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 positive integers $q_0, q_1,q_2,\ldots \in\mathbb Z$ of the following form:



Algorithms: 1
Lemmas: 2
Proofs: 3 4

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  1. Schnorr, C.P.: "Lecture Notes Diskrete Mathematik", Goethe University Frankfurt, 2001
  2. Kraetzel, E.: "Studienbücherei Zahlentheorie", VEB Deutscher Verlag der Wissenschaften, 1981


  1. This is only a formal definition, since the question, whether an infinite continued fraction is convergent, is not answered yet.