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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:


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]

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Bibliography (further reading)

[8189] Kraetzel, E.: “Studienb├╝cherei Zahlentheorie”, VEB Deutscher Verlag der Wissenschaften, 1981

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

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