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 Category: Miscellaneous Miscellaneous
 Published: 15 September 2008 15 September 2008
Another article on quick head calculations explains how to quiclky divide a number by 9.

This is another article on simple ways to make calculations on your head. Think of all the calculator batteries you will be saving with this stuff:)
First of all, I must give credit to where it is due. This article was inspired by something a roommate of mine in college taught me. His name was Abel and he is one of the sharpest mathematical minds I have ever met.
The rule is very simple and you may already have heard about it. What is original is the way you prove it, in a very simple and elegant way.
the rule
When you divide by 9 any number a between 1 and 9 you get 0.aaaaaa...
For example, 2/9 = 0.2222... and 7/9 = 0.7777... you get my drift. You probably also learned how to prove this type of relation in basic Calculus. Most of the times, people teach you this using the sum of all the terms of a geometrical progression. But there's a much easier way to prove this. And this is what I'll show you next.
the proof
So, let's say number x = 0.aaaaaa...
then it is easy to see that 10.x = a.aaaaaa...
Next we will subtract the two equations above to get the value of 10.x  x
10.x  x =  a.aaaaaa...  
  0.aaaaaa...  
=  a.00000... = a 
So, if :10.x  x = a <=> 9x = a <=> x = a/9
And there you have it! We just showed that x = 0.aaaa... = a/9. This is our divide by nine rule.
beyond divide by 9
Now, using the same idea, it's very easy to see that we can apply the same rule to any fraction where numbers repeat. Example, if x=0.27272727... then 100x = 27.27272727... and 100xx = 99x = 27
<=> x = 27/99.
Now you have a rule to divide by 99 that says dividing any number ab between 10 and 99 by 99 you get 0.abababab...
Comments, questions, suggestions? You can reach me at: contact (at sign) paulorenato (dot) com
And you can do the same for divide by 999, 9999 etc.