# Continued Fractions

• Richard Zippel
Part of the The Springer International Series in Engineering and Computer Science book series (SECS, volume 241)

## Abstract

By a continued fraction we mean an expression of the form
$${a_0} + \frac{1}{{{a_1} + \frac{1}{{{a_2} + \ddots }}}},$$
When this expression is finite, it represents a rational number. In its infinite form, the continued fraction is interpreted as the limiting value of the sequence
$${a_0},{a_0} + \frac{{{b_1}}}{{{a_1}}},{a_0} + \frac{{{b_1}}}{{{a_1} + \frac{{{b_2}}}{{{a_2}}}}}, \ldots ,$$
if the limit exists. Assuming this sequence converges, denote its limit by a. The elements of this sequence are called the continued fraction convergents of α. When the bi are equal to 1, the elements of the above sequence are quite good approximations to α and, in a certain sense, all of the “best” approximations to α are elements of the sequence.

## Keywords

Rational Number Continue Fraction Integer Part Irrational Number Continue Fraction Expansion
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.