Reversing the Digits

Date: 07/28/99 at 08:15:15
From: Kelly Wray
Subject: Reversing the digits

We have investigated this question, and we know ho to do it, but we do 
not know why it is so, which is a major part of our maths 
investigation. We would greatly appreciate it if you could explain why 
when you:

   1. Take any two-digit number,
   2. Reverse the digits of this number,
   3. Subtract the reversed number from the original one,
   4. Divide the result by the difference of the two digits of the
      original number

the answer is always nine. 

Example: 73 - 37 = 36/(7-3) = 9

Thank-you very much.
Kelly

Date: 07/28/99 at 10:29:19
From: Doctor Rick
Subject: Re: Reversing the digits

Hi, Kelly.

I don't know what level of math you can work with. I hope you know 
some algebra, because that's the best way to understand this 
phenomenon. Here is what to do.

Call the tens digit of the bigger number, a.

Call the ones digit of the bigger number, b.

Then the bigger number is 10a + b. What do you get when you reverse 
the digits?

Follow the steps you listed, using these algebraic expressions instead 
of particular numbers. You will get an expression that you can 
simplify. See what happens then.

By the way, will this really work with ANY two-digit number? Your step 
3 should be changed to "subtract the smaller from the greater of the 
numbers from steps 1 and 2." Even then, it won't always work.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/   

Date: 07/29/99 at 04:38:04
From: Kelly Wray
Subject: Re: Reversing the digits

Thank you very much for your reply. Unfortunately, I still do not 
understand why the answer always equals nine. I understand the concept 
that the bigger number is 10a + b, and when you reverse that it is 
10b + a.

So 10a-b - 10b-a = x/(a-b) = 9

But I still do not understand why this is the case. Also I am aware 
that there are many exceptions to this rule, such as the multiples of 
eleven. Any further help would be greatly appreciated.

Thank you.
Kelly Wray

Date: 07/29/99 at 11:40:15
From: Doctor Rick
Subject: Re: Reversing the digits

Hi, Kelly.

Yes, there are exactly 9 exceptions, because 0/0 does not equal 9; it 
is "indeterminate."

You start out well in writing out the process as an expression, but 
you need to follow through and be careful. Watch as I follow the 
steps:

1. Take any two-digit number:

     10a + b

2. Reverse the digits of this number:

     10b + a

3. Subtract the reversed number from the original one:

     (10a + b) - (10b + a)   [Note the parentheses you omitted]

4. Divide the result by the difference in the two digits of the
   original number:

     (10a + b) - (10b + a)
     ---------------------
            (a - b)

Now simplify this expression and see what you get.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/