Every number, rational or irrational, unfolds into a chain of whole numbers: a₀ + 1/(a₁ + 1/(a₂ + …)). Cut the chain early and you get the best fraction there is for the size — 355/113 for π, Pell equations solved for √d. Type a decimal, a fraction like 355/113, or sqrt(13), and watch the convergents snap shut on the value.
Useful for gear ratios, music tuning, “how big does the wheel need to be” questions — and it's provably the best fraction at that size, not just a good guess.
φ = [1;1,1,1,…] has the slowest-converging continued fraction of all, which is why golden-ratio phyllotaxis never doubles up on itself — nature's irrational packing.
| n | aₙ | convergent | decimal | rel. error | p²−d·q² |
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