The Engel expansion of a positive real number x is the unique non-decreasing sequence of positive integers such that For instance, Euler's number e has the Engel expansion 1, 1, 2, 3, 4, 5, 6, 7, 8, ... corresponding to the infinite series 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. Engel expansions, continued fractions, and Fibonacci 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. Algorithm for computing Engel expansions To find the Engel expansion of x, let and where is the ceiling function (the smallest integer not less than r). If for any i, halt the algorithm. Iterated functions for computing Engel expansions Another equivalent method is to consider the map and set where and Yet another equivalent method, called the modified Engel expansion calculated by and The transfer operator of the Engel map The Frobenius–Perron transfer operator of the Engel map acts on functions with since and the inverse of the n-th component is which is found by solving for . Relation to the Riemann ζ function The Mellin transform of the map is related to the Riemann zeta function by the formula Example To find the Engel expansion of 1.175, we perform the following steps. The series ends here. Thus, 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 ui is a rational number x/y, then ui +&hairsp;1 = (−y mod x)/y. Therefore, at each step,