Real and Rational NumbersDate: 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/ |
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