Egyptian Fractions Calculator
Solve egyptian fractions problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
n/d = 1/a1 + 1/a2 + ... + 1/ak (where a1 < a2 < ... < ak)
Where n/d is a proper fraction decomposed into a sum of distinct unit fractions 1/a1, 1/a2, etc. The greedy algorithm finds each unit fraction by computing ceiling(d/n) to get the largest possible unit fraction not exceeding the remaining value.
Worked Examples
Example 1: Decomposing 3/7 into Egyptian Fractions
Problem:Express 3/7 as a sum of distinct unit fractions using the greedy algorithm.
Solution:Step 1: ceiling(7/3) = 3, so first unit fraction is 1/3\n3/7 - 1/3 = (9 - 7)/21 = 2/21\n\nStep 2: ceiling(21/2) = 11, so next unit fraction is 1/11\n2/21 - 1/11 = (22 - 21)/231 = 1/231\n\nStep 3: 1/231 is already a unit fraction\n\nResult: 3/7 = 1/3 + 1/11 + 1/231\nVerification: 1/3 + 1/11 + 1/231 = 77/231 + 21/231 + 1/231 = 99/231 = 3/7
Result:3/7 = 1/3 + 1/11 + 1/231
Example 2: Decomposing 5/8 for Fair Division
Problem:Express 5/8 as Egyptian fractions to divide 5 items among 8 people.
Solution:Step 1: ceiling(8/5) = 2, so first unit fraction is 1/2\n5/8 - 1/2 = (5 - 4)/8 = 1/8\n\nStep 2: 1/8 is already a unit fraction\n\nResult: 5/8 = 1/2 + 1/8\nPractical meaning: Each person gets 1/2 of an item plus 1/8 of an item\nVerification: 1/2 + 1/8 = 4/8 + 1/8 = 5/8
Result:5/8 = 1/2 + 1/8 (each person gets half plus one-eighth)
Frequently Asked Questions
What are Egyptian fractions and where do they come from?
Egyptian fractions are a way of representing rational numbers as sums of distinct unit fractions, where each unit fraction has 1 as its numerator. This notation originated in ancient Egypt around 1650 BCE, as documented in the Rhind Mathematical Papyrus and the Egyptian Mathematical Leather Roll. The ancient Egyptians used this system for all their fractional calculations, with the sole exception of 2/3 which had its own special symbol. For example, they would write 3/4 as 1/2 + 1/4 rather than using the single fraction 3/4. This system persisted for over 2,000 years in Egyptian mathematics and influenced Greek and medieval European mathematical traditions as well.
How does the greedy algorithm for Egyptian fractions work?
The greedy algorithm, also known as the Fibonacci-Sylvester algorithm, works by always choosing the largest possible unit fraction at each step. Given a fraction n/d, find the smallest integer k such that 1/k is less than or equal to n/d (k = ceiling of d/n). Subtract 1/k from n/d to get a new, smaller fraction, and repeat. For example, for 3/7: ceiling(7/3) = 3, so take 1/3. Then 3/7 - 1/3 = 2/21. Ceiling(21/2) = 11, so take 1/11. Then 2/21 - 1/11 = 1/231. The result is 3/7 = 1/3 + 1/11 + 1/231. This algorithm always terminates because the numerator strictly decreases at each step, guaranteeing convergence to zero.
Why did ancient Egyptians use unit fractions instead of regular fractions?
Historians believe the Egyptians preferred unit fractions for practical and conceptual reasons. Unit fractions are intuitive for fair division: 1/3 means one part out of three equal parts. The Egyptians may have found it easier to conceptualize sharing as taking successive unit portions rather than working with abstract numerators greater than one. Their hieroglyphic notation also made unit fractions simple to write: an eye symbol (or mouth symbol) was placed over the denominator. Additionally, comparison between fractions is straightforward with unit fractions since 1/n is always greater than 1/m when n is less than m. Some scholars also argue the system was a deliberate mathematical choice that enabled efficient computation.
Is the Egyptian fraction representation of a number unique?
No, Egyptian fraction representations are not unique. Most fractions can be decomposed into unit fractions in multiple ways. For example, 2/3 can be written as 1/2 + 1/6, or as 1/3 + 1/4 + 1/12, or even as 1/3 + 1/5 + 1/9 + 1/45. Different algorithms produce different decompositions. The greedy algorithm tends to produce fewer terms but can result in very large denominators. Other methods, like the splitting identity (1/n = 1/(n+1) + 1/(n(n+1))), can be used to generate alternative representations. The Erdos-Straus conjecture, one of the famous unsolved problems in mathematics, asks whether every fraction 4/n can be written as a sum of exactly three unit fractions.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy