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| A '''non-integer representation''' uses non-[[integer]] numbers as the [[radix]], or bases, of a [[positional notation|positional numbering system]]. For a non-integer radix β > 1, the value of
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| :<math>x=d_n\dots d_2d_1d_0.d_{-1}d_{-2}\dots d_{-m}</math>
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| :<math>x=\beta^nd_n + \cdots + \beta^2d_2 + \beta d_1 + d_0 + \beta^{-1}d_{-1} + \beta^{-2}d_{-2} + \cdots + \beta^{-m}d_{-m}.</math>
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| 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.
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| There are applications of β-expansions in [[coding theory]] {{harv|Kautz|1965}} and models of [[quasicrystal]]s {{harv|Burdik|Frougny|Gazeau|Krejcar|1998}}.
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| ==Construction==
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| β-expansions are a generalization of [[decimal expansion]]s. 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<sup>2</sup> 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 {{harvtxt|Rényi|1957}} and formulated as given here by {{harvtxt|Frougny|1992}}.
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| 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
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| :<math>d_k = \lfloor x/\beta^k\rfloor</math>
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| and
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| :<math>r_k = \{x/\beta^k\}.\,</math>
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| For {{nowrap|''k'' − 1 ≥ ''j'' > −∞}}, put
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| :<math>d_j = \lfloor\beta r_{j+1}\rfloor, \quad r_j = \{\beta r_{j+1}\}.</math>
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| 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''.
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| 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.
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| ==Examples==
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| ===Base φ===
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| See [[Golden ratio base]]; 11<sub>φ</sub> = 100<sub>φ</sub>.
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| ===Base e===
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| With base [[e (mathematical constant)|e]] the [[natural logarithm]] behaves like the [[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.
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| The base ''e'' is the most economical choice of radix β > 1 {{harv|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.
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| ===Base π===
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| Base [[pi|π]] can be used to more easily show the relationship between the [[diameter]] of a [[circle]] to its [[circumference]]; 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]] = π × [[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>.
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| ===Base √2===
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| Base [[Square root of 2|√2]] behaves in a very similar way to [[binary numeral system|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<sub>10</sub> = 11101110111<sub>2</sub> becomes 101010001010100010101<sub>√2</sub> and 5118<sub>10</sub> = 1001111111110<sub>2</sub> becomes 1000001010101010101010100<sub>√2</sub>. 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 [[edge (geometry)|side]] of a [[square (geometry)|square]] to its [[diagonal]] as a square with a side length of 1<sub>√2</sub> will have a diagonal of 10<sub>√2</sub> and a square with a side length of 10<sub>√2</sub> will have a diagonal of 100<sub>√2</sub>. Another use of the base is to show the [[silver ratio]] as its representation in base √2 is simply 11<sub>√2</sub>.
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| == Properties ==
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| In no positional number system can every number be expressed uniquely. For example, in base 10, the number 1 has two representations: 1.000... and [[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}}.
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| 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.
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| == See also ==
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| * [[Beta encoder]]
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| * [[Non-standard positional numeral systems]]
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| * [[Decimal expansion]]
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| * [[Power series]]
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| * [[Ostrowski numeration]]
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| == References ==
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| *{{citation | last=Bugeaud | first=Yann | title=Distribution modulo one and Diophantine approximation | series=Cambridge Tracts in Mathematics | volume=193 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2012 | isbn=978-0-521-11169-0 | zbl=pre06066616 }}
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| *{{Citation | last1=Burdik | first1=Č. | last2=Frougny | first2=Ch. | last3=Gazeau | first3=J. P. | last4=Krejcar | first4=R. | title=Beta-integers as natural counting systems for quasicrystals | doi=10.1088/0305-4470/31/30/011 | mr=1644115 | year=1998 | journal=Journal of Physics. A. Mathematical and General | issn=0305-4470 | volume=31 | issue=30 | pages=6449–6472}}.
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| *{{citation|series=Lecture Notes in Computer Science
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| |publisher=Springer Berlin / Heidelberg
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| |ISSN=0302-9743
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| |volume=583/1992
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| |title=LATIN '92
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| |doi=10.1007/BFb0023826
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| |year=1992
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| |ISBN=978-3-540-55284-0
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| |pages=154–164
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| |chapter=How to write integers in non-integer base
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| |first=Christiane
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| |last=Frougny
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| |url=http://books.google.com/books?id=I3fC6batwokC&lpg=PA154&pg=PA154#v=onepage&q=&f=false
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| }}.
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| *{{Citation | last1=Glendinning | first1=Paul | author1-link=Paul Glendinning | last2=Sidorov | first2=Nikita | title=Unique representations of real numbers in non-integer bases | url=http://intlpress.com/site/pub/pages/journals/items/mrl/content/vols/0008/0004/00019835/index.html | mr=1851269 | year=2001 | journal=Mathematical Research Letters | issn=1073-2780 | volume=8 | issue=4 | pages=535–543}}.
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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}}.
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| *{{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 | pages=284–292}}.
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| *{{Citation | last1=Parry | first1=W. | authorlink=Bill Parry (mathematician) | title=On the β-expansions of real numbers | mr=0142719 | year=1960 | journal=Acta Mathematica Academiae Scientiarum Hungaricae | issn=0001-5954 | volume=11 | pages=401–416}}.
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| *{{Citation | last1=Petkovšek | first1=Marko | author1-link=Marko Petkovšek | 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}}.
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| *{{Citation | last1=Rényi | first1=Alfréd | authorlink=Alfréd Rényi | 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 | pages=477–493}}.
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| *{{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}}.
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| ==External links==
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| * {{mathworld|title=Base|urlname=Base}}
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| {{DEFAULTSORT:Non-Integer Representation}}
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| [[Category:Non-standard positional numeral systems]]
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