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^_].](/images/equations/NobleNumber/NumberedEquation1.svg) |
The prototype is the inverse of the golden ratio 
, whose continued
fraction is composed entirely of 1s (except for the 
term), ![[0,1^_]](/images/equations/NobleNumber/Inline4.svg)
.
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]](/images/equations/NobleNumber/Inline8.svg)
.
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^_]](/images/equations/NobleNumber/Inline14.svg)
.
