Casting out nines

Casting out nines is a sanity check to ensure that hand computations of sums, differences, products, and quotients of integers are correct. By looking at the digital roots of the inputs and outputs, the casting-out-nines method can help one check arithmetic calculations. The method is so simple that most schoolchildren can apply it without understanding its mathematical underpinnings.

Examples

As the explanation can be hard for many to understand, below are examples for using casting out nines to check addition, subtraction, multiplication, and division.

Addition

^*8 times 8 is 64; 6 and 4 are 10; 1 and 0 are 1

Division

How it works

Formally, casting out nines is a valid method of checking equations because of a property of modular arithmetic. Specifically, if "x" and "x" (respectively, "y" and "y") have the same remainder modulo 9, then so do "x" + "y" and "x" + "y","x" − "y" and "x" − "y" and "x" × "y" and "x" × "y".

For an equation utilizing only integers to be correct, the following must be true: the sum of the digits of the decimal writing of an integer has the same remainder, modulo 9, as this integer. Because of this, one can add all digits in the original number to obtain another number, and so on repeatedly until one gets a 1-digit number, which is necessarily equal to the original number. Also, nines can be tossed out before this, because 9 is equal to 0 modulo 9.

In a correct equation, one side equals the other. If the equation was correct before, performing the above operation on both sides preserves correctness. However, it is possible that two previously unequal integers will be identical modulo 9 (on average, a ninth of the time).

One should note that the operation does not work on fractions, since a given fractional number does not have a unique representation.

History

Casting out nines was known to the Roman bishop Hippolytos as early as the third century. It was employed by Twelfth-century Hindu mathematicians. [Cajori, Florian (1991, 5e) "A History of Mathematics", AMS. ISBN . p.91] In his book "Synergetics", R. Buckminster Fuller claims to have used casting out nines "before World War I." [Fuller, R. Buckminster: "Synergetics, Explorations in the Geometry of Thinking." New York: Macmillan Publishing Company. ISBN p.765.] Fuller explains how to cast out nines and makes other claims about the resulting 'indigs,' but he fails to note that casting out nines can result in false positives.

References

External links

* [http://www.rwgrayprojects.com/synergetics/s12/p2200.html "Numerology"] by R. Buckminster Fuller
* [http://niquette.com/puzzles/paranump.html "Paranormal Numbers"] by Paul Niquette
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