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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

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)

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

*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*, 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*.

Following the steps above, we can create a *β*-expansion for a real number (the steps are identical for an , 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 , where is n written in base β). The k value for n and β can be written as:

After a k value is found, can be written as d, where

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 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 and , 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.

- 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(""); }

- 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; }

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…

In the golden base, some numbers have more than one decimal base equivalent: they are **ambiguous**. For example:
11_{φ} = 100_{φ}.

There are some numbers in base ψ that are also ambiguous. For example, 101_{ψ} = 1000_{ψ}.

This section may have misleading content. Please help clarify the content. (May 2024)

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 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}.^{[5]}

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.

- Beta encoder
- Non-standard positional numeral systems
- Decimal expansion
- Power series
- Ostrowski numeration

**^**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

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*AMS Colloquium Lectures*

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