Continued fractions and rational approximation

For a positive real α, take its integer part a₀. If the remainder is nonzero, invert that remainder, take its integer part a₁, and continue. Every aₙ after the first is a positive integer. A rational input eventually produces a zero remainder and stops, precisely because Euclidean remainders stop. An irrational input cannot terminate, since a finite expression of integer parts and reciprocals would be rational.

The notation [a₀; a₁, a₂, …] records these operations with nested reciprocals. Terminating expansions have an ambiguity: a final entry greater than one can be replaced by one less followed by a final one. Choosing the last entry greater than one gives a standard convention except for integer inputs. This convention changes notation, not the rational value. For irrational inputs the positive-entry expansion is uniquely determined by the floor-and-reciprocal algorithm.

Truncate after aₙ to obtain the n-th convergent pₙ/qₙ. Initialize p₋₂ = 0, p₋₁ = 1, q₋₂ = 1, and q₋₁ = 0. Then each new numerator and denominator is the current entry times the preceding value plus the value before that. These initial conditions are part of the recurrence, not optional bookkeeping: they produce p₀/q₀ = a₀ correctly.

Consecutive convergents obey pₙqₙ₋₁ − pₙ₋₁qₙ = (−1) to the power n − 1 for n at least one. It follows that each fraction is reduced, because a common divisor of its numerator and denominator would divide the determinant one. Dividing the determinant by the two denominators gives the exact distance between consecutive convergents. Denominators eventually strictly increase, making their separation shrink.

For an irrational α, let αₙ₊₁ be the positive complete tail beginning at aₙ₊₁. Substitution expresses α as (pₙαₙ₊₁ + pₙ₋₁)/(qₙαₙ₊₁ + qₙ₋₁). Subtracting pₙ/qₙ and using the determinant identity gives an error of size one divided by qₙ(qₙαₙ₊₁ + qₙ₋₁). Since the tail exceeds its integer part, this is smaller than 1/(qₙqₙ₊₁).

In particular, for n at least one the error is smaller than 1/qₙ². This is a theorem about convergents of an irrational number, not an error guarantee for an arbitrary fraction with denominator qₙ. A large next entry makes the current convergent especially good by enlarging the omitted tail. More refined best-approximation statements require specifying whether one minimizes absolute error or denominator-weighted error; do not blur those different objectives into a slogan about being the best fraction.