Noble Number

A noble number nu 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 phi^(-1), whose continued fraction is composed entirely of 1s (except for the a_0 term), [0,1^_].

Any noble number can be written as


where A_k and B_k are the numerator and denominator of the kth convergent of [0,a_1,a_2,...,a_n].

The noble numbers are a subset of Q(sqrt(5)) but not a subfield, since there is no subfield lying properly between Q and Q(sqrt(5)). To see this, consider sqrt(5)=2phi-1, which must be contained in the same field as phi but is not a noble number since its continued fraction is [2,4^_].

See also

Near Noble Number, Periodic Continued Fraction

Explore with Wolfram|Alpha


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.

Referenced on Wolfram|Alpha

Noble Number

Cite this as:

Weisstein, Eric W. "Noble Number." From MathWorld--A Wolfram Web Resource.

Subject classifications