Every real number has a continued fraction [a₀; a₁, a₂, …]. The convergents pₙ/qₙ alternate above and below the target — each is the best rational approximation with denominator ≤ qₙ. φ needs all 1s; π's [3;7,15,1,292,…] gives 355/113 ≈ π to 7 decimal places.