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Anon-integer representation uses non-integer numbers as theradix, or base, of apositional numeral system. For a non-integer radixβ > 1, the value of

${\displaystyle x=d_n\dots d_2{d}_1{d}_0.d_{-1}{d}_{-2}\dots d_{-m}}$

is

The numbersd_i are non-negative integers less thanβ. This is also known as aβ-expansion, a notion introduced byRényi (1957) and first studied in detail byParry (1960). Everyreal number has at least one (possibly infinite)β-expansion. Theset of allβ-expansions that have a finite representation is asubset of theringZ[β, β⁻¹].

There are applications ofβ-expansions incoding theory^[1] and models ofquasicrystals.^[2]

β-expansions are a generalization ofdecimal expansions. While infinite decimal expansions are not unique (for example, 1.000... =0.999...), all finite decimal expansions are unique. However, even finiteβ-expansions are not necessarily unique, for exampleφ + 1 =φ² forβ =φ, thegolden ratio. A canonical choice for theβ-expansion of a given real number can be determined by the followinggreedy algorithm, essentially due toRényi (1957) and formulated as given here byFrougny (1992).

Letβ > 1 be the base andx a non-negative real number. Denote by⌊x⌋ thefloor function ofx (that is, the greatest integer less than or equal tox) and let{x} =x − ⌊x⌋ be the fractional part ofx.There exists an integerk such thatβ^k ≤x <β^k+1. Set

${\displaystyle d_k=\lfloor x/{\beta }^k\rfloor }$

and

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

Fork − 1 ≥  j > −∞, put

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

In other words, the canonicalβ-expansion ofx is defined by choosing the largestd_k such thatβ^kd_k ≤x, then choosing the largestd_k−1 such thatβ^kd_k + β^k−1d_k−1 ≤x, and so on. Thus it chooses thelexicographically largest string representingx.

With an integer base, this defines the usual radix expansion for the numberx. This construction extends the usual algorithm to possibly non-integer values ofβ.

Following the steps above, we can create aβ-expansion for a real number${\displaystyle n\ge 0}$ (the steps are identical for an, althoughn 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 ourk value (the exponent of the nearest power ofβ greater thann, as well as the amount of digits in${\displaystyle \lfloor n_{\beta }\rfloor }$, where${\displaystyle n_{\beta }}$ isn written in baseβ). Thek value forn andβ can be written as:

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

After ak value is found,${\displaystyle n_{\beta }}$ can be written asd, where

${\displaystyle d_j=\lfloor (n/{\beta }^j)mod\beta \rfloor ,n=n-d_j\ast {\beta }^j}$

fork − 1 ≥  j > −∞. The firstk values ofd appear to the left of the decimal place.

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

functiontoBase(n,b){k=floor(log(b,n))+1precision=8result=""for(i=k-1,i>-precision-1,i--){if(result.length==k)result+="."digit=floor((n/b^i)modb)n-=digit*b^iresult+=digit}returnresult}

Note that the above code is only valid for and${\displaystyle n\ge 0}$, as it does not convert each digits to their correct symbols or correct negative numbers. For example, if a digit's value is10, it will be represented as10 instead ofA.

• JavaScript:^[3]

  functiontoBasePI(num,precision=8){letk=Math.floor(Math.log(num)/Math.log(Math.PI))+1;if(k<0)k=0;letdigits=[];for(leti=k-1;i>(-1*precision)-1;i--){letdigit=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,".");returndigits.join("");}

• #"#cite_note-DecimalSystem-3">[3]

  functionfromBasePI(num){letnumberSplit=num.split(/\./g);letnumberLength=numberSplit[0].length;letoutput=0;letdigits=numberSplit.join("");for(leti=0;i<digits.length;i++){output+=digits[i]*Math.pow(Math.PI,numberLength-i-1);}returnoutput;}

Base√2 behaves in a very similar way tobase 2 as all one has to do to convert a number frombinary 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 radix point. The base can also be used to show the relationship between theside of asquare to itsdiagonal 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 thesilver ratio as its representation in base√2 is simply 11_√2. In addition, the area of aregular 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 areambiguous. For example, 11_φ = 100_φ, since φ² = φ + 1.

There are some numbers in baseψ, the supergolden ratio, that are also ambiguous. For example, 101_ψ = 1000_ψ, since ψ³ = ψ² + 1.

With basee thenatural logarithm behaves like thecommon logarithm in base 10, as ln(1_e) = 0, ln(10_e) = 1, ln(100_e) = 2 and ln(1000_e) = 3 (or more precisely the representation in basee of 3, which is of course a non-terminating number). This means that the integer part of the natural logarithm of a number in basee counts the number of digits before the separating point in that number, minus one.

The basee is the most economical choice of radixβ > 1,^[4] where theradix 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. A binary number uses only two different digits, but it needs a lot of digits for representing a number; base 10 writes shorter numbers, but it needs 10 different digits to write them. The balance between those is basee, which therefore would store numbers optimally.

Baseπ can be used to more easily show the relationship between thediameter of acircle to itscircumference, which corresponds to itsperimeter; 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 thearea = 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]

In every positional number system, not all numbers are expressed uniquely. For example, in base 10, the number 1 has two representations: 1.000... and0.999.... The set of numbers with two different representations isdense in the reals,^[6] but the question of classifying real numbers with uniqueβ-expansions is considerably more subtle than that of integer bases.^[7]

Another problem is to classify the real numbers whoseβ-expansions are periodic. Letβ > 1, andQ(β) be the smallestfield extension of therationals containingβ. Then any real number in [0,1) having a periodicβ-expansion must lie inQ(β). On the other hand, theconverse need not be true. The converse does hold ifβ is aPisot number,^[8] although necessary and sufficient conditions are not known.

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

• Sidorov, Nikita (2003), "Arithmetic dynamics", in Bezuglyi, Sergey; Kolyada, Sergiy (eds.),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., vol. 310, Cambridge:Cambridge University Press, pp. 145–189,ISBN 978-0-521-53365-2,Zbl 1051.37007

• Weisstein, Eric W."Base".MathWorld.

• Number theory
• Ring theory
• Coding theory
• Non-standard positional numeral systems

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