非整数进位制

非整数进位制是指底数不是正整数的进位制。对于一个非正整数的底数β > 1，以下的数值: $x=d_{n}\dots d_{2}d_{1}d_{0}.d_{-1}d_{-2}\dots d_{-m}$ 为

x=\beta ^{n}d_{n}+\cdots +\beta ^{2}d_{2}+\beta d_{1}+d_{0}+\beta ^{-1}d_{-1}+\beta ^{-2}d_{-2}+\cdots +\beta ^{-m}d_{-m}.

而数字d_i为小于β的非负整数。此进位制可以配合所使用β，称为β进制或β展开，后者的名称是数学家Rényi在1957年开始使用^[1]，而数学家Parry在1960年第一个进行相关的研究^[2]。每一个实数至少有一个β进位制的表示方式（也可能是无限多个）。

β进制可以应用在编码理论^[3]及准晶体模型的描述^[4]^[5]。

参考文献

[1]
Rényi, Alfréd, Representations for real numbers and their ergodic properties, Acta Mathematica Academiae Scientiarum Hungaricae, 1957, 8 (3–4): 477–493, ISSN 0001-5954, MR 0097374, S2CID 122635654, doi:10.1007/BF02020331, hdl:10338.dmlcz/102491 
[2]
Parry, W., On the β-expansions of real numbers, Acta Mathematica Academiae Scientiarum Hungaricae, 1960, 11 (3–4): 401–416, ISSN 0001-5954, MR 0142719, S2CID 116417864, doi:10.1007/bf02020954, hdl:10338.dmlcz/120535 
[3]
Kautz, William H., Fibonacci codes for synchronization control, Institute of Electrical and Electronics Engineers. Transactions on Information Theory, 1965, IT–11 (2): 284–292, ISSN 0018-9448, MR 0191744, doi:10.1109/TIT.1965.1053772
[4]
Burdik, Č.; Frougny, Ch.; Gazeau, J. P.; Krejcar, R., Beta-integers as natural counting systems for quasicrystals, Journal of Physics A: Mathematical and General, 1998, 31 (30): 6449–6472, Bibcode:1998JPhA...31.6449B, CiteSeerX 10.1.1.30.5106 , ISSN 0305-4470, MR 1644115, doi:10.1088/0305-4470/31/30/011
[5]
Thurston, W.P., Groups, tilings and finite state automata, AMS Colloquium Lectures, 1989