A continued fraction writes a number as an integer plus a reciprocal of an integer plus a reciprocal, nesting indefinitely. It produces the cleanest possible ladder of rational approximations — the convergents — which is why it underlies calendar design, gear ratios, and the best rational approximations of mathematical constants.
What continued fractions are and why they matter
A simple continued fraction represents any real number as:
x = a0 + 1 / (a1 + 1 / (a2 + 1 / (a3 + ...)))Written compactly as [a0; a1, a2, a3, ...], where each ak is a non-negative
integer. This representation has a remarkable property: truncating after any term
gives you the best possible rational approximation of the original number for
any fraction with a denominator that size or smaller. No other fraction with that
denominator gets closer to the true value.
This is why continued fractions appear across applied mathematics:
- Pi: [3; 7, 15, 1, 292, …] — truncating after 7 gives 22/7, after 15 gives 355/113, both famous pi approximations
- Calendar design: The Gregorian calendar’s 97-leap-years-in-400 structure comes from the continued fraction expansion of the solar year’s fractional day
- Gear ratios: Mechanical engineers use convergents to find gear tooth counts that approximate a desired ratio with small integers
- Music theory: Tuning systems use continued fractions to approximate just intervals with small-integer frequency ratios
How the algorithm works step by step
The expansion algorithm is a simple iteration:
pi = 3.14159265...
Step 1: a0 = floor(3.14159) = 3, remainder = 0.14159...
Step 2: reciprocal = 1/0.14159 = 7.0625..., a1 = 7, remainder = 0.0625...
Step 3: reciprocal = 1/0.0625 = 15.99..., a2 = 15, remainder = 0.99...
Step 4: reciprocal = 1/0.99 = 1.0006..., a3 = 1
pi ≈ [3; 7, 15, 1, 292, ...]Each convergent is computed from the sequence using the recurrence:
p_k = a_k × p_(k-1) + p_(k-2)
q_k = a_k × q_(k-1) + q_(k-2)with starting values p_{-1} = 1, p_0 = a_0, q_{-1} = 0, q_0 = 1. The convergent at step k is p_k / q_k. Convergents alternate above and below the true value and get successively closer — the error at step k is bounded by 1/(q_k × q_{k+1}).
Why a large term means a very good approximation
When a term in the continued fraction is large — like the 292 in pi’s expansion or the 15 two steps earlier — the next convergent is dramatically more accurate than the previous one. Intuitively, a large term means the “current” convergent was already very close, and the correction from the next term is correspondingly tiny. This is why 22/7 (stopping at a1 = 7) is a good approximation of pi: the next term is 15, a large number, signalling that 22/7 was already nearly exact and 355/113 is even better.
Worked examples of important constants
| Number | Expansion | Notable convergents |
|---|---|---|
| pi | [3; 7, 15, 1, 292, …] | 22/7, 333/106, 355/113 |
| sqrt(2) | [1; 2, 2, 2, 2, …] | 1/1, 3/2, 7/5, 17/12, 41/29 |
| phi (golden ratio) | [1; 1, 1, 1, 1, …] | 1/1, 2/1, 3/2, 5/3, 8/5, 13/8 |
| e (Euler’s number) | [2; 1, 2, 1, 1, 4, 1, 1, 6, …] | 3/1, 8/3, 11/4, 19/7 |
The golden ratio has all 1s because it converges as slowly as possible — it is the “most irrational” number, which is why Fibonacci numbers (its convergent numerators and denominators) appear in plant growth patterns.
Tips and notes
The output uses standard bracket notation with a semicolon after the integer
part, for example [3; 7, 15, 1]. Each convergent is listed with its decimal
value and absolute error. Because the input is read as a floating-point decimal,
very long expansions eventually reflect floating-point rounding rather than the
true mathematical value; keep the term count modest for typical inputs, and for
high-precision work provide a high-precision decimal string.