce to 1. So, we have 8 chances in 9 of catching an error. 1/9 of all our errors will go uncaught. That is pretty good, really. Here's a multiplication problem:
137 2 x 92 x 2 --- -- 12604 4
Again, we get the same answer in both directions. We can be moderately confident of our answer.
Why does this work? Why doesn't casting out sevens work? Well, first of all, nine is not a magical number. It only works because we use a base-10 number system. If we were using base-8, then casting out 7's would work just fine.
Let's look at one digit plus one digit. 2+3=5 gives us no problems. 2+3=5 is the same, whether we are adding the actual numbers or their casting-out-nines equivalents. It is only when there is a carry that we have something different: 3+8=11 and 3+8=2. This works just by the definition of casting out nines. The answer 11 converts to 2. And so does 3+8, because 11 is the first step in casting out nines for 3+8. All of the other possible sums of one digit plus one digit work just as simply.
Casting out sevens (or elevens) would not work very well, because one digit plus one digit does not always work as they do (using nines) in the previous paragraph. The carries mess things up. See Casting Out Elevens.
Well basically, every addition problem involves adding many pairs of single digits (in some of these, a carry is one or more of the digits). Well, the whole addition problem should work, because every one of its smaller operations (between pairs of digits) works. Similarly, multiplication should work, because multiplication is just repeated addition. And subtraction and division should work, because we can change these problems into addition and multiplication problems.
Note: This article involved Modular Arithmetic. I avoided mentioning that, and I avoided using the methods of modular arithmetic, so I wouldn't scare anyone away. Modular arithmetic is fairly simple, in general.
My spell checker did not like the word "+." Did I misspell that? It didn't like "uncaught" either. And I don't have a grammar checker. So, I got away with "The carries mess things up."
Someone asked me about casting out nines and division. My response:
Well, I suspect that it is not very useful with division, but it seems to work just fine:
22 4 ------ ---- 17 ) 377 8 ) 8 34 5 -- - 37 3 34 -- 3
The 8/8 doesn't work, but the multiplication works, giving the right remainder. We could also verify every step (they are just multiplication and subtraction) with casting out nines. I guess we can't do any dividing of the numbers after we have cast out nines. But we can do all of the other operations involved (multiplications and subtractions). So x/y=z with a remainder of w means that x-w=yz (definition of remainder). There is no division there, so we can cast out nines to see if x, y, z, and w are correct. I think that works.
How about negative numbers, like 4-19? 4-19=-15. Casting out nines we get 4-1=3. So -15 is 3 casting out nines. Just add a couple of nines (-15+18) to see that 3 is right.
Here is a product:
87 6 x 13 x 4 --- -- 1041 6
Is the answer right? Even though casting out nines suggests that the answer is right, it is wrong. Casting out nines does not catch every error. Here the answer is 1131. Some problems are more error prone than others. In fact multiplication is always somewhat error prone, as one of the two original numbers may be a product of 9. Then almost any answer will have a casting-out-nines equivalent of 0.
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