Convergent (continued Fraction) - Representation of Real Numbers

Representation of Real Numbers

Every real number can be expressed as a regular continued fraction in canonical form. Each convergent of that continued fraction is in a sense the best possible rational approximation to that real number, for a given number of digits. Such a convergent is usually about as accurate as a finite decimal expansion having as many digits as the total number of digits in the nth numerator and nth denominator. For example, the third convergent 333/106 for π (Pi) is roughly 3.1415094, which is not quite as accurate as the 6-digit 3.14159; the fourth convergent 355/113 = 3.14159292 is more accurate than the 6-digit decimal.

By the determinant formula it appears that the successive convergents Ak/Bk of a regular continued fraction are connected by the formula

A_{k-1}B_k - A_kB_{k-1} = (-1)^k \,

This implies, in particular, that the greatest common divisor (Ak, Bk) = 1; in other words, each convergent of a regular continued fraction, as given by the fundamental recurrence formulas, is automatically expressed in lowest terms.

More detailed properties of best rational approximations and convergents of π are discussed in the continued fraction article.

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