Continued Fraction

continued fraction

[kən′tin·yüd ′frak·shən] (mathematics) The sum of a number and a fraction whose denominator is the sum of a number and a fraction, and so forth; it may have either a finite or an infinite number of terms.

Continued Fraction

one of the most important ways of representing numbers and functions. A continued fraction is an expression of the form

where a₀ is any integer and a₁, a₂, . . . , a_n, . . . are natural numbers, called the partial quotients of the given continued fraction.

We can arrive at the continued fraction representing some number α by writing this number in the form α = a₀ + (l/α₁), where a o is an integer and 0 < 1/α1 < 1; we then write a₁ in this form, and we continue in this manner. The number of quotients of a continued fraction may be finite or infinite, and the continued fraction is correspondingly called finite or infinite. The continued fraction (1) is often denoted symbolically as

(2) [a₀;a₁,a₂, ... , a_n, ... ] infinite continued fraction or

(3) [a₀;a₁, a₂, ... , a_n] finite continued fraction

A finite continued fraction is always a rational number; conversely, every rational number can be represented in the form of a finite continued fraction (3). Such a representation is unique if we require that a_n ≠ 1. The continued fraction [a₀; a₁, a₂, . . . , a_k (k ≤ n) written in the form of an irreducible fraction Pk/qk is called the kth convergent of the continued fraction (2). The numerators and denominators of convergents of continued fractions are related by the recursion formulas

Pk + ₁ = a_k + ₁P_k + P_k - ₁ and q_k - ₁ = ^ak + q_k + q_k - ₁

which serve as the foundation for the entire theory of continued fractions. These formulas directly imply the important equation

p_kq_k + ₁ − q_kp_k −₁ = ± ₁

For every infinite continued fraction, there exists a limit

called the value of the given continued fraction. Every irrational number is the value of a unique infinite continued fraction that can be obtained by expanding α in the above manner; for example, (e - l)/2 = [0, 1, 6, 10, 14, 18, ... ] and = [1, 2, 2, . . . ]. Quadratic irrationalities yield periodic continued fractions.

The chief value of continued fractions for applications is that the convergents of a continued fraction are the best approximations of a number α in the sense that for any other fraction m/n whose denominator is at most qk we have the inequality |na —m\\ q_kα − pk |. Here,|q_k a —p_k < 1/q_k + 1- The odd convergents of a continued fraction are greater than α, and the even convergents are less than α. With increasing k, the odd convergents of a continued fraction decrease and the even convergents increase.

Continued fractions are used to approximate irrational numbers by rational numbers. For example, the well-known approximations 22/7 and 355/113 for the number π (ratio of the circumference of a circle to the diameter) are convergents of the continued fraction expansion of π. It should be noted that the first proof that the numbers e and Π are irrational was given in 1766 by the German mathematician J. Lambert using continued fractions. The French mathematician J. Liouville proved that for any algebraic number α of degree n it is possible to find a constant λ such that |a —x/y | > λ/yⁿ for any fraction x/y. By means of continued fractions, it is possible to construct numbers a such that the difference |α —p_k/q_k| can be made less than λ/q_k for an arbitrary choice of the constant λ. It is thus possible to construct transcendental numbers by using continued fractions. A major drawback of continued fractions lies in the extraordinary difficulty (in practice, the impossiblity) of performing arithmetic operations on them. For example, given the elements of two continued fractions, we have no simple way of obtaining the elements of their sum or product.

Continued fractions had already been encountered in the 16th century in the work of R. Bombelli. In the 17th century, continued fractions were studied by J. Wallis, and a number of important properties of continued fractions were discovered by C. Huygens, who investigated them in connection with the theory of gears. In the 18th century, L. Euler made many contributions to the theory of continued fractions.

In the 19th century, many researchers, including P. L. Chebyshev and A. A. Markov, applied continued fractions whose elements are polynomials to the study of orthogonal polynomials.

REFERENCES

Chebyshev, P. L. Polnoe sobranie sochinenii, 2nd ed., vol. 1. Moscow-Leningrad, 1946.
kinchín. A. Ia. Tsepnye drobi, 2nd ed. Moscow-Leningrad, 1949.
Euler, L. Vvedenie v analiz beskonechno malykh, vol. 1. Moscow-Leningrad, 1936. (Translated from Latin.)
Stieltjes, T. J. Issledovaniia o nepreryvnykh drobiakh. Kharkov-Kiev, 1936. (Translated from French.)
Perron, O. Die Lehre von den Kettenbrüchen, 2nd ed. Leipzig-Berlin, 1929.
Wall, H. S. Analytic Theory of Continued Fractions. Toronto-New York-London, 1948.

continued fraction