Continued fraction

Definition

A continued fraction is given by a number $b_0$ and a finite or infinite sequence of pairs $(a_1, b_1), (a_2, b_2), \dots$ of numbers. It stands for the expression $$b_0 + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cdots}}}$$ The $a_n$ are the partial numerators and the $b_n$ are the partial denominators. The $n$-th convergent is the value of the expression cut off after $(a_n, b_n)$. The value of an infinite continued fraction is the limit of its convergents, when this limit exists.

Remarks

When $a_n = 1$ for all $n$, $b_0$ is an integer and $b_n$ is a positive integer for $n \ge 1$, the continued fraction is a simple continued fraction. For example, $\sqrt{2} = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cdots}}$. Other continued fractions give formulas for $\pi$ and for functions such as $\tan x$.

Explanations

L1 human

A continued fraction is a fraction whose denominator contains another fraction, and so on. Cut it off at some level, and you get an ordinary fraction. These fractions often approximate a number very well.