Simple continued fraction

Definition

A simple continued fraction is a continued fraction whose partial numerators are all $1$. Its first term $a_0$ is an integer, and its later terms $a_1, a_2, \dots$ are positive integers: $$[a_0; a_1, a_2, \dots] = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cdots}}$$ The sequence of terms can be finite or infinite.

Remarks

The $a_n$ are called the partial quotients. For example, $[2; 3, 4] = 2 + \frac{1}{3 + 1/4} = \frac{30}{13}$. An infinite simple continued fraction always converges, by the convergence of the convergents. Each real number has a continued fraction expansion of this form.

Explanations

L1 human

A simple continued fraction is a list of integers $[a_0; a_1, a_2, \dots]$ that encodes a number. To read it, start at the end, take a reciprocal, add the next term, and repeat.