Engel expansion
The Engel expansion of a positive real number x is the unique non-decreasing sequence of positive integers such that
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 Friedrich Engel, who studied them in 1913.
An expansion analogous to an Engel expansion, in which alternating terms are negative, is called a Pierce expansion.
Kraaikamp & Wu (2004) observe that an Engel expansion can also be written as an ascending variant of a continued fraction:
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:
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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