Non-integer representation

A non-integer representation uses non-integer numbers as the radix, or bases, of a positional numeral system. For a non-integer radix β > 1, the value of

[math]\displaystyle{ x=d_n\dots d_2d_1d_0.d_{-1}d_{-2}\dots d_{-m} }[/math]

is

[math]\displaystyle{ \begin{align} x&=\beta^nd_n + \cdots + \beta^2d_2 + \beta d_1 + d_0 \\ &\qquad + \beta^{-1}d_{-1} + \beta^{-2}d_{-2} + \cdots + \beta^{-m}d_{-m}. \end{align} }[/math]

The numbers d_i are non-negative integers less than β. This is also known as a β-expansion, a notion introduced by (Rényi 1957) and first studied in detail by (Parry 1960). Every real number has at least one (possibly infinite) β-expansion.

There are applications of β-expansions in coding theory (Kautz 1965) and models of quasicrystals (Burdik et al. 1998; Thurston 1989).

Construction

β-expansions are a generalization of decimal expansions. While infinite decimal expansions are not unique (for example, 1.000... = 0.999...), all finite decimal expansions are unique. However, even finite β-expansions are not necessarily unique, for example φ + 1 = φ² for β = φ, the golden ratio. A canonical choice for the β-expansion of a given real number can be determined by the following greedy algorithm, essentially due to (Rényi 1957) and formulated as given here by (Frougny 1992).

Let β > 1 be the base and x a non-negative real number. Denote by ⌊x⌋ the floor function of x, that is, the greatest integer less than or equal to x, and let {x} = x − ⌊x⌋ be the fractional part of x. There exists an integer k such that β^k ≤ x < β^k+1. Set

[math]\displaystyle{ d_k = \lfloor x/\beta^k\rfloor }[/math]

and

[math]\displaystyle{ r_k = \{x/\beta^k\}.\, }[/math]

For k − 1 ≥ j > −∞, put

[math]\displaystyle{ d_j = \lfloor\beta r_{j+1}\rfloor, \quad r_j = \{\beta r_{j+1}\}. }[/math]

In other words, the canonical β-expansion of x is defined by choosing the largest d_k such that β^kd_k ≤ x, then choosing the largest d_k−1 such that β^kd_k + β^k−1d_k−1 ≤ x, etc. Thus it chooses the lexicographically largest string representing x.

With an integer base, this defines the usual radix expansion for the number x. This construction extends the usual algorithm to possibly non-integer values of β.

Examples

Base √2

Base √2 behaves in a very similar way to base 2 as all one has to do to convert a number from binary into base √2 is put a zero digit in between every binary digit; for example, 1911₁₀ = 11101110111₂ becomes 101010001010100010101_√2 and 5118₁₀ = 1001111111110₂ becomes 1000001010101010101010100_√2. This means that every integer can be expressed in base √2 without the need of a decimal point. The base can also be used to show the relationship between the side of a square to its diagonal as a square with a side length of 1_√2 will have a diagonal of 10_√2 and a square with a side length of 10_√2 will have a diagonal of 100_√2. Another use of the base is to show the silver ratio as its representation in base √2 is simply 11_√2. In addition, the area of a regular octagon with side length 1_√2 is 1100_√2, the area of a regular octagon with side length 10_√2 is 110000_√2, the area of a regular octagon with side length 100_√2 is 11000000_√2, etc…

Base φ

11_φ = 100_φ.

Base ψ

101_ψ = 1000_ψ

Base e

With base e the natural logarithm behaves like the common logarithm as ln(1_e) = 0, ln(10_e) = 1, ln(100_e) = 2 and ln(1000_e) = 3.

The base e is the most economical choice of radix β > 1 (Hayes 2001), where the radix economy is measured as the product of the radix and the length of the string of symbols needed to express a given range of values.

Base π

Base π can be used to more easily show the relationship between the diameter of a circle to its circumference, which corresponds to its perimeter; since circumference = diameter × π, a circle with a diameter 1_π will have a circumference of 10_π, a circle with a diameter 10_π will have a circumference of 100_π, etc. Furthermore, since the area = π × radius², a circle with a radius of 1_π will have an area of 10_π, a circle with a radius of 10_π will have an area of 1000_π and a circle with a radius of 100_π will have an area of 100000_π.

Properties

In no positional number system can every number be expressed uniquely. For example, in base ten, the number 1 has two representations: 1.000... and 0.999.... The set of numbers with two different representations is dense in the reals (Petkovšek 1990), but the question of classifying real numbers with unique β-expansions is considerably more subtle than that of integer bases (Glendinning Sidorov).

Another problem is to classify the real numbers whose β-expansions are periodic. Let β > 1, and Q(β) be the smallest field extension of the rationals containing β. Then any real number in [0,1) having a periodic β-expansion must lie in Q(β). On the other hand, the converse need not be true. The converse does hold if β is a Pisot number (Schmidt 1980), although necessary and sufficient conditions are not known.

See also

• Beta encoder
• Non-standard positional numeral systems
• Decimal expansion
• Power series
• Ostrowski numeration

References

• Bugeaud, Yann (2012), Distribution modulo one and Diophantine approximation, Cambridge Tracts in Mathematics, 193, Cambridge: Cambridge University Press, ISBN 978-0-521-11169-0 
• Burdik, Č.; Frougny, Ch.; Gazeau, J. P.; Krejcar, R. (1998), "Beta-integers as natural counting systems for quasicrystals", Journal of Physics A: Mathematical and General 31 (30): 6449–6472, doi:10.1088/0305-4470/31/30/011, ISSN 0305-4470 .
• Frougny, Christiane (1992), "How to write integers in non-integer base", LATIN '92, Lecture Notes in Computer Science, 583/1992, Springer Berlin / Heidelberg, pp. 154–164, doi:10.1007/BFb0023826, ISBN 978-3-540-55284-0, ISSN 0302-9743, https://books.google.com/books?id=I3fC6batwokC&lpg=PA154&pg=PA154#v=onepage&q=&f=false .
• Glendinning, Paul; Sidorov, Nikita (2001), "Unique representations of real numbers in non-integer bases", Mathematical Research Letters 8 (4): 535–543, doi:10.4310/mrl.2001.v8.n4.a12, ISSN 1073-2780, http://intlpress.com/site/pub/pages/journals/items/mrl/content/vols/0008/0004/00019835/index.html .
• Hayes, Brian (2001), "Third base", American Scientist 89 (6): 490–494, doi:10.1511/2001.40.3268, http://www.americanscientist.org/issues/pub/third-base/2 .
• Kautz, William H. (1965), "Fibonacci codes for synchronization control", Institute of Electrical and Electronics Engineers. Transactions on Information Theory IT-11 (2): 284–292, doi:10.1109/TIT.1965.1053772, ISSN 0018-9448, http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?tp=&arnumber=1053772&isnumber=22626 .
• Parry, W. (1960), "On the β-expansions of real numbers", Acta Mathematica Academiae Scientiarum Hungaricae 11 (3–4): 401–416, doi:10.1007/bf02020954, ISSN 0001-5954 .
• Petkovšek, Marko (1990), "Ambiguous numbers are dense", The American Mathematical Monthly 97 (5): 408–411, doi:10.2307/2324393, ISSN 0002-9890 .
• Rényi, Alfréd (1957), "Representations for real numbers and their ergodic properties", Acta Mathematica Academiae Scientiarum Hungaricae 8 (3–4): 477–493, doi:10.1007/BF02020331, ISSN 0001-5954 .
• Schmidt, Klaus (1980), "On periodic expansions of Pisot numbers and Salem numbers", The Bulletin of the London Mathematical Society 12 (4): 269–278, doi:10.1112/blms/12.4.269, ISSN 0024-6093 .
• Thurston, W.P. (1989), "Groups, tilings and finite state automata", AMS Colloquium Lectures 

Further reading

• Sidorov, Nikita (2003), "Arithmetic dynamics", in Bezuglyi, Sergey; Kolyada, Sergiy, Topics in dynamics and ergodic theory. Survey papers and mini-courses presented at the international conference and US-Ukrainian workshop on dynamical systems and ergodic theory, Katsiveli, Ukraine, August 21–30, 2000, Lond. Math. Soc. Lect. Note Ser., 310, Cambridge: Cambridge University Press, pp. 145–189, ISBN 978-0-521-53365-2 

External links

• Weisstein, Eric W.. "Base". http://mathworld.wolfram.com/Base.html.