Combining Numbers Reversibly

Date: 06/04/2003 at 16:29:14
From: Dene
Subject: A formula to obtain two numbers from the formula result.

Hello,

I have been trying to find a way of getting two different numbers, 
combining them through a formula that will give a final number. From 
that new number it should be possible to get the two numbers back.

For example, say you have the two numbers 49 and 637. You take the 
greatest number and multiply it by 10^n, where n is the number of 
digits within that number. Therefore, 637 * 10^3 = = 637000. Then, 
you add the the smaller number, which will give you the final number, 
637049.  The reverse to get the two numbers would be:

637049 / 1000 = 637.049 //first number
637049 mod 1000 = 49

This is the only method I have been able to come up with so far, and 
if you use further formulas and feed them into further formula 
calculations, the results go up exponentially, which is not what I am 
looking for. I need a formula where the result is not really more than 
double the sum of the two numbers, if there is one. Could you help me?

Date: 06/04/2003 at 23:17:24
From: Doctor Peterson
Subject: Re: A formula to obtain two numbers from the formula result.

Hi, Dene.

The fact is, the numbers you get HAVE to go up exponentially when you 
do something like this. That's because the information contained in a 
numeral is determined by the number of digits, and when you combine 
two numbers into one you are combining the information they contain, 
which means essentially multiplying the size of the numbers, or adding 
the number of digits. In general, combining their digits is the most 
efficient way to do this, and gives a way to measure the information 
content.

For example, suppose we combine two two-digit numbers by concatenating 
their digits (in base ten): xx, yy --> xxyy. The total number of 
digits in the result is the sum of the number of digits in the input 
numbers. And it has to be this way, because there are 100 
possibilities for the first number xx, and 100 for the second, yy; so 
there are 100*100 = 10000 possible combinations, and it takes four 
digits to distinguish them all. If you had a formula that would 
produce only three digits when it combined any two two-digit numbers, 
then there would not be enough different outputs to distinguish all 
input pairs; some combinations of inputs would have to produce the 
same output, and you could not reverse the formula.

There are other specific ways to combine numbers reversibly. One that 
does not require you to decide ahead of time how big the numbers can 
be is to interleave their digits. For example, 049 and 637 might 
combine to make 064397. This can also be done in binary rather than 
decimal, which is more suitable for computers. But however you do it, 
the result is adding the numbers of digits.

Here is a discussion of another approach to the same problem:

   Coding Pairs of Numbers
   http://mathforum.org/library/drmath/view/56036.html 

If you have any further questions, feel free to write back.

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