Noble Number
A noble number 
is defined as an irrational
number having a continued fraction that
becomes an infinite sequence of 1s at some point,
![nu=[0,a_1,a_2,...,a_n,1^_].](/National_Library/20161222123739im_/https://mathworld.wolfram.com/images/equations/NobleNumber/NumberedEquation1.gif) |
The prototype is the inverse of the golden ratio 
, whose continued
fraction is composed entirely of 1s (except for the 
term), ![[0,1^_]](/National_Library/20161222123739im_/https://mathworld.wolfram.com/images/equations/NobleNumber/Inline4.gif)
.
Any noble number can be written as
 |
where 
and 
are the numerator
and denominator of the 
th convergent
of ![[0,a_1,a_2,...,a_n]](/National_Library/20161222123739im_/https://mathworld.wolfram.com/images/equations/NobleNumber/Inline8.gif)
.
The noble numbers are a subset of 
but not
a subfield, since there is no subfield lying properly
between 
and 
. To see
this, consider 
, which must be contained
in the same field as 
but is not a noble number since its
continued fraction is ![[2,4^_]](/National_Library/20161222123739im_/https://mathworld.wolfram.com/images/equations/NobleNumber/Inline14.gif)
.
Contact the MathWorld Team
continued fractions