Number of the day: 9

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.

Voila!

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