Best Approximation Theorem

The best approximation theorem for continued fractions states that if pₙ/qₙ is the nth convergent of the continued fraction expansion of an irrational number α, then for any rational number p/q with 0 < q ≤ qₙ, the inequality |α − pₙ/qₙ| ≤ |α − p/q| holds. In other words, no rational with denominator no larger than qₙ approximates α more closely than the nth convergent does.

This property makes convergents the canonical sequence of rational approximations to an irrational number. The theorem is not merely a statement about optimality; it is a structural characterization. It says that the continued fraction algorithm does not merely produce approximations — it produces the best possible approximations at each scale, where scale is measured by denominator size.

A deeper result, known as Legendre's theorem, provides a converse: if a rational p/q satisfies |α − p/q| < 1/(2q²), then p/q must be a convergent of α's continued fraction. Together, these theorems establish that the convergents are not merely good approximations; they are exactly the approximations that are better than a certain threshold.

The best approximation property extends to higher dimensions through the theory of simultaneous Diophantine approximation, where the problem is to approximate several real numbers by rationals with a common denominator. The multi-dimensional analogues of continued fractions — including the Jacobi–Perron algorithm — generalize the best approximation property to systems of linear forms.