Every real number has a unique continued fraction expansion [a₀; a₁, a₂, ...]. Rational numbers terminate; quadratic irrationals have periodic expansions (Lagrange 1770). The golden ratio φ=[1;1,1,1,...] has the "worst" rational approximations — maximally irrational.
| n | aₙ | pₙ/qₙ | error |
|---|