r in calculating the check digit); but if the check digits agree, your work could still be wrong, such as if you switched two digits when you were copying. In fact, I use a variant of this method when I balance my checkbook. If I get the wrong balance, I know my calculator didn't add wrong, but I may have entered something wrong. If the check digit for my balance is the same as what the bank says, I can guess that I reversed two digits somewhere; if they are different, it's more likely that I dropped a digit, or perhaps a whole transaction. If you want to know what this has to do with nines, or why it works, check out the other answers I referred to above. The basic idea is that the check digit is essentially the remainder after you divide by 9. (A slightly more advanced way of working with these check digits is to treat a result of 9 as a zero, so that check digits are always between 0 and 8 rather than 1 and 9, making it a genuine remainder.) You may notice that when you add the digits of 6395, if you just ignore the 9, and the 6+3 = 9, you still end up with 5 as your check digit. This is because any 9's make no difference in the result. That's why the process is called "casting out" nines. Also, at any step in the process, you can add digits, not just at the end: to do 8051647, I can say 8 + 5 = 13, which gives 4; plus 1 is 5, plus 6 is 11, which gives 2, plus 4 is 6, plus 7 is 13 which gives 4. I never have to work with numbers bigger than 18. I hope this clarifies what you're doing. It takes a lot of words to explain, but it's really easy to do. Keep at it and you'll get the idea. If you want a simpler explanation of WHY the method works than we have in our archives, write back and I can send that to you too. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 03/23/2003 at 13:28:21 From: Sarah Dodge Subject: Casting out nines Dear Dr. Math, I don't understand casting out nines. Can you help me? Here is an example 3942 - 1581 ------- I know that it equals 2361 I think that you would add the numbers together and you would throw out the nines, so it would be 3942 (0) - 1581 (6) so what wwould the answer be? (3?) How do you get that though ?
Date: 03/24/2003 at 12:40:37 From: Doctor Peterson Subject: Re: Casting out nines Hi, Sarah. I would usually check a subtraction by checking the equivalent addition; here 2361 + 1581 = 3942. In this case, the reduced digit sums are 3, 6, and 0, which is correct since 3+6=9 which reduces to 0. But you can cast out nines to check subtraction directly, if you add one step. In this example you want to subtract 6 from 0. To make the 0 big enough to subtract, you can "borrow" a 9, by adding 9 to the zero. (Just as you reduce by casting out nines, you can grab an extra nine when you need one.) Now you are subtracting 6 from 9, which gives you the 3 you expect. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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