## Casting Out Nines

April 4th, 2013 by Math Tricks | 1 Comment | Filed in Math Tricks

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I thought that now would be a good time to introduce a very useful math trick.  Why is it a good time?  Well, because at the time of this writing, many of you are getting ready to get down and do your taxes!  So why is it useful?   Well, it will help you to catch any mistakes you will make while adding up your tax data – perhaps even preventing you to experience a dreaded IRS audit!

Actually, this handy math trick will be useful for many situations in which arithmetic operations are carried out.  It will help you to check for errors when you add up bowling scores, perform inventory counts, totaling card game scores – the list goes on.  This trick, called casting out nines, is simple to learn and very easy to implement.

Digit Sums

Before I explain about casting out nines, let me tell you about digit sums.  A digit sum is simply the result that you get when you add up all the digits in a number to the point where only one digit remains.  For instance, if you add up all the digits in the number 23, you get the digit sum 5 (i.e., 2 + 3 = 5).

The get the digit sum of 349, you first add

3 + 4 + 9  = 16

The result, 16, is not the digit sum – you must continue the process until only one digit remains.  So, continuing

1 + 6 = 7

So, the digit sum of 349 is in fact 7.

For another example, to see that the digit sum of 5725 is equal to 1:

5 + 7 + 2 + 5 = 19

1 + 9 = 10

1 + 0 = 1

The important property to keep in mind about digit sums is that you can remove any nines or any combination of numbers that add up to nine while you are summing up your numbers.  For example, the digit sum of 3954 is:

3 + 9 + 5 + 4 = 21

2 + 1 = 3

You can arrive at the digit sum much faster if you drop out the “9” and the “5 + 4” that adds up to 9, and get the digit sum of “3” directly.

Another example what is the digit sum of 452362?  This looks a bit intimidating, but if you drop out the numbers that add up to 9, you are left with 22, and the digit sum of 22 is 4.

The ability to drop out the 9s and the numbers that add up to 9 is where the term “casting out nines” comes from.  I will not go into depth here as to why this works – for a good explanation, I recommend that you check out this link.

Error Checking

You can use digit sums to check for arithmetic errors very quickly.  This method is easy to remember – it only takes a little bit of practice to get good at it.

Let’s start with an example.  Suppose you are totaling bowling scores in order to get 3-game series data.  Your first set of data may look like this:

132

169

181

482

I don’t know about you, but whenever I went out bowling, I would typically have a couple of beers, socialize with friends, eat junk food – in other words, a LOT of distractions!

So, tallying scores would be very open to errors, and you would need to double-check your results.  To do this in our example, simple determine the digit sums of each figure we are adding together (addends) and get the digit sum of each digit sum.  If the resulting digit sum matches the digit sum of the answer you arrived at, then there is a very good chance that there were no errors.  It is best to see an example, so let’s apply this method to our example:

132 – – – – – – – –  6

169 – – – – – – – –  7

181 – – – – – – – –  1    6 + 7 + 1 = 14; 1 + 4 = 5

482 – – – – – – – – 14; 1 + 4 = 5

So the digit sum of the addend digit sums is equal to the digit sum of the sum (that is some sentence!).  Therefore, the answer you arrived at is most probably correct.  If you made an error that was a multiple of 9 off of the correct answer, then you would not notice any indication of an error using this method, so be sure to carefully compare your sum to the addends before using this method; chances are that any large errors will be obvious.

Go ahead and try some examples on your own.  Try to see what happens when you have a mistake in your sum, and also see what happens when mistakes are in multiples of nines.  With a little bit of practice, you will get really good at removing nines and numbers that add up to nine, which will then make you lightning-fast at checking summations!

I’ll show you later in another post how to use casting out nines to check for errors in subtractions, multiplications, and divisions too!