Continued Fractions

• The typography is a bit overwhelming

• We write: π = [3; 7, 15, 1, 292, 1, ...]

• In general, we have a = [a₀; a₁, a₂, ...]

  • a₀ is the integer part of a; α is the fractional part a - a₀

  • a₁, a₂, ... is the continued fraction expansion of 1/(a-a₀)

  • So [a₀; a₁, a₂, ...] = a₀ + 1/[a₁; a₂, ...]

• a₀ can be any integer

• But for i ≥ 1, a_i is necessarily positive

• With this restriction, the continued fraction representation is unique (almost)