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

x=d_{n}\dots d_{2}d_{1}d_{0}.d_{-1}d_{-2}\dots d_{-m}

is

{\begin{aligned}x&=\beta ^{n}d_{n}+\cdots +\beta ^{2}d_{2}+\beta d_{1}+d_{0}\\&\qquad +\beta ^{-1}d_{-1}+\beta ^{-2}d_{-2}+\cdots +\beta ^{-m}d_{-m}.\end{aligned}}

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. The set of all β-expansions that have a finite representation is a subset of the ring Z[β, β⁻¹].

There are applications of β-expansions in coding theory^[1] and models of quasicrystals.^[2]

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 $β k \leq x < β k +1$ . Set

d_{k}=\lfloor x/\beta ^{k}\rfloor

and

r_{k}=\{x/\beta ^{k}\}.\,

For $k - 1 \geq j > -\infty$ , 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 d_k such that $β k d k \leq x$ , then choosing the largest d_k−1 such that $β k d k + β k -1 d k -1 \leq x$ , and so on. 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.

Conversion

Following the steps above, we can create a β-expansion for a real number $n\geq 0$ (the steps are identical for an $n<0$ , although $n$ must first be multiplied by −1 to make it positive, then the result must be multiplied by −1 to make it negative again).

First, we must define our $k$ value (the exponent of the nearest power of $β$ greater than $n$ , as well as the amount of digits in $\lfloor n_{\beta }\rfloor$ , where $n_{\beta }$ is $n$ written in base $β$ ). The $k$ value for $n$ and $β$ can be written as:

k=\lfloor \log _{\beta }(n)\rfloor +1

After a $k$ value is found, $n_{\beta }$ can be written as $d$ , where

d_{j}=\lfloor (n/\beta ^{j}){\bmod {\beta }}\rfloor ,\quad n=n-d_{j}*\beta ^{j}

for $k - 1 \geq j > -\infty$ . The first $k$ values of $d$ appear to the left of the decimal place.

This can also be written in the following pseudocode:^[3]

function toBase(n, b) {
	k = floor(log(b, n)) + 1
	precision = 8
	result = ""

	for (i = k - 1, i > -precision-1, i--) {
		if (result.length == k) result += "."
		
		digit = floor((n / b^i) mod b)
		n -= digit * b^i
		result += digit
	}

	return result
}

Note that the above code is only valid for $1<\beta \leq 10$ and $n\geq 0$ , as it does not convert each digits to their correct symbols or correct negative numbers. For example, if a digit's value is 10, it will be represented as 10 instead of $A$ .

Example implementation code

To base $π$

JavaScript:^[3]

function toBasePI(num, precision = 8) {    
    let k = Math.floor(Math.log(num)/Math.log(Math.PI)) + 1;
    if (k < 0) k = 0;

    let digits = [];

    for (let i = k-1; i > (-1*precision)-1; i--) {
        let digit = Math.floor((num / Math.pow(Math.PI, i)) % Math.PI);
        num -= digit * Math.pow(Math.PI, i);
        digits.push(digit);

        if (num < 0.1**(precision+1) && i <= 0)
            break;
    }

    if (digits.length > k)
        digits.splice(k, 0, ".");

    return digits.join("");
}

From base $π$

JavaScript:^[3]

function fromBasePI(num) {
    let numberSplit = num.split(/\./g);
    let numberLength = numberSplit[0].length;

    let output = 0;
    let digits = numberSplit.join("");

    for (let i = 0; i < digits.length; i++) {
        output += digits[i] * Math.pow(Math.PI, numberLength-i-1);
    }

    return output;
}

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…

Golden base

In the golden base, some numbers have more than one decimal base equivalent: they are ambiguous. For example: 11_φ = 100_φ.

Base ψ

There are some numbers in base ψ that are also ambiguous. For example, 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,^[4] 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.^[5]

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,^[6] but the question of classifying real numbers with unique β-expansions is considerably more subtle than that of integer bases.^[7]

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,^[8] although necessary and sufficient conditions are not known.

References

Footnotes

^ Kautz 1965
^ Burdik et al. 1998; Thurston 1989
^ ^a ^b ^c "Home", decimalsystem.js.org
^ Hayes 2001
^ "Weird Number Bases", DataGenetics, retrieved 2018-02-01
^ Petkovšek 1990
^ Glendinning & Sidorov 2001
^ Schmidt 1980

Sources

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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, Bibcode:1998JPhA...31.6449B, CiteSeerX 10.1.1.30.5106, 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, vol. 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, archived from the original on 2016-03-24.
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, MR 0191744.
Parry, W. (1960), "On the β-expansions of real numbers", Acta Mathematica Academiae Scientiarum Hungaricae, 11 (3–4): 401–416, doi:10.1007/bf02020954, hdl:10338.dmlcz/120535, ISSN 0001-5954, MR 0142719, S2CID 116417864.
Petkovšek, Marko (1990), "Ambiguous numbers are dense", The American Mathematical Monthly, 97 (5): 408–411, doi:10.2307/2324393, ISSN 0002-9890, JSTOR 2324393, MR 1048915.
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, hdl:10338.dmlcz/102491, ISSN 0001-5954, MR 0097374, S2CID 122635654.
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, hdl:10338.dmlcz/141479, ISSN 0024-6093, MR 0576976.
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