# Non-integer representation

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 numbering system. For a non-integer radix β > 1, the value of

is

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).

## 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

and

For *k* − 1 ≥ *j* > −∞, put

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 φ

See Golden ratio base; 11_{φ} = 100_{φ}.

### 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}.

### 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}.

## 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 2001).

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, Zbl pre06066616 - 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, MR 1644115. - 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. - 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, MR 1851269. - Hayes, Brian (2001), "Third base",
*American Scientist*,**89**(6): 490–494, doi:10.1511/2001.40.3268. - Kautz, William H. (1965), "Fibonacci codes for synchronization control",
*Institute of Electrical and Electronics Engineers. Transactions on Information Theory*, IT-11: 284–292, ISSN 0018-9448, MR 0191744. - Parry, W. (1960), "On the β-expansions of real numbers",
*Acta Mathematica Academiae Scientiarum Hungaricae*,**11**: 401–416, doi:10.1007/bf02020954, ISSN 0001-5954, MR 0142719. - Petkovšek, Marko (1990), "Ambiguous numbers are dense",
*The American Mathematical Monthly*,**97**(5): 408–411, doi:10.2307/2324393, ISSN 0002-9890, MR 1048915. - Rényi, Alfréd (1957), "Representations for real numbers and their ergodic properties",
*Acta Mathematica Academiae Scientiarum Hungaricae*,**8**: 477–493, doi:10.1007/BF02020331, ISSN 0001-5954, MR 0097374. - 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, MR 576976.

## 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 0-521-53365-1, Zbl 1051.37007