Continued fractions

• Many many fascinating properties

• Enough material for several colloquium talks

• Connections with:

  • Euclid's algorithm

  • Farey series

  • Fibonacci numbers

  • Möbius transformations

  • Measure theory

  • Stern-Brocot representation

• We will have to content ourselves with just a few of the most important properties

  1. Every real number has a (nearly) unique continued fraction representation

  2. The representation is finite if and only if the number is rational

  3. Simple rationals have representations with few terms and small terms

  4. It's easy to decide which of two continued fractions represents the larger number

  5. Truncating the representation of a gets you a very good rational approximation to a