View source for Non-integer representation

{{inline citations|date=March 2019}}
{{Numeral systems}}
A '''non-integer representation''' uses non-[[Integer|integer]] numbers as the [[Radix|radix]], or bases, of a [[Positional notation|positional numeral system]]. For a non-integer radix β > 1, the value of
:<math>x=d_n\dots d_2d_1d_0.d_{-1}d_{-2}\dots d_{-m}</math>
is
:<math>
\begin{align}
x&=\beta^nd_n + \cdots + \beta^2d_2 + \beta d_1 + d_0 \\
&\qquad + \beta^{-1}d_{-1} + \beta^{-2}d_{-2} + \cdots + \beta^{-m}d_{-m}.
\end{align}
</math>
The numbers ''d''<sub>''i''</sub> are non-negative integers less than β.  This is also known as a '''β-expansion''', a notion introduced by {{harvtxt|Rényi|1957}} and first studied in detail by {{harvtxt|Parry|1960}}.  Every real number has at least one (possibly infinite) β-expansion.

There are applications of β-expansions in [[Coding theory|coding theory]] {{harv|Kautz|1965}} and models of [[Physics:Quasicrystal|quasicrystal]]s ({{harvnb|Burdik|Frougny|Gazeau|Krejcar|1998}}; {{harvnb|Thurston|1989}}).

==Construction==
β-expansions are a generalization of decimal expansions. While infinite decimal expansions are not unique (for example, 1.000... = [[0.999...|0.999...]]), all finite decimal expansions are unique. However, even finite β-expansions are not necessarily unique, for example φ + 1 = φ<sup>2</sup> for β = φ, the [[Golden ratio|golden ratio]]. A canonical choice for the β-expansion of a given real number can be determined by the following [[Greedy algorithm|greedy algorithm]], essentially due to {{harvtxt|Rényi|1957}} and formulated as given here by {{harvtxt|Frougny|1992}}.

Let {{nowrap|β > 1}} be the base and ''x'' a non-negative real number.  Denote by {{nowrap|⌊''x''⌋}} the floor function of ''x'', that is, the greatest integer less than or equal to ''x'', and let {{nowrap begin}}{''x''} = ''x'' − ⌊''x''⌋{{nowrap end}} be the fractional part of ''x''. There exists an integer ''k'' such that {{nowrap|β<sup>''k''</sup> ≤ ''x'' < β<sup>''k''+1</sup>}}. Set
:<math>d_k = \lfloor x/\beta^k\rfloor</math>
and
:<math>r_k = \{x/\beta^k\}.\,</math>
For {{nowrap|''k'' − 1 ≥ ''j'' > −∞}}, put
:<math>d_j = \lfloor\beta r_{j+1}\rfloor, \quad r_j = \{\beta r_{j+1}\}.</math>

In other words, the canonical β-expansion of ''x'' is defined by choosing the largest ''d''<sub>''k''</sub> such that {{nowrap|β<sup>''k''</sup>''d''<sub>''k''</sub> ≤ ''x''}}, then choosing the largest ''d''<sub>''k''−1</sub> such that {{nowrap|β<sup>''k''</sup>''d''<sub>''k''</sub> + β<sup>''k''−1</sup>''d''<sub>''k''−1</sub> ≤ ''x''}}, etc. Thus it chooses the [[Lexicographical order|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 {{radic|2}}===
Base [[Square root of 2|{{radic|2}}]] behaves in a very similar way to base 2 as all one has to do to convert a number from binary into base {{radic|2}} is put a zero digit in between every binary digit; for example, 1911<sub>10</sub> = 11101110111<sub>2</sub> becomes 101010001010100010101<sub>{{radic|2}}</sub> and 5118<sub>10</sub> = 1001111111110<sub>2</sub> becomes 1000001010101010101010100<sub>{{radic|2}}</sub>. This means that every integer can be expressed in base {{radic|2}} without the need of a decimal point. The base can also be used to show the relationship between the [[Edge (geometry)|side]] of a square to its [[Diagonal|diagonal]] as a square with a side length of 1<sub>{{radic|2}}</sub> will have a diagonal of 10<sub>{{radic|2}}</sub> and a square with a side length of 10<sub>{{radic|2}}</sub> will have a diagonal of 100<sub>{{radic|2}}</sub>. Another use of the base is to show the [[Silver ratio|silver ratio]] as its representation in base {{radic|2}} is simply 11<sub>{{radic|2}}</sub>. In addition, the area of a regular octagon with side length 1<sub>{{radic|2}}</sub> is 1100<sub>{{radic|2}}</sub>, the area of a regular octagon with side length 10<sub>{{radic|2}}</sub> is 110000<sub>{{radic|2}}</sub>, the area of a regular octagon with side length 100<sub>{{radic|2}}</sub> is 11000000<sub>{{radic|2}}</sub>, etc…

===Base φ===
{{main|Golden ratio base}}
11<sub>φ</sub> = 100<sub>φ</sub>.

===Base ψ===
101<sub>ψ</sub> = 1000<sub>ψ</sub>

===Base e===
With base [[E (mathematical constant)|e]] the [[Natural logarithm|natural logarithm]] behaves like the [[Common logarithm|common logarithm]] as ln(1<sub>e</sub>) = 0, ln(10<sub>e</sub>) = 1, ln(100<sub>e</sub>) = 2 and ln(1000<sub>e</sub>) = 3.

The base ''e'' is the most economical choice of radix β > 1 {{harv|Hayes|2001}}, where the [[Radix economy|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 [[Pi|π]] can be used to more easily show the relationship between the [[Diameter|diameter]] of a [[Circle|circle]] to its [[Circumference|circumference]], which  corresponds to its [[Perimeter|perimeter]]; since circumference = diameter × π, a circle with a diameter 1<sub>π</sub> will have a circumference of 10<sub>π</sub>, a circle with a diameter 10<sub>π</sub> will have a circumference of 100<sub>π</sub>, etc. Furthermore, since the [[Area|area]] = π × [[Radius|radius]]<sup>2</sup>, a circle with a radius of 1<sub>π</sub> will have an area of 10<sub>π</sub>, a circle with a radius of 10<sub>π</sub> will have an area of 1000<sub>π</sub> and a circle with a radius of 100<sub>π</sub> will have an area of 100000<sub>π</sub>.<ref>{{Cite web|url=http://datagenetics.com/blog/december22015/index.html|title=Weird Number Bases|website=DataGenetics|access-date=2018-02-01}}</ref>

== 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...|0.999...]].  The set of numbers with two different representations is [[Dense set|dense]] in the reals {{harv|Petkovšek|1990}}, but the question of classifying real numbers with unique β-expansions is considerably more subtle than that of integer bases {{harv|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 {{harv|Schmidt|1980}}, although necessary and sufficient conditions are not known.

== See also ==
* [[Beta encoder]]
* [[Non-standard positional numeral systems]]
* Decimal expansion
* [[Power series]]
* [[Ostrowski numeration]]

== References ==
{{Reflist}}
*{{citation | last=Bugeaud | first=Yann | title=Distribution modulo one and Diophantine approximation | series=Cambridge Tracts in Mathematics | volume=193 | location=Cambridge | publisher={{wipe|Cambridge University Press}} | year=2012 | isbn=978-0-521-11169-0 | zbl= 1260.11001}}
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*{{citation|series=Lecture Notes in Computer Science
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|title=LATIN '92
|doi=10.1007/BFb0023826
|year=1992
|isbn=978-3-540-55284-0
|pages=154–164
|chapter=How to write integers in non-integer base
|first=Christiane
|last=Frougny
|chapter-url=https://books.google.com/books?id=I3fC6batwokC&lpg=PA154&pg=PA154#v=onepage&q=&f=false
}}.
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*{{citation | first=Brian|last=Hayes|title=Third base|journal=American Scientist|url=http://www.americanscientist.org/issues/pub/third-base/2|year=2001|volume=89|issue=6|pages=490–494|doi=10.1511/2001.40.3268 | archive-url = https://web.archive.org/web/20160324100419/http://www.americanscientist.org/issues/pub/2001/11/third-base/2 | archive-date = 2016-03-24 | url-status = dead }}.
*{{Citation | last1=Kautz | first1=William H. | title=Fibonacci codes for synchronization control | url=http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?tp=&arnumber=1053772&isnumber=22626 | mr=0191744 | year=1965 | journal=Institute of Electrical and Electronics Engineers. Transactions on Information Theory | issn=0018-9448 | volume=IT-11 | issue=2 | pages=284–292| doi=10.1109/TIT.1965.1053772 }}.
*{{Citation | last1=Parry | first1=W. |  title=On the β-expansions of real numbers | mr=0142719 | year=1960 | journal=Acta Mathematica Academiae Scientiarum Hungaricae | issn=0001-5954 | volume=11 | issue=3–4 | pages=401–416 | doi=10.1007/bf02020954}}.
*{{Citation | last1=Petkovšek | first1=Marko |  title=Ambiguous numbers are dense | doi=10.2307/2324393 | mr=1048915 | year=1990 | journal=[[American Mathematical Monthly|The American Mathematical Monthly]] | issn=0002-9890 | volume=97 | issue=5 | pages=408–411| jstor=2324393 }}.
*{{Citation | last1=Rényi | first1=Alfréd |  title=Representations for real numbers and their ergodic properties | mr=0097374 | doi=10.1007/BF02020331 | year=1957 | journal=Acta Mathematica Academiae Scientiarum Hungaricae | issn=0001-5954 | volume=8 | issue=3–4 | pages=477–493}}.
*{{Citation | last1=Schmidt | first1=Klaus | title=On periodic expansions of Pisot numbers and Salem numbers | doi=10.1112/blms/12.4.269 | mr=576976 | year=1980 | journal=The Bulletin of the London Mathematical Society | issn=0024-6093 | volume=12 | issue=4 | pages=269–278}}.
*{{Citation | last1=Thurston | first1=W.P. | title=Groups, tilings and finite state automata |journal=AMS Colloquium Lectures | year=1989}}

==Further reading==
* {{citation | last=Sidorov | first=Nikita | chapter=Arithmetic dynamics | zbl=1051.37007 | editor1-last=Bezuglyi | editor1-first=Sergey | editor2-last=Kolyada | editor2-first=Sergiy | title=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 | location=Cambridge | publisher={{wipe|Cambridge University Press}} | isbn=978-0-521-53365-2 | series=Lond. Math. Soc. Lect. Note Ser. | volume=310 | pages=145–189 | year=2003 }}

==External links==
* {{mathworld|title=Base|urlname=Base}}

{{DEFAULTSORT:Non-Integer Representation}}
[[Category:Non-standard positional numeral systems]]

{{Sourceattribution|Non-integer representation|1}}