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

• If the sequence $(q_n)_{n\in\mathbb N}$ is infinite, then $[q_0;q_1,q_2,\ldots]$ is called an infinite continued fraction.1
• A the case of a finite sequence, $[q_0;q_1,q_2,\ldots,q_n]$ is called a finite continued fraction.
• In a continued fraction (both finite or infinite), we sometimes consider the first $k+1$ elements $[q_0;q_1,q_2,\ldots,q_k]$ for a $k\ge 0$ and call them the $k$-th convergent of the continued fraction. If this is a section of a finite continued fraction, then we require $0\le k\le n.$
• See continued fraction. Python algorithm for practical calculation of the $k$-th section.

Algorithms: 1
Lemmas: 2
Proofs: 3 4

Github: ### References

#### Bibliography

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

#### Footnotes

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