is called the -expansion,
or -expansion (Parry 1957), of the positive
real number
if, for all ,
, where is the floor function
and need not be an integer. Any real number
such that has such an expansion, as can be found
using the greedy algorithm (Allouche and Cosnard
2000).

The special case of ,
, and or 1 is sometimes called a -development (Komornik and Loreti 1998). gives the only 2-development. However, for almost all
, there are an infinite number
of different -developments.
Even more surprisingly though, there exist exceptional for which there exists only a single -development (Erdős et al. 1990, 1991, Komornik
and Loreti 1998). Furthermore, there is a smallest number known as the Komornik-Loreti
constant for which there exists a unique -development (Komornik and Loreti 1998).

Allouche, J.-P. and Cosnard, M. "The Komornik-Loreti Constant Is Transcendental." Amer. Math. Monthly107, 448-449,
2000.Erdős, P.; Horváth, M.; and Joó, I. "On
the Uniqueness of the Expansions ." Acta. Math. Hungar.58, 333-342,
1991.Erdős, P.; Joó, I.; and Komornik, V. "Characterization
of the Unique Expansions
and Related Problems." Bull. Soc. Math. France118, 377-390, 1990.Komornik,
V. and Loreti, P. "Unique Developments in Non-Integer Bases." Amer.
Math. Monthly105, 636-639, 1998.Parry, W. "On the -Expansions of Real Numbers."
Acta Math. Acad. Sci. Hungar.11, 401-416, 1960.Rényi,
A. "Representations for Real Numbers and Their Ergodic Properties." Acta
Math. Acad. Sci. Hungar.8, 477-493, 1957.