2.2 Continued Fractions

To discuss angles in integer geometry, we need to introduce the classical notion of continued fractions.

Definition 2.2.1.

Let $\alpha$ be a real number. Then for real numbers $a_{0},a_{1},a_{2},a_{3},\ldots$ , a continued fraction expansion of $\alpha$ (or simply a continued fraction of $\alpha$ ) is a sum of the form

\alpha=a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{a_{3}+\ldots}}}.

When this expression has infinite number of terms $a_{i}$ , we call it an infinite continued fraction.

We use the following classical notation $[a_{0},a_{1},a_{2},a_{3},\ldots]$ .

Definition 2.2.2.

Let $[a_{0},a_{1},a_{2},\ldots,a_{n}]$ be a continued fraction expansion of $\alpha$ . This continued fraction is called odd/even if it has an odd/even number elements (i.e., $n\equiv 0\mod 2$ or respectively $n\equiv 1\mod 2$ ).

Example 2.2.3.

A continued fraction for $\pi$ is

[\pi-1;3,\frac{-1}{2}].

Definition 2.2.4.

A continued fraction is called regular if $a_{0}$ is an integer and each element $a_{i}$ is a positive integer.

Theorem 2.2.5.

Let $\alpha$ be a real number. Then

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if $\alpha\in\mathbb{Q}$ , there is one unique odd regular continued fraction and one unique even regular continued fraction of $\alpha$ ;
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otherwise, there is some unique infinite continued fraction such that the limit as $k\to\infty$ of $[a_{0},a_{1},\ldots,a_{k}]$ converges to $\alpha$ . ∎

For the proof, we refer to.

Example 2.2.6.

The regular continued fraction for $\pi$ is

[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,2,...].

Let us now consider the geometric construction of a continued fraction.

Definition 2.2.7.

Let $\alpha$ be a real number.

Consider the ray $\{y=\alpha x:x\geq 0\}$ . The complement of this ray has two connected components in $\mathbb{R}^{2}_{+}$ . We construct two convex hulls in each connected component.

The two sails of a ray are the boundaries of these convex hulls. The vertices of the upper/left sail are denoted $B_{0},B_{1},\ldots$ , and those of the lower/right sail are denoted $A_{0},A_{1},\ldots$ .

Theorem 2.2.8.

Let $\alpha\geq 1$ be a real number with sails $A_{0},A_{1},\ldots$ and $B_{0},B_{1},\ldots$ and continued fraction $[a_{0},a_{1},\ldots]$ . Then the following equalities hold:

\operatorname{l\ell}(A_{i},A_{i+1})=a_{2i},\\ \operatorname{l\ell}(B_{i},B_{i+1})=a_{2i+1}.

For the proof we refer to.

Example 2.2.9.

Let $\alpha=5/3$ . Then the regular continued fractions for $\alpha$ are:

[1;1,2]\text{ and }[1;1,1,1].

From Figure 3, we see that this agrees with Theorem 2.2.8.