Friday, August 26, 2011

Casting out nines

You are sitting somewhere and a question of calculation comes up.  You owe this much on an account and there is this much pending to be added.  What is the total?

You own calculators and you know where one is but you are not interested in disturbing yourself to get up and get it.  You want to add two numbers or make some other arithmetic calculation.  You want to be fairly sure you have worked out the answer correctly.  What do you do?

You "cast out nines."  I was rather looking forward to explaining just what doing so is, but I try to train myself to check with Google to see what it can turn up on any subject I am thinking of.  Sure enough, a fine example, nicely worded and nicely laid out, is here

I believe some users of this blog are reluctant to click on any link for fear of what doing so might unleash on their computer.  I urge people to experiment with links, learning which to trust and gaining experience reading and decoding them.  One tool is to copy the link and insert it in the Google search window.  Google and other internet companies are on the lookout for bad links.

Ok, back to casting out nines.  The basic idea is to get a substitute number for each one in the calculation, including your version of the answer.  Do the calculation with the substitutes and see if that answer equals the substitute of your answer.

Like this:  312 x 45  = 14040  Get substitutes for all three number by casting out the nines in each, that is, simply add the digits of each number and keep on adding digits until you are left with a single digit.  So, 312 = 6 when the 3, the 1 and the 2 are summed.  45 = 9 when its digits are added but casting out nines, means that all nines are immeditately zero.  We have 6 and 0, when multiplied, result in 0.  The answer I have 14040 also sums to 9, which we cast out, leaving 0.  

Using casting out nines results in my thinking I have the correct answer.  I do not have a perfect proof that I am correct since I could make corresponding errors, read the numbers incorrectly, etc.  But it is a pretty good check.  If the answers do not match, there is an error. This subject relates to error checks, detection and correction.

The method works with addition, subtraction, multiplication and division.  
  1. Get substitute figures for each one in the calculation.  
  2. Perform the calculation with the substitutes and
  3. See if the substitute of your answer = the answer of the substitutes
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