gen-in-3-4-sentences-explain-continued-fractions-state-any-key-f-6481

In 3-4 sentences, explain continued fractions. State any key formula.

A continued fraction is an expression of the form $a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{a_3 + \cdots}}}$, often written compactly as $[a_0; a_1, a_2, a_3, \ldots]$. Any real number can be represented as a continued fraction, and for rational numbers this representation is finite, while irrational numbers have infinite continued fractions. The key recursive formula for the convergents (rational approximations) $\frac{p_n}{q_n}$ is: $p_n = a_n p_{n-1} + p_{n-2}$ and $q_n = a_n q_{n-1} + q_{n-2}$, with initial values $p_{-1}=1, p_0=a_0, q_{-1}=0, q_0=1$. Continued fractions provide the best rational approximations to real numbers and reveal deep properties of numbers like the golden ratio $\phi = [1; 1, 1, 1, \ldots]$.