A noble number is defined as an irrational number having a continued fraction that becomes an infinite sequence of 1s at some point,
The prototype is the inverse of the golden ratio , whose continued fraction is composed entirely of 1s (except for the term), .
Any noble number can be written as
where and are the numerator and denominator of the th convergent of .
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 .