Concept
Representing reals as nested fractions. φ = [1;1,1,1,...], π = [3;7,15,1,292,...]. The most efficient rational approximations of irrationals.
Understand it in one breath
An integer plus the reciprocal of the rest, recursively. π ≈ 3 + 1/(7 + 1/(15 + …)) gives 3, 22/7, 333/106, 355/113 — each step a strictly better rational approximation. The fact that 22/7 is famous as π is no accident.
At a glance
Real number | Continued-fraction expansion | Notable approximation | Distinctive feature |
|---|---|---|---|
√2 ≈ 1.4142 | [1; 2, 2, 2, …] | 7/5, 17/12, 41/29, 99/70 | Periodic |
φ (golden ratio) ≈ 1.618 | [1; 1, 1, 1, …] | Ratios of Fibonacci numbers | The most irrational number |
π ≈ 3.1416 | [3; 7, 15, 1, 292, …] | 22/7, 333/106, 355/113 | Highly irregular |
e ≈ 2.7183 | [2; 1, 2, 1, 1, 4, 1, 1, 6, …] | A specific pattern (1, 2k, 1) | Has regular structure |
√n (n an integer) | Always periodic | Lagrange, 1770 | Connected to Pell’s equation |
The best rational approximations to an irrational number arise naturally from continued fractions. It is no accident that π ≈ 22/7 is famous — it is an exceptionally accurate approximation with a small denominator.
Key formula
[a₀; a₁, a₂, a₃, …]
Modern applications
Musical tuning, astronomical calendar correction, and exceptionally accurate rational approximations of irrational numbers.
Beyond MathVoyage
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