Egyptian Fraction Calculator
An Egyptian fraction expresses a rational number as a sum of distinct unit fractions, each of the form 1/n. Ancient Egyptians wrote fractions this way for thousands of years, with the Rhind Mathematical Papyrus (c. 1550 BC) containing tables of Egyptian fraction expansions. Enter a fraction p/q where p is less than q and both are positive integers. The calculator uses the Fibonacci-Sylvester greedy algorithm: at each step, take the largest unit fraction 1/n that does not exceed the remaining value, where n = ceiling(q/p), subtract it, and repeat until nothing remains. For example, 2/5: ceiling(5/2) = 3, so 1/3 is the first term. The remainder is 2/5 - 1/3 = 1/15, which is already a unit fraction. So 2/5 = 1/3 + 1/15. The algorithm always terminates, but it sometimes produces large denominators; alternative decompositions may exist with fewer terms or smaller denominators. This calculator shows each greedy step and the final sum notation.
Greedy algorithm steps
| Step | Remaining fraction | n = ceil(q/p) | Unit fraction | New remainder |
|---|
Greedy algorithm
Given p/q (with 0 < p < q):
n = ceiling(q / p)
Subtract: p/q - 1/n = (p*n - q) / (q*n)
Simplify remainder. Repeat until remainder = 0.
Result: p/q = 1/n1 + 1/n2 + ... (all distinct)
Examples from the Rhind Papyrus
| Fraction | Egyptian fraction form |
|---|---|
| 2/3 | 1/2 + 1/6 |
| 2/5 | 1/3 + 1/15 |
| 2/7 | 1/4 + 1/28 |
| 3/4 | 1/2 + 1/4 |
| 4/5 | 1/2 + 1/4 + 1/20 |
Egyptian fractions: frequently asked questions
What is an Egyptian fraction?
An Egyptian fraction is a representation of a rational number as a sum of distinct unit fractions (fractions of the form 1/n). Ancient Egyptians used this notation extensively, as evidenced by the Rhind Mathematical Papyrus from around 1550 BC. For example, 2/5 = 1/3 + 1/15, and 3/7 = 1/3 + 1/11 + 1/231.
What is the greedy algorithm for Egyptian fractions?
The Fibonacci-Sylvester greedy algorithm: given a fraction p/q, find the smallest unit fraction 1/n that is less than or equal to p/q (so n = ceiling(q/p)). Subtract 1/n from p/q. Repeat with the remainder until the fraction reaches 0. This always terminates for any rational number in (0,1).
Is every fraction between 0 and 1 expressible as an Egyptian fraction?
Yes. Every positive rational number less than 1 can be written as a sum of distinct unit fractions. The Erdos-Straus conjecture asks whether 4/n can always be expressed as a sum of three unit fractions; it has been verified for all n up to very large values but not yet proved in general.
Why did ancient Egyptians use unit fractions?
Ancient Egyptian mathematics used a positional-style notation for fractions but only naturally represented unit fractions. Most non-unit fractions were expressed as sums. The only exception was 2/3, which had its own special symbol. Tables in the Rhind Papyrus provided Egyptian fraction expansions for 2/n for odd n from 3 to 101.
What are the limitations of this calculator?
This calculator accepts fractions p/q where p is less than q (proper fractions) and both are positive integers, with q up to about 1,000. The greedy algorithm always finds a valid expansion, but it may not be the shortest or simplest one. Some fractions have multiple valid Egyptian fraction representations.
Sources
- Rhind Mathematical Papyrus: British Museum collection EA10057.
- Egyptian fractions: Wolfram MathWorld, Egyptian Fraction.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.