Every real number has a unique simple continued fraction expansion [a₀; a₁, a₂, …]. The partial truncations (convergents) are the best possible rational approximations. The patterns inside the brackets are surprising — e has an obvious one, π has none.
p_n/q_n is closer to α than any fraction with denominator ≤ q_n. The CF coefficients aₖ encode how "irrational" α is: large coefficients mean α has an exceptionally good rational approximation right around that step.
[2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, …] — the pattern 1, 2k, 1 repeats forever, proven by Euler in 1737. π's CF is [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, …] — that 292 is why 355/113 (Zu Chongzhi, 5th century) approximates π to 7 digits. The golden ratio is [1; 1, 1, 1, …] — all 1s, the slowest-converging possible CF, which is why φ is "the most irrational number."