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| The '''Engel expansion''' of a positive [[real number]] ''x'' is the unique non-decreasing sequence of [[natural number|positive integer]]s <math>\{a_1,a_2,a_3,\dots\}</math> such that
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| :<math>x=\frac{1}{a_1}+\frac{1}{a_1a_2}+\frac{1}{a_1a_2a_3}+\cdots.\;</math>
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| [[Rational number]]s have a finite Engel expansion, while [[irrational number]]s have an infinite Engel expansion. If ''x'' is rational, its Engel expansion provides a representation of ''x'' as an [[Egyptian fraction]]. Engel expansions are named after [[Friedrich Engel (mathematician)|Friedrich Engel]], who studied them in 1913.
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| An expansion analogous to an '''Engel expansion''', in which alternating terms are negative, is called a [[Pierce expansion]].
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| == Engel expansions, continued fractions, and Fibonacci ==
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| Kraaikamp and Wu (2004) observe that an Engel expansion can also be written as an ascending variant of a [[continued fraction]]:
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| :<math>x = \frac{\displaystyle 1+\frac{\displaystyle 1+\frac{\displaystyle 1+\cdots}{\displaystyle a_3}}{\displaystyle a_2}}{\displaystyle a_1}.</math>
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| They claim that ascending continued fractions such as this have been studied as early as [[Fibonacci]]'s [[Liber Abaci]] (1202). This claim appears to refer to Fibonacci's compound fraction notation in which a sequence of numerators and denominators sharing the same fraction bar represents an ascending continued fraction:
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| :<math>\frac{a\ b\ c\ d}{e\ f\ g\ h} = \dfrac{d+\dfrac{c+\dfrac{b+\dfrac{a}{e}}{f}}{g}}{h}.</math>
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| If such a notation has all numerators 0 or 1, as occurs in several instances in [[Liber Abaci]], the result is an Engel expansion. However, Engel expansion as a general technique does not seem to be described by Fibonacci.
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| ==Algorithm for computing Engel expansions==
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| To find the Engel expansion of ''x'', let
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| :<math>u_1=x,</math>
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| :<math>a_k=\left \lceil \frac{1}{u_k} \right \rceil,</math>
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| and
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| :<math> u_{k+1}=u_ka_k-1</math>
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| where <math>\left \lceil r \right \rceil</math> is the [[floor function#The ceiling function|ceiling function]] (the smallest integer not less than ''r'').
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| If <math>u_i=0</math> for any ''i'', halt the algorithm.
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| == Example ==
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| To find the Engel expansion of 1.175, we perform the following steps. | |
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| :<math>u_1 = 1.175, a_1=\left \lceil \frac{1}{1.175} \right\rceil = 1; \, </math>
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| :<math>u_2 = u_1a_1-1=1.175\cdot1-1=0.175, a_2=\left\lceil\frac{1}{0.175}\right\rceil=6 \, </math>
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| :<math>u_3 = u_2a_2-1=0.175\cdot6-1=0.05, a_3=\left\lceil\frac{1}{0.05}\right\rceil=20 \, </math>
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| :<math>u_4 = u_3a_3-1=0.05\cdot20-1=0 \, </math>
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| The series ends here. Thus,
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| :<math>1.175=\frac{1}{1}+\frac{1}{1\cdot6}+\frac{1}{1\cdot6\cdot20}</math>
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| and the Engel expansion of 1.175 is {1, 6, 20}.
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| ==Engel expansions of rational numbers==
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| Every positive rational number has a unique finite Engel expansion. In the algorithm for Engel expansion, if ''u<sub>i</sub>'' is a rational number ''x''/''y'', then ''u''<sub>''i''+1</sub> = (−''y'' mod ''x'')/''y''. Therefore, at each step, the numerator in the remaining fraction ''u<sub>i</sub>'' decreases and the process of constructing the Engel expansion must terminate in a finite number of steps. Every rational number also has a unique infinite Engel expansion: using the identity
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| :<math>\frac{1}{n}=\sum_{r=1}^{\infty}\frac{1}{(n+1)^r}.</math>
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| the final digit ''n'' in a finite Engel expansion can be replaced by an infinite sequence of (''n'' + 1)s without changing its value. For example
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| :<math>1.175=\{1,6,20\}=\{1,6,21,21,21,\dots\}.\;\;</math>
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| This is analogous to the fact that any rational number with a finite decimal representation also has an infinite decimal representation (see [[0.999...]]).
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| An infinite Engel expansion in which all terms are equal is a [[geometric series]].
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| [[Paul Erdős|Erdős]], [[Alfréd Rényi|Rényi]], and Szüsz asked for nontrivial bounds on the length of the finite Engel expansion of a rational number ''x''/''y''; this question was answered by Erdős and [[Jeffrey Shallit|Shallit]], who proved that the number of terms in the expansion is O(''y''<sup>1/3 + ε</sup>) for any ε > 0.<ref>{{harvtxt|Erdős|Rényi|Szüsz|1958}}; {{harvtxt|Erdős|Shallit|1991}}.</ref>
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| ==Engel expansions for some well-known constants==
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| :<math>\pi</math> = {1, 1, 1, 8, 8, 17, 19, 300, 1991, 2492,...} {{OEIS|id=A006784}}
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| :<math>\sqrt{2}</math> = {1, 3, 5, 5, 16, 18, 78, 102, 120, 144,...} {{OEIS|id=A028254}}
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| :<math>e</math> = {1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,...} {{OEIS|id=A000027}}
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| And in general,
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| :<math>e^{1/r}-1=\{1r, 2r, 3r, 4r, 5r, 6r, \dots\}\;</math>
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| More Engel expansions for constants can be found [http://oeis.org/wiki/Index_to_OEIS:_Section_El#Engel here].
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| ==Growth rate of the expansion terms==
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| The coefficients ''a<sub>i</sub>'' of the Engel expansion typically exhibit [[exponential growth]]; more precisely, for [[almost all]] numbers in the interval (0,1], the limit <math>\lim_{n\rightarrow\infty} a_n^{1/n}</math> exists and is equal to [[e (mathematical constant)|''e'']]. However, the subset of the interval for which this is not the case is still large enough that its [[Hausdorff dimension]] is one.<ref>{{harvtxt|Wu|2000}}. Wu credits the result that the limit is almost always ''e'' to [[Janos Galambos]].</ref>
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| The same typical growth rate applies to the terms in expansion generated by the [[greedy algorithm for Egyptian fractions]]. However, the set of real numbers in the interval (0,1] whose Engel expansions coincide with their greedy expansions has measure zero, and Hausdorff dimension 1/2.<ref>{{harvtxt|Wu|2003}}.</ref>
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| ==Notes==
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| {{reflist}}
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| ==References==
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| {{refbegin}}
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| *{{citation
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| | last = Engel | first = F.
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| | contribution = Entwicklung der Zahlen nach Stammbruechen
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| | title = Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg
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| | pages = 190–191
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| | year = 1913}}.
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| * {{cite journal
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| |first1=T. A. | last1=Pierce | title=On an algorithm and its use in approximating roots of algebraic equations
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| |year=1929
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| |journal=Am. Math. Monthly
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| |volume=36
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| |number=10
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| |pages=523–525
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| |jstor=2299963}}
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| *{{citation
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| | authorlink1 = Paul Erdős | last1 = Erdős | first1 = Paul
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| | authorlink2 = Alfréd Rényi | last2 = Rényi | first2 = Alfréd
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| | last3 = Szüsz | first3 = Peter
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| | title = On Engel's and Sylvester's series
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| | journal = Ann. Univ. Sci. Budapest. Eötvös Sect. Math.
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| | url = http://www.renyi.hu/~p_erdos/1958-07.pdf
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| | volume = 1
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| | year = 1958
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| | pages = 7–32}}.
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| *{{citation
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| | authorlink1 = Paul Erdős | last1 = Erdős | first1 = Paul
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| | authorlink2 = Jeffrey Shallit | last2 = Shallit | first2 = Jeffrey
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| | title = New bounds on the length of finite Pierce and Engel series
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| | journal = Journal de théorie des nombres de Bordeaux
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| | volume = 3 | issue = 1 |year = 1991 | pages = 43–53
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| | url = http://jtnb.cedram.org/item?id=JTNB_1991__3_1_43_0
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| | mr = 1116100
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| | doi = 10.5802/jtnb.41 }}.
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| *{{cite journal| first1= J. | last1=Paradis|
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| |first2=P. | last2=Viader | first3=L. | last3=Bibiloni
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| |title= Approximation to quadratic irrationals and their Pierce expansions
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| |journal= Fib. Quart.
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| |year=1998
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| |url=http://www.fq.math.ca/36-2.html
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| |volume=36 | number =2 | pages=146–153
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| }}
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| *{{citation
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| | last1 = Kraaikamp | first1 = Cor | last2 = Wu | first2 = Jun
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| | title = On a new continued fraction expansion with non-decreasing partial quotients
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| | journal = Monatshefte für Mathematik
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| | year = 2004
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| | volume = 143
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| | pages = 285–298
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| | doi = 10.1007/s00605-004-0246-3
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| | issue = 4}}.
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| *{{cite journal
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| | last = Wu | first = Jun
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| | issue = 4
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| | journal = Acta Arithmetica
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| | mr = 1760244
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| | pages = 383–386
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| | title = A problem of Galambos on Engel expansions
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| | volume = 92
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| | year = 2000}}.
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| *{{citation
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| | last = Wu | first = Jun
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| | doi = 10.1016/S0022-314X(03)00017-9
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| | issue = 1
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| | journal = Journal of Number Theory
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| | mr = 2008063
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| | pages = 16–26
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| | title = How many points have the same Engel and Sylvester expansions?
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| | volume = 103
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| | year = 2003}}.
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| {{refend}}
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| ==External links==
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| * {{cite web
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| | author = Weisstein, Eric W
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| | authorlink = Eric W. Weisstein
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| | title = Engel Expansion
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| | publisher = MathWorld–A Wolfram Web Resource
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| | url = http://mathworld.wolfram.com/EngelExpansion.html}}
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| [[Category:Mathematical analysis]]
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| [[Category:Continued fractions]]
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| [[Category:Egyptian fractions]]
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