In the following, we will introduce *continued fractions*, which will help us to find a representation of real numbers, which is independent of any basis and with the help of which we will be able to reveal the structure of a given real number.

**Definition**: Continued Fractions

A **continued fraction** is a ratio built from the real number $x_0\in\mathbb R$ and any positive real numbers $x_1,x_2,\ldots\in\mathbb R$ of the following form:

$$[x_0;x_1,x_2,\ldots]:=x_0+\frac{1}{x_1+\frac{1}{x_2+\frac{1}\ddots}}.$$

If the sequence $(x_n)_{n\in\mathbb N}$ is infinite and convergent, then $[x_0;x_1,x_2,\ldots]$ is called an **infinite continued fraction**. A the case of a finite sequence, then $[x_0;x_1,x_2,\ldots,x_n]$ is called a **finite continued fraction**.

| | | | | created: 2019-05-11 08:28:24 | modified: 2019-05-11 17:58:56 | by: *bookofproofs* | references: [8186], [8189]

[8189] **Kraetzel, E.**: “Studienbücherei Zahlentheorie”, VEB Deutscher Verlag der Wissenschaften, 1981

[8186] **Schnorr, C.P.**: “Lecture Notes Diskrete Mathematik”, Goethe University Frankfurt, 2001

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