Two Enumerations of the Rationals

Here are two methods for enumerating the rational numbers. The Calkin–Wilf method starts with corresponding to the binary number 1. If the binary number corresponds to the fraction , then the binary number corresponding to is 0 appended to , and the one corresponding to is 1 appended to . According to this scheme, the Calkin–Wilf enumeration begins with , where semicolons separate the rows of an array.
The Stern–Brocat method is more complicated, but it winds up reversing the Calkin–Wilf binary number except for the leading 1. The corresponding enumeration then reads .
Since either of the above arrays can be put into one-to-one correspondence with the set of natural numbers (positive integers) ℕ, the rationals must also comprise a set of cardinality .



  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+