Why Casting Out Nines Works
Date: 10/13/2008 at 17:19:51 From: Las Subject: Why Casting Out Nines Works Hello. I have to do a math project for school, and one of the things I must know is why the method of "casting out nines" works. I know it has to do with modular arithmetic but can you please explain the math behind this method AS SIMPLY AS POSSIBLE. I would like it if you could explain it in simple terms, not college math. NOTE: I know how to do this method, so please do not include that with your response. I cannot find any place that clearly defines why this method works to me, because I don't understand what the web site is trying to say. I have tried reading about modular arithmetic but that is only part of the reason why this works and I really need someone to explain it loud and clear to me.
Date: 10/14/2008 at 11:50:42 From: Doctor Peterson Subject: Re: Why Casting Out Nines Works Hi, Las. Modular arithmetic provides a language in which it is easy to explain what is happening; without that language, it will take more words to make it clear (and even more if I wanted to give a complete proof), but it can be done. The basic idea of modular arithmetic is that two numbers are congruent modulo 9 when they leave the same remainders on division by 9; you'll be seeing phrases like that all through what I write here! Let's take a simple example; I'll check the addition 157 + 246. This way we can avoid using a lot of variables, but you can follow the ideas and see that they apply to any number. (This is how people talked about algebra before they had the idea of variables, so I'm following an old tradition.) The check digit for 157 is 1+5+7 = 13, 1+3 = 4; and the check digit for 246 is 2+4+6 = 12, 1+2 = 3. The first thing to ask is, what do these numbers mean? The answer is, the check digit gives the remainder when you divide by 9. In our example, when you divide 157 by 9 you get 17 with a remainder of 4 (our check digit), and when you divide 246 by 9 you get 27 with remainder 3 (again our check digit). Why is that? What can adding the digits of a number have to do with dividing and getting a remainder? Well, let's look at 157, which we can write as 1*100 + 5*10 + 7. Notice that 10 = 9+1 and 100 = 99+1; any power of ten is 1 more than a multiple of 9, and therefore will leave a remainder of 1 when divided by 9. So we can rewrite 157 as 157 = 1*100 + 5*10 + 7 = 1(99 + 1) + 5(9 + 1) + 7 = (1*99 + 5*9) + (1 + 5 + 7) Since the first part is a multiple of 9, the second part (the sum of the digits) will have the same remainder; whatever remainder you get when you divide 1+5+7 by 9 is the remainder when you divide 157 by 9. So we can repeat the process with 1+5+7 = 13, adding its digits and again getting the same remainder. Once we get this down to one digit, it IS the remainder. So what have we learned? The check digits are remainders; now we have to consider what happens to remainders when you add numbers. This will take more of the same kind of thinking. Take our two numbers, 157 = 9*17 + 4 and 246 = 9*27 + 3. I've written each as a multiple of 9 plus its remainder (which is its check digit). Now let's add them: 157 + 246 = (9*17 + 4) + (9*27 + 3) = (9*17 + 9*27) + (4 + 3) = 9*(17 + 27) + (4 + 3) So the remainder when you divide the sum by 9 is the sum of the remainders, 4+3 = 7. Well, not always--if the sum of the remainders had been greater than 9, you'd have to divide by 9 again and take the final remainder. But in all cases the remainder of the sum is the same as THE REMAINDER OF the sum of the remainders. This is where the language of modular arithmetic saves a lot of words! This means that the check digit for the sum is the same as the check digit for the sum of the check digits--and that is what casting out nines is. If you find that this is not true, you know that the sum is incorrect. Here is an attempt I made once before to explain this in even simpler terms: Casting Out Nines for 2nd Graders http://mathforum.org/library/drmath/view/65299.html If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 10/15/2008 at 20:33:17 From: Las Subject: Thank you (Why Casting Out Nines Works) Thank you so very much. This was the most helpful thing any web site did for me. I have a feeling I'm going to get a 100 on this math project! Thanks again for taking your time to help me out. Without you, I wouldn't understand this.
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