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

$$[q_0;q_1,q_2,\ldots]:=q_0+\cfrac{1}{q_1+\cfrac{1}{q_2+\cfrac{1}\ddots}}.$$

Notes

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

| | | | | created: 2019-05-11 08:28:24 | modified: 2019-07-14 10:34:59 | by: bookofproofs | references: [8186], [8189]

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Bibliography (further reading)

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

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