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 .

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.

CITE THIS AS:

Weisstein, Eric W. "Noble Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NobleNumber.html