Casting out Nines

Last Up date: 2005 November 12
Recent changes: Mention of "Casting out Elevens" and link to a table, with answers.
Casting out Nines is an old technique used to check for arithmetic errors. By and large since machines do most arithmetic today, it has all but been forgotten. I think it has enough use and novelty appeal to be worth the effort to learn.
(This web page started on 2004 September 30.)

Casting out Nines is basically just finding the remainder of a number when divided by nine, and using it as a "check digit". Because nine is just one less than ten, the base of the decimal number system, there are many short cuts to finding that remainder; which will be a single digit zero through 8. I will list some of these "short cuts" as "Rules":

Simply, add the digits together. If you get a number 10 or greater add those digits together and continue.
You can replace the digit nine, or any combination of digits that add to nine, with zero (effectively ignoring 9's). This is probably the reason for the name "Casting out nines".
You can subtract one instead of adding eight, also you can subtract 2 instead of adding seven.
If you keep all the digits you can in general ignore decimal points.

Now, before I show you how to use the "check digits" we need some practice in calculating the check digits.

The following is the first 100 decimal places of the mathematical constant "e", blocked in groups of four digits. We will use these for "typical" numbers to practice casting out 9's. I suggest you to copy these to a work sheet. If you want to work on the CRT you could cut and and paste them into something like: Notepad; and work there.

Numbers	Check
2.7182 8182 8459 0452 3536 0287 4713 5266 2497 7572 4709 3699 9595 7496 6967 6277

I will explain my thought process for calculating the check digits for the first three lines above:

1st line: 2 + 7 add to nine so I ignore them, likewise for 1 + 8 thus the check digit for the first line is 2. If you have printed or copied these numbers, write the check digit to the right of the number.
2nd line: again I ignore the first two digits, the last two add to 10 so I add the 1 and 0 getting 1 for the check digit for the 2nd line.
3rd line: I immediately see the 4 + 5 in the middle and the 9 at the end and ignore them leaving the check digit as 8.

Now you need the practice. Calculate the check digits for the remaining lines.

Checking arithmetic by Casting Out 9's

You have calculated for each number

For the arithmetic operations of addition, subtraction and multiplications. Do the same operations on the check digits. (If that results in a check value greater than eight apply the above rules to reduce it to a value less than 9.) If the result is 9 replace it with zero, in the world of Casting out 9's, nine is equivalent to zero; and visa versa.
If you add the first two lines, then add the check digits: The correct answer is 3.5364 and the sum of the check digits is 3. I calculate the check digit for the sum by seeing the 3 + 6 and the 5 + 4 thus I can ignore all four so the check digit is 3; which agrees! If it don't, you have made an arithmetic error somewhere!
Multiply the first two lines, and multiply the check digits getting 2. The check digit for the product better be 2.

Practice by adding several of the numbers and their check digits. Also try subtraction and multiplication doing the same operation with the check digits. With a bit of practice you will find the mental effort in calculating check digits and using them is almost trivial.

Fortunately, Casting Out 9's will never tell you there is an error when no error exists!
Casting out 9's works for Addition, Subtraction, and Multiplication but not Division.

To check division you do it the same as you were taught in school: You multiply the divisor by the result and add the remainder. Simply do that with the check digits.

Checking arithmetic by Casting Out 9's ain't perfect!
It will not catch all arithmetic errors.

Obviously, any single digit error in the answer will produce a check digit that does not agree. But a simple transposition of digits in the answer will not be caught; multiple digit errors will be caught 90% of the time. So checking by "casting out nines" is not perfect, but it is so easy you will probably want to use it for hand done arithmetic. Also with non-printing calculators a 90% check for keyboard entry errors is better than no check at all.

I should point out there are better arithmetic checks based on Congruence Arithmetic, but they are a bit more involved. One is "Casting Out Elevens" which involves: Starting on the right and alternately adding then subtracting digits, and if you don't like negative values at any time you can add eleven to the check digit. Casting out Elevens will detect simple transposition of digits.

Finally, for those who want much more practice I have a table much larger than the one above, with the check values for both (mod 9) and (mod 11).

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