Welcome guest
You're not logged in.
224 users online, thereof 0 logged in

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]

(none)