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/