This is another thing I learned about from *Capitalism and Arithmetic*. It's a way to check arithmetic.

The ways we are taught to check our math are usually pretty lousy: the dim bulbs in the school emphasize doing the math over again (as if we are incapable of making the same mistake twice).

Casting out nines is easy, quick, and won't generally put you in the position to make the same mistake twice. The exception to that is that it won't catch a transposition error.

Of course, most people use a calculator. They use this as a convenient excuse not to check their work. Fortunately, casting out nines works on a calculator as well.

Here's how it works for addition. Suppose you have to add 5,059 to 9,004 and 4,790. If you add those up the traditional way you'll get 18,853. What you do is take all the *digits* "above the line", that is :

5 0 5 9 9 0 0 4 4 7 9 0

Now, "cast out the nines", that is delete them to get:

5 0 5 0 0 4 4 7 0

You can delete the zeros too, to get:

5 5 4 4 7

Now add those up, casting out nines as you go. To do this, you could 1) recognize that a 5 and a 4 sum to 9, or 2) recognize that 5 and 5 sum to 10 (which is more than 9). In the first case, you'd just delete the 5 and 4. In the second one, you'd remove the 9 from the 10 leaving 1. Either way, what you are left with is 7.

This is clearly different from the traditional addition you did to get an answer. What makes it useful as a check is that you should get the same result if you cast out nines from the answer to the addition problem. There, the digits are:

1 8 8 5 3

Cast out the 1 and 8 since they sum to 9, yielding:

8 5 3

Sum them to get 16. Cast out the 9 and you have 7 left.

That example was too easy, because you could use a calculator for it. But, most calculators end at 8 digits. What if you have to add something larger, like:

909,090,912 + 88,888,888 = 997,979,800

To check the answer using casting out nines, you would write all the digits on the left of the equals sign as:

9 0 9 0 9 0 9 1 2 8 8 8 8 8 8 8 8

Cast out the nines (and zeros) to get:

1 2 8 8 8 8 8 8 8 8

Sum those up to get 67, and cast out the seven nines in 67 leaving 4.

Doing the same thing for the right hand side of the equals will go like this:

9 9 7 9 7 9 8 0 0

Cast out the nines to get:

7 7 8

Add those up to get 22, and cast out the 2 nines in 22, leaving 4. *QED.*

There are 2 ways that casting out nines can fail. The first is if you transpose some numbers. For example,

909,090,921 + 88,888,888 = 997,979,800

This checks according to casting out nines, even though my sum is too low by 9.

The second way is if you substitute a 0 for a 9 or vice versa. So, this is incorrect, but it will pass a casting out nines test:

909,000,912 + 88,888,888 = 997,979,800

Interestingly, the technique changes a bit, but casting out nines works for subtraction, multiplication and division as well.

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Posted by: hogan online | August 07, 2013 at 06:58 AM