# Non-integer representation

This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. (March 2019) (Learn how and when to remove this template message) |

Numeral systems |
---|

Hindu–Arabic numeral system |

East Asian |

Alphabetic |

Former |

Positional systems by base |

Non-standard positional numeral systems |

List of numeral systems |

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 = φ^{2} 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^{k}*d*_{k} ≤ *x*, then choosing the largest *d*_{k−1} such that β^{k}*d*_{k} + β^{k−1}*d*_{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, 1911_{10} = 11101110111_{2} becomes 101010001010100010101_{√2} and 5118_{10} = 1001111111110_{2} 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 × radius^{2}, 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}.^{[1]}

## 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