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.

Read more about Engel Expansion: Engel Expansions, Continued Fractions, and Fibonacci, Algorithm For Computing Engel Expansions, Example, Engel Expansions of Rational Numbers, Engel Expansions For Some Well-known Constants, Growth Rate of The Expansion Terms

### Other articles related to "engel expansion, expansion, engel expansions, expansions":

**Engel Expansion**- Growth Rate of The Expansion Terms

... The coefficients ai of the

**Engel expansion**typically exhibit exponential growth more precisely, for almost all numbers in the interval (0,1], the limit exists and is equal to e ... 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 ...

... see Erdős–Graham conjecture, Znám's problem, and

**Engel expansion**... of bounding the length or maximum denominator in Egyptian fraction representations, finding

**expansions**of certain special forms or in which the denominators are all of some special type, the termination of ... represented using fewer fractions, as any

**expansion**with repeated fractions can be converted to an Egyptian fraction of equal or smaller length by repeated application of the replacement if k ...

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