# Non-integer representation

Non-integer representation

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

$x=d_n\dots d_2d_1d_0.d_{-1}d_{-2}\dots d_{-m}$

is

$x=\beta^nd_n + \cdots + \beta^2d_2 + \beta d_1 + d_0 + \beta^{-1}d_{-1} + \beta^{-2}d_{-2} + \cdots + \beta^{-m}d_{-m}.$

The numbers di 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 = 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 βkx < βk+1. Set

$d_k = \lfloor x/\beta^k\rfloor$

and

$r_k = \{x/\beta^k\}.\,$

For k − 1 ≥ j > −∞, put

$d_j = \lfloor\beta r_{j+1}\rfloor, \quad r_j = \{\beta r_{j+1}\}.$

In other words, the canonical β-expansion of x is defined by choosing the largest dk such that βkdkx, then choosing the largest dk−1 such that βkdk + βk−1dk−1x, 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; 11φ = 100φ.

### Base e

With base e the natural logarithm behaves like the common logarithm as ln(1e) = 0, ln(10e) = 1, ln(100e) = 2 and ln(1000e) = 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; since circumference = diameter × π, a circle with a diameter 1π will have a circumference of 10π, a circle with a diameter 10π will have a circumference of 100π, etc. Furthermore, since the area = π × radius2, a circle with a radius of 1π will have an area of 10π, a circle with a radius of 10π will have an area of 1000π and a circle with a radius of 100π will have an area of 100000π.

### 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.

## Properties

In no positional number system can every number be expressed uniquely. For example, in base 10, 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.

## References

• Hayes, Brian (2001), "Third base", American Scientist 89 (6): 490–494 .
• Parry, W. (1960), "On the β-expansions of real numbers", Acta Mathematica Academiae Scientiarum Hungaricae 11: 401–416, ISSN 0001-5954, MR0142719 .

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