 Non-integer Base of Numeration
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Non-integer Base of Numeration

A non-integer representation uses non-integer numbers as the radix, or base, of a positional numeral system. For a non-integer radix ? > 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 di are non-negative integers less than ?. This is also known as a ?-expansion, a notion introduced by Rényi (1957) and first studied in detail by Parry (1960). Every real number has at least one (possibly infinite) ?-expansion. The set of all ?-expansions that have a finite representation is a subset of the ring Z[?,?-1].

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

## Construction

?-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 ?-expansions are not necessarily unique, for example ? + 1 = ?2 for ? = ?, the golden ratio. A canonical choice for the ?-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 ? > 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 ?k x < ?k+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 ?-expansion of x is defined by choosing the largest dk such that ?kdk x, then choosing the largest dk-1 such that ?kdk + ?k-1dk-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 ?.

### Conversion

Following the steps above, we can create a ?-expansion for a real number $n\geq 0$ . The steps are identical for an $n<0$ , although n must first be multiplied by to make it positive, then the result must be multiplied by 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 >= j > -?. The first k values of d appear to the left of the decimal place.

This can also be written in the following pseudo-code:

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 , it will be represented as instead of A.

### Example implementation code

#### To base ?

• Javascript:
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)
break;
}

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

return digits.join("");
}


#### From base ?

• Javascript:
function fromBasePI(num) {
let numberSplit = num.split(/\./g);
let numberLength = numberSplit.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

Base behaves in a very similar way to base 2 as all one has to do to convert a number from binary into base is put a zero digit in between every binary digit; for example, 191110 = 111011101112 becomes 101010001010100010101 and 511810 = 10011111111102 becomes 1000001010101010101010100. This means that every integer can be expressed in base 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 will have a diagonal of 10 and a square with a side length of 10 will have a diagonal of 100. Another use of the base is to show the silver ratio as its representation in base is simply 11. In addition, the area of a regular octagon with side length 1 is 1100, the area of a regular octagon with side length 10 is 110000, the area of a regular octagon with side length 100 is 11000000, etc...

### Golden base

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

### Base ?

There are also some numbers in base ? are also ambiguous. For example, 101? = 1000?.

### 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 ? > 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 ?

Base ? 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 × ?, a circle with a diameter 1? will have a circumference of 10?, a circle with a diameter 10? will have a circumference of 100?, etc. Furthermore, since the area = ? × radius2, a circle with a radius of 1? will have an area of 10?, a circle with a radius of 10? will have an area of 1000? and a circle with a radius of 100? will have an area of 100000?.

## 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 (Petkov?ek 1990), but the question of classifying real numbers with unique ?-expansions is considerably more subtle than that of integer bases (Glendinning & Sidorov 2001).

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