Ancient Egyptian scribes wrote fractions such as 4/13 as sums of different unit fractions: 1/4 + 1/18 + 1/468. The Rhind papyrus contains a table of such decompositions. Today Egyptian fractions are a rich topic in number theory.
How to use the Egyptian Fractions Calculator
- Numerator x
- Denominator y
- Algorithm
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How it works
The greedy algorithm (Fibonacci–Sylvester) always takes the largest unit fraction that fits; splitting repeatedly replaces duplicate 1/a with 1/(a + 1) + 1/(a(a + 1)); pairing combines equal pairs. Exact integer arithmetic is used throughout.
Formula
Greedy step: x/y = 1/⌈y/x⌉ + ((−y) mod x)/(y·⌈y/x⌉)
Worked example
Inputs — Numerator x: 4; Denominator y: 13; Algorithm: Greedy.
Results — Egyptian fraction = 1/4 + 1/18 + 1/468; Number of unit fractions = 3; Method = Greedy: repeatedly subtract the largest unit fraction 1/⌈d/n⌉ that fits..
Key terms
- Unit fraction
- A fraction with numerator 1.
Frequently asked questions
What is an Egyptian fraction?
A sum of distinct unit fractions, such as 3/4 = 1/2 + 1/4.
Does every fraction have one?
Yes – the greedy algorithm always terminates.
Is the representation unique?
No – 1/2 = 1/3 + 1/6 = 1/4 + 1/6 + 1/12, among others.
Why does the greedy method sometimes give huge denominators?
Each step is locally optimal, which can leave awkward remainders.