Noble number

As noble numbers is called such irrational numbers whose infinite continued fraction representation from any location contains only ones.

They are closely related to the golden number and are characterized by the fact that they are particularly difficult to approximate using rational numbers . Noble numbers are used in the theory of dynamic systems . ${\displaystyle \Phi }$

The "classiest" number

The infinite continued fraction for the golden number is:

${\displaystyle \Phi ={\frac {1}{2}}\left(1+{\sqrt {5}}\right)=1+{\frac {1}{1+{\cfrac {1}{1+{\frac {1}{1+{\frac {1}{1+...}}}}}}}}.}$

The golden number can therefore be called the “noblest” number - its continued fraction representation contains only ones from the start.

Countability

The set of noble numbers is a subset of the algebraic numbers and therefore countable .

Their countability can also be easily shown using the continued fraction expansion (the shorthand for a regular continued fraction is on the right):

The image

${\displaystyle [1]\rightarrow [\Phi ]}$

${\displaystyle [a_{0}]\rightarrow [a_{0};\Phi ]\;\;(1\neq a_{0}\in \mathbb {Z} )}$

${\displaystyle [a_{0};a_{1},\ldots ,a_{n}]\rightarrow [a_{0};a_{1},\ldots ,a_{n},\Phi ]\quad (a_{0}\in \mathbb {Z} ;a_{k}\in \mathbb {N} _{>0}\ \mathrm {f{\ddot {u}}r} \ 1\leq k\leq n;a_{n}\neq 1;n>0)}$

of on the set of noble numbers is bijective . ${\displaystyle \mathbb {Q} }$

Almost noble numbers

As almost noble numbers such real numbers in the interval will be referred to, the chain breaking developments are periodic (period length be with hereinafter) and in which: after each of ones followed by a fixed natural number . Therefore applies to every almost noble number ${\displaystyle [0,1]}$ ${\displaystyle p}$ ${\displaystyle p-1}$ ${\displaystyle n>1}$ ${\displaystyle u}$

${\displaystyle u=[0;1,1,\ldots ,1,n,u^{-1}]}$.

Literature and web links

• Manfred Schroeder : Number theory in science and communication , 5th edition, Springer, 2009
• Eric W. Weisstein: Noble numbers and Near Noble Numbers on MathWorld .

Individual evidence

1. ↑ According to the Caroline Series: The Geometry of Markoff Numbers , Math. Intell. 7 (1985) by IC Percival .