Casting out Nines: How Is 9 the Same As 0? Why Bother?
Date: 09/25/2009 at 10:40:02 From: Samantha Subject: Casting out 9's in a simple way and what's the use? Dear Dr. Math, I'm homeschooled, and in 5th grade, and am having trouble casting out nines. I've read the directions, and all your other information, and it's too confusing. My parents don't get it either. Also, what's the use? One of my problems is 7326 + 5037 + 2765 + 9932 + 8416. I don't get how 7 + 3 + 2 + 6 = 0, which is the answer that my Teacher's Edition gives.
Date: 09/25/2009 at 11:03:09 From: Doctor Peterson Subject: Re: Casting out 9's in a simple way and what's the use? Hi, Samantha. Casting out nines may not be quite as useful today when everyone uses calculators; but then, it can catch errors in entering the data into the calculator, so it might not be a bad idea to use it more than we do! Casting out nines is a fascinating (and rather old) method for checking manual calculations. If you look into why it works, you'll see that it ties into some important bits of number theory. Since I may well have written some of what you wrote that was too complicated, let's try just using your example rather than trying to explain it fully in general terms. The basic idea is that we can make what is called a "digital root" by adding all the digits of a number, and then repeating the process until we have a single digit. Let's do that with each of your numbers, and also with their sum: 7326 -> 7 + 3 + 2 + 6 = 18 -> 1 + 8 = 9 5037 -> 5 + 0 + 3 + 7 = 15 -> 1 + 5 = 6 2765 -> 2 + 7 + 6 + 5 = 20 -> 2 + 0 = 2 9932 -> 9 + 9 + 3 + 2 = 23 -> 2 + 3 = 5 + 8416 -> 8 + 4 + 1 + 6 = 19 -> 1 + 9 = 10 -> 1 + 0 = 1 ------ 33476 -> 3 + 3 + 4 + 7 + 6 = 23 -> 2 + 3 = 5 Before we continue ... I now see the specific question you are asking. For the first addend, above, I wrote 7 + 3 + 2 + 6 = 18 -> 1 + 8 = 9 Your book says it's 0, not 9. Why? This is actually where the name "casting out nines" comes from. If you work with this method enough, you find that anywhere you have a 9, you can just "cast it out" (throw it away) because it doesn't affect the digital root. For example, for the number 19, you get 1 + 9 = 10 -> 1 + 0 = 1, which is what you'd get if you ignored the 9 in the first place. But how can 0 and 9 really be the same answer, in our specific case? That's because all this work is based on the remainder you would get if you divided a number by 9. For example, if you divide 19 by 9, you get 2 with a remainder of 1, and 1 is the digital root! So the digital root is the remainder when you divide by 9 ... except when you get a 9! The remainder has to be less than the divisor. So when you get 9, in order to really find the remainder, you have to divide by 9 again -- and now the remainder is 0. So when we cast out nines, we treat 0 and 9 as the same thing; if we get a 9 we can "cast it out" and use 0 as the digital root. Back to our process. The digital roots of the addends are now 0 + 6 + 2 + 5 + 1 = 14 We take the digital root yet again, since this has two digits; and find that the digital root of the sum is 5. If we had gotten a different digital root, we'd have known that we made a mistake. But since this agrees with the 5 we got by using the sum itself, we've confirmed our addition. Now, having the same digital root doesn't prove the answer is right -- we could have added wrong but gotten the same result by coincidence -- but it does give us more confidence. I hope that helps a bit. And I hope your text explained this idea of ignoring 9's, and didn't just tell you to add digits, as I did at first. That simplified explanation is actually good enough to use the method (it would have worked fine if we had used 9 rather than 0 for that first digital root). But ignoring 9's makes the work a bit easier. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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