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==History== |
==History== |
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Casting out nines was known to the Roman bishop Hippolytos as early as the third century. It was employed by Twelfth-century Hindu mathematicians.<ref>[[Cajori, Florian]] (1991, 5e) ''A History of Mathematics'', AMS. ISBN . p.91</ref> |
Casting out nines was known to the Roman bishop Hippolytos as early as the third century. It was employed by Twelfth-century Hindu mathematicians.<ref>[[Cajori, Florian]] (1991, 5e) ''A History of Mathematics'', AMS. ISBN . p.91</ref> in the 17th century, [[Gottfried Wilhelm Leibniz]] not only used the method extensively, but presented it frequently as a model for rational process, in which errors can be monotored throughout the computation <ref> G.W Leibniz, The Art of Controversies, Marcelo Dascal (ed.), Springer 2008, p. 22 </ref>. [[Synergetics|''Synergetics'']], [[Buckminster Fuller|R. Buckminster Fuller]] claims to have used casting out nines "before World War I."<ref>Fuller, R. Buckminster: ''Synergetics, Explorations in the Geometry of Thinking.'' New York: Macmillan Publishing Company. ISBN p.765.</ref> 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. |
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The method bears striking resemblance to standard [[signal processing]] and computational [[error detection]] and [[error correction]] methods, typically using similar modal arithmetic. |
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==References== |
==References== |
Revision as of 23:51, 18 March 2009
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
* | First, cross out all 9's and pairs of digits that total 9 in each addend (italicized). | ||
† | Add up leftover digits for each addend until one digit is reached. | ||
‡ | These new values are called excesses. | ||
** | Do to the excesses what you did to the addends, coming to a single digit. | ||
Now follow the same procedure with the sum, coming to a single digit. | |||
†† | The excess from the sum should equal the final excess from the addends. |
*2 and 4 add up to 6.
†There are no digits left.
‡2, 4, and 6 make 12; 1 and 2 make 3.
**2 and 0 are 2.
††7, 3, and 1 make 11; 1 and 1 add up to 2.
Subtraction
First, cross out all 9's and digits that total 9 in both minuend and subtrahend (italicized). | |||
Add up leftover digits for each value until one digit is reached. | |||
Now follow the same procedure with the difference, coming to a single digit. | |||
Because subtracting 2 from zero gives a negative number, borrow a 9 from the minuend. | |||
The difference between the minuend and the subtrahend excesses should equal the difference excess. |
Multiplication
First, cross out all 9's and digits that total 9 in each factor (italicized). | |||
Add up leftover digits for each multiplicand until one digit is reached. | |||
Multiply the two excesses, and then add until one digit is reached. | |||
Do the same with the product, crossing out 9's and getting one digit. | |||
* | The excess from the product should equal the final excess from the factors. |
*8 times 8 is 64; 6 and 4 are 10; 1 and 0 are 1
Division
Cross out all 9's and digits that total 9 in the divisor, quotient, and remainder. | ||||||||
Add up all uncrossed digits from each value to a single digits. | ||||||||
Multiply the divisor and quotient excesses, and add the remainder excess. | ||||||||
Do the same with the dividend, crossing out 9's and getting one digit. | ||||||||
The dividend excess should equal the final excess from the other values. |
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.
A variation on the explanation
A nice trick for very young children to learn their nine facts for addition is to add ten to the digit and to count back one. Since we are adding 1 to the ten's digit and subtracting one from the unit's digit, the sum of the digits should remain the same. For example 9+2=11 with 1+1=2. When adding 9 to itself, we would thus expect the sum of the digits to be 9 as follows: 9+9=18 (1+8=9) and 9+9+9=27 (2+7=9). Let us look at a simple multiplication: 5x7=35 (3+5=8). Now consider (7+9)x5=16x5=80 (8+0=8) or 7x(9+5)=7x14=98 (9+8=17 1+7=8).
Any non-zero positive integer can be written as 9 x n+a where 'a' is a single digit 1 to 8 and 'n' is zero or any positive integer. Thus, using the distributive rule (9 x n + a)x(9 x m + b)= 9 x 9 x n x m + 9 x(am+bn) +ab. Since the first two factors are multiplied by 9, their sums will end up being 9 or 0, leaving us with 'ab'. In our example, 'a' was 7 and 'b' was 5. We would expect in any base system the number before that base would behave just like the nine.
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.[1] in the 17th century, Gottfried Wilhelm Leibniz not only used the method extensively, but presented it frequently as a model for rational process, in which errors can be monotored throughout the computation [2]. Synergetics, R. Buckminster Fuller claims to have used casting out nines "before World War I."[3] 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.
The method bears striking resemblance to standard signal processing and computational error detection and error correction methods, typically using similar modal arithmetic.
References
- ^ Cajori, Florian (1991, 5e) A History of Mathematics, AMS. ISBN . p.91
- ^ G.W Leibniz, The Art of Controversies, Marcelo Dascal (ed.), Springer 2008, p. 22
- ^ Fuller, R. Buckminster: Synergetics, Explorations in the Geometry of Thinking. New York: Macmillan Publishing Company. ISBN p.765.
External links
- "Numerology" by R. Buckminster Fuller
- "Paranormal Numbers" by Paul Niquette
- Weisstein, Eric W. "Casting Out Nines". MathWorld.