Farey Fractions

Neighbours, mediants and kissing circles

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 Farey sequence Fₙ lists all fractions in [0, 1] with denominator ≤ n.
• Neighbours a/b < c/d satisfy bc − ad = 1; the first fraction between them is the mediant (a+c)/(b+d).
• Ford circles of neighbouring fractions touch, and never overlap.
• Mediants find the simplest fraction in any interval.

Steps

1. Squeeze one in
2. Farey sequences and mediants
3. Count them
4. Ford circles
5. In the gap
6. Batting average