Covering the Rationals

Dense, yet taking up no room

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• The rationals are dense, but can be covered by intervals of total length ε for any ε.
• So they have measure zero: a random real is irrational with probability 1.
• Countability and measure are different notions of size.

Steps

1. Everywhere, but nowhere
2. Adding the covers
3. Shrinking covers
4. A random number
5. Small denominators
6. Size versus length