Engel expansion

The Engel expansion of a positive real number x is the unique non-decreasing sequence of positive integers $\{a_{1},a_{2},a_{3},\dots \}$ such that

$x={\frac {1}{a_{1}}}+{\frac {1}{a_{1}a_{2}}}+{\frac {1}{a_{1}a_{2}a_{3}}}+\cdots .\;$

Rational numbers have a finite Engel expansion, while irrational numbers 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 F. Engel, who studied them in 1913.

Engel expansions, continued fractions, and Fibonacci

Kraaikamp and Wu (2004) observe that an Engel expansion can also be written as an ascending variant of a continued fraction:

$x={\frac {\displaystyle 1+{\frac {\displaystyle 1+{\frac {\displaystyle 1+\cdots }{\displaystyle a_{3}}}}{\displaystyle a_{2}}}}{\displaystyle a_{1}}}.$

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:

${\frac {a\ b\ c\ d}{e\ f\ g\ h}}={\frac {d+{\frac {c+{\frac {b+{\frac {a}{e}}}{f}}}{g}}}{h}}.$

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.

Algorithm for computing Engel expansions

To find the Engel expansion of x, let

$u_{1}=x,$

$a_{k}=\left\lceil {\frac {1}{u_{k}}}\right\rceil ,$

and

$u_{k+1}=u_{k}a_{k}-1$

where $\left\lceil r\right\rceil$ is the ceiling function (the smallest integer not less than r).

If $u_{i}=0$ for any i, halt the algorithm.

Example

To find the Engel expansion of 1.175, we perform the following steps.

$u_{1}=1.175,a_{1}=\left\lceil {\frac {1}{1.175}}\right\rceil =1;$

$u_{2}=u_{1}a_{1}-1=1.175\cdot 1-1=0.175,a_{2}=\left\lceil {\frac {1}{0.175}}\right\rceil =6$

$u_{3}=u_{2}a_{2}-1=0.175\cdot 6-1=0.05,a_{3}=\left\lceil {\frac {1}{0.05}}\right\rceil =20$

$u_{4}=u_{3}a_{3}-1=0.05\cdot 20-1=0$

The series ends here. Thus,

$1.175={\frac {1}{1}}+{\frac {1}{1\cdot 6}}+{\frac {1}{1\cdot 6\cdot 20}}$

and the Engel expansion of $1.175$ is $\{1,6,20\}\;$

Engel expansions of rational numbers

Every positive rational number has a unique finite Engel expansion. In the algorithm for Engel expansion, if u_i is a rational number x/y, then u_i+1 = (-y mod x)/y. Therefore, at each step, the numerator in the remaining fraction u_i 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

${\frac {1}{n}}=\sum _{r=1}^{\infty }{\frac {1}{(n+1)^{r}}}.$

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

$1.175=\{1,6,20\}=\{1,6,21,21,21,\dots \}.\;\;$

This is analogous to the fact that any rational number with a finite decimal representation also has an infinite decimal representation (see 0.999...).

Engel expansions for some well-known constants

$\pi =\{1,1,1,8,8,17,19,300,1991,2492,\dots \}\;$ (sequence A006784 in the OEIS)

$e=\{1,1,2,3,4,5,6,7,8,9,10,11,12,13,\dots \}\;$ (sequence A000027 in the OEIS)

${\sqrt {2}}=\{1,3,5,5,16,18,78,102,120,144,\dots \}\;$ (sequence A028254 in the OEIS)

More Engel expansions for constants can be found here.

References

• Engel, F. (1913). "Entwicklung der Zahlen nach Stammbruechen". Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg. pp. 190–191. {{cite conference}}: Unknown parameter |booktitle= ignored (|book-title= suggested) (help)

• Kraaikamp, Cor; Wu, Jun (2004). "On a new continued fraction expansion with non-decreasing partial quotients". Monatshefte für Mathematik. 143: 285–298. doi:10.1007/s00605-004-0246-3.{{cite journal}}: CS1 maint: multiple names: authors list (link)

External links

• Weisstein, Eric W. "Engel Expansion". MathWorld–A Wolfram Web Resource.