Real and Rational Numbers

Date: 02/27/2001 at 14:02:38
From: Eileen Bach
Subject: Real and rational numbers

How can I show that the number of rational numbers between 0 and 1 is 
the same as the number of natural numbers? (considering the following 
ordering of these fractions: 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5...)

Date: 02/28/2001 at 14:17:57
From: Doctor Floor
Subject: Re: Real and rational numbers

Hi, Eileen,

Thanks for writing.

By having the ordering as you present it, you know you can count the 
rationals between 0 and 1. But counting here is the same as making a 
function 

  f: (0,1)/\Q ---> N

[ (0,1)/\Q is the open interval from 0 to 1 of the rational numbers ]

which is to say, a function giving for each rational number between 0 
and 1 a natural number.

By this function each natural number is reached exactly once. And of 
course each rational between 0 and 1 is mapped to a natural number. 
Such a function we call a 'bijection'.

If two sets are mapped to each other by a bijection, then their number 
of elements is equal.

If you have more questions, just write back.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/