Positive Real and Rational Numbers

Date: 09/21/2002 at 15:05:25
From: Corey Sassaman
Subject: Positive real numbers and positive rational numbers?

Is there a positive real number that is smaller than all positive 
rational numbers?

I've been trying to see what would work for this problem and can't 
figure it out.

Date: 09/23/2002 at 04:02:25
From: Doctor Mike
Subject: Re: Positive real numbers and positive rational numbers?

Corey,

The answer to your question is "no," and here is one way to see why 
this is so. Let's say you have a real number somewhere between zero 
and one, and you think it might be smaller than all rational numbers.  
This real number could be written in decimal form something like this:
 
    0.abcdefghijklmnopqrstuv.....
   
I'm representing how it starts using the letters in the alphabet, 
which obviously are limited, but it goes on forever without repeating.  
Why?  Because if it repeated like

    0.7654321000560005600056000560005600056.......

then it would be a rational number. The rational numbers all either 
have a repeating pattern or have zero from some point on.
  
Go back to the real number above. If you stop it somewhere, like

   0.abcdefghijk       or     0.abcdefghijk000000000000.....

then that would be a rational number. But the really interesting thing 
is that it is SMALLER THAN the original real! Why is it less? It is 
less because we subtracted something from it. What did we subtract?  
We subtracted the decimal

    0.00000000000lmnopqrstuv....
  
We could write the subtraction in standard form like this: 
   
     0.abcdefghijklmnopqrstuv........
   - 0.00000000000lmnopqrstuv........
 -------------------------------------- 
     0.abcdefghijk00000000000........
   
What this shows is that for ANY positive real, you can get a smaller 
positive rational number by discarding (or subtracting) the part of 
the real number that is the part of the decimal from some "place" on 
out. That shows there couldn't possibly be a real number that is 
smaller than all rational numbers.  
  
I hope this clears things up.  
  
- Doctor Mike, The Math Forum
  http://mathforum.org/dr.math/