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# A different way to divide by nine.

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### A different way to divide by nine.

1. 1. A completely different way of dividing by nine Dave Coulson, 2013 dtcoulson@gmail.com
2. 2. Check this out: 5202 9 520 52 5 1 578 6012 9 601 60 6 1 668What I’m doing is repeatedly stripping off the last digit until there arenone left, and then adding the numbers together. Finally, add 1 more.
3. 3. The procedure is accurate to within 0.1% for all numbers of 4 digits orless. Usually this means rounding the answer to the nearest integer. 2345 9 234 23 2 1 260For multiples of nine in this range, the answer is usually (though notalways) exact.
4. 4. For very large numbers, the procedure is quite tedious, to the pointwhere it is no longer faster than the method you learned at school. 123 , 456 9 12 ,345 1, 234 123 12 1 1 13 ,704 ( Should be 13,716) (0.1% error )
5. 5. The procedure is laughably easy for two digit numbers. 54 9 5 1 6 81 9 8 1 9 27 9 2 1 3 etcBut most of us already know the answers to these because we rememberthe nine times table from school (er...right?).
6. 6. Therefore the procedure is at its best with numbers in the hundreds. 641 9 64 6 1 71 321 9 32 3 1 36 981 9 98 9 1 108
7. 7. Why the procedure works can be shown in a number of ways.Perhaps the best and simplest is to begin with the decimal version ofone-ninth, which is 0.1111....
8. 8. Why the procedure works can be shown in a number of ways.Perhaps the best and simplest is to begin with the decimal version ofone-ninth, which is 0.1111....Therefore dividing by nine is the same as multiplying by 0.1111...
9. 9. Why the procedure works can be shown in a number of ways.Perhaps the best and simplest is to begin with the decimal version ofone-ninth, which is 0.1111....Therefore dividing by nine is the same as multiplying by 0.1111...This is the same as adding a tenth to a hundredth to a thousandth, etc
10. 10. A quick and dirty way of doing this is to strip off the last digits of theoriginal number one at a time and add them together. 4321 / 9 = 432+43+4+1Adding a 1 at the end compensates for the downward rounding incurredby truncating the numbers.
11. 11. Is the procedure useful?Hardly! Most of us have no difficulty doing calculations in theconventional way.The procedure is offered simply as a mathematical oddity, a peculiarcharacteristic of the nine times table that seems to have been overlookedall of these years.
12. 12. Are there other divisors that can be handled in the same way?Theoretically yes, but none of them in practise are easy.Dividing by eight, for example is the same as adding a tenth to one-tenthof a quarter of the original number. (0.125)Dividing by six is the same as adding two-thirds of the number to itselfand then dividing the lot by ten. (0.16667)NONE of these alternative approaches are memorable or practical.It seems as if the procedure for ninths is the only one that works wellenough to bother sharing with others.
13. 13. [END]