Continued Fractions & The Golden Ratio
Every real number has a unique CF expansion — the most irrational is φ = [1;1,1,1,...]
Expand
φ (golden)
π
e
√2
√3
1/φ
CF will appear here
Convergents (p/q → x)
Key Facts
x = a₀ + 1/(a₁ + 1/(a₂ + ...))
Convergents p_n/q_n are best rational approximations
φ = [1;1,1,1,...] — slowest converging CF
π = [3;7,15,1,292,...] — next coeff is huge!
|x − p_n/q_n| < 1/q_n·q_{n+1}