Continued Fraction - Infinite Continued Fractions

Infinite Continued Fractions

Every infinite continued fraction is irrational, and every irrational number can be represented in precisely one way as an infinite continued fraction.

An infinite continued fraction representation for an irrational number is mainly useful because its initial segments provide excellent rational approximations to the number. These rational numbers are called the convergents of the continued fraction. Even-numbered convergents are smaller than the original number, while odd-numbered ones are bigger.

For a continued fraction, the first four convergents (numbered 0 through 3) are

\frac{a_0}{1},\qquad \frac{a_1a_0 + 1}{a_1},\qquad \frac{ a_2(a_1a_0+1)+a_0}{a_2a_1+1},\qquad \frac{a_3(a_2(a_1a_0+1)+a_0)+(a_1a_0+1)}{a_3(a_2a_1+1)+a_1}.

In words, the numerator of the third convergent is formed by multiplying the numerator of the second convergent by the third quotient, and adding the numerator of the first convergent. The denominators are formed similarly. Therefore, each convergent can be expressed explicitly in terms of the continued fraction as the ratio of certain multivariate polynomials called continuants.

If successive convergents are found, with numerators h1, h2, ... and denominators k1, k2, ... then the relevant recursive relation is:

h_n=a_nh_{n-1}+h_{n-2},\qquad k_n=a_nk_{n-1}+k_{n-2}.

The successive convergents are given by the formula

\frac{h_n}{k_n}= \frac{a_nh_{n-1}+h_{n-2}}{a_nk_{n-1}+k_{n-2}}.

Thus to incorporate a new term into a rational approximation, only the two previous convergents are necessary. The initial "convergents" (required for the first two terms) are 0⁄1 and 1⁄0. For example, here are the convergents for .

n −2 −1 0 1 2 3 4
an 0 1 5 2 2
hn 0 1 0 1 5 11 27
kn 1 0 1 1 6 13 32

When using the Babylonian method to generate successive approximations to the square root of an integer, if one starts with the lowest integer as first approximant, the rationals generated all appear in the list of convergents for the continued fraction. Specifically, the approximants will appear on the convergents list in positions 0, 1, 3, 7, 15, … , … For example, the continued fraction expansion for √3 is . Comparing the convergents with the approximants derived from the Babylonian method:

n −2 −1 0 1 2 3 4 5 6 7
an 1 1 2 1 2 1 2 1
hn 0 1 1 2 5 7 19 26 71 97
kn 1 0 1 1 3 4 11 15 41 56

Read more about this topic:  Continued Fraction

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