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 .

REFERENCES:

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, p. 236, 1979.

Schroeder, M. "Noble and Near Noble Numbers." In Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. New York: W. H. Freeman, pp. 392-394, 1991.