◀ ▲ ▶Branches / Algebra / Definition: Continued Fractions
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.
- 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.
Mentioned in:
Algorithms: 1
Lemmas: 2
Proofs: 3 4
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References
Bibliography
- Schnorr, C.P.: "Lecture Notes Diskrete Mathematik", Goethe University Frankfurt, 2001
- Kraetzel, E.: "Studienbücherei Zahlentheorie", VEB Deutscher Verlag der Wissenschaften, 1981
Footnotes
