**Canceling Nines**

I've always had a problem with mathematics. I guess my mind just doesn’t function like a mathematician’s who can follow abstract concepts that refuse to be clear for me. I think in pictures. This always gave me trouble in school and I tell my friends that the only course I ever took which had tests that I failed with some regularity was Accounting 101. However, I've always been intrigued with the number of magical things you could do with mathematics and one of the interesting things I learned from a fellow classmate at the time was the problem of the “accountant’s error” and a rather simple and unique way of discovering it.

It seems that accountants can make mistakes too and they most frequently occur when dealing with long numbers or long columns of numbers. The eyes and the mind get tired and a 5 can easily be mistaken for a 6, especially if the numbers are hand written. The funny thing is that when the accountant goes over the figures once more, to check them, he or she is prone to make exactly the same error again. This can even happen when using a calculator. In fact it can continue to even a third or fourth time in a row. That is what the “accountant’s error” is, the inability to see the error that was made.

If another person checks the calculation the error is quickly brought to light, but how can the accountant find it himself if he’s doomed to consistently and repeatedly make the same error? Well, there is a way explained my friend and it is called “canceling nines”. If you don’t already know it, here is how it works.

Let’s say you've added up this column and you want to make sure your answer is correct:

$ 444.63

10.52

216.11

1,376.88

150.07

__361.22__

__$2,559.43__

OK, the trick is to cancel all the nines in the column and

__all__the numbers that, when added together and are reduced to a single number, add up to nine; excluding those that are in the total, of course. Whatever number you have left over, keep in mind. Then do the same thing with the total you had arrived at originally and the number you have left over should be the same. If it isn’t, your total is incorrect.

You can do this in any manner you find easiest and quickest. You can look at each number horizontally or go down the column vertically. Read along carefully as I mentally go through the numbers horizontally.

In the first number I see a 9 (the 6 and the 3 added together). Cancel it, throw it out, chuck it, forget it. The three numbers left add up to 12 which I add together to reduce to a 3. I bring the 3 down to the next number. Immediately I see another 9 (the 3 I brought down plus the 1 plus the 5) and cancel it which leaves a 2. As I look at the next number I again see a 9 (2 + 1 + 6) and cancel it. I add the 2 I brought down to the remaining two 1s and get 4 which I bring down to the next line. There I see two 9s, a 1 and an 8 and a 6 and a 3. Left in that line are a 7 and an 8. I add them together for a 15 and add the 4 left from the previous line which results in a 19 The 9 gets canceled and left is a 1 (or I could have added the left over 4 to the 7 for 11 which reduced to a 2 and added to the 8 gives me 10 which reduces to 1 because zeros are nothing). That 1 with the 1 and 7 in the next line is another canceled 9 and 5 remains. The last line gives me another immediate 9 (the 3 and 6) and when I bring the 5 down and add it to the two 2s there’s another. All that’s left is a 1.

Now for the total -

There are two nines in the total, the 9 and the 5 + 4. The remaining 2, 5 and 3 add up to 10 which is a 1. The 1 from the column and the 1 from the total are the same so the total is correct. If they were not to match we would know our total is incorrect although we don’t know where the error occurred but a recalculation is called for.

This looks very complicated but it really isn’t. After you've used it a few times you’ll be surprised how fast you can do this – much faster than the original calculation – and it’s absolutely accurate.

As I mentioned you can follow any pattern you want although I find doing it line by line makes it easier to keep track of what you’re doing.

Canceling nines also works in subtractions and multiplication. It doesn’t seem to work for division; at least I've never been able to make it do so. I’m guessing that is because division can end in “left overs” or fractions.

Here are multiplication and subtraction examples:

1,476,029

__X 143__

211,072,147

When you cancel all the nines in the multiplicand you end up with 2, ergo.

Eliminate 1 + 6 + 2 and 9; 4 + 7 = 11 = 2

The multiplier is 8 (1 + 4 + 3)

8 X 2 = 16 = 7

Remove the nines from the answer (7 + 1+ 1 and 7 +2) and you have left 2 + 1 + 4 which comes to 7; so, 7 = 7 and the calculation was correct.

1,476,029

__- 310,428__

__1,165,601__

The first line reduces to 2 and the second to 9. You can’t subtract 9 from 2 but no, never mind, since 9s don’t exist the result remains 2.

When you reduce the answer you’ll get 2

If the second line had come out to a number different than 9 but higher than 2 you need to create a higher number for line 1 to have something to subtract from and you do this by keeping the 9 and adding it to the 2 to get 11.

Cool!