Egyptian Fraction Calculator
Express any proper fraction as a sum of distinct unit fractions using the greedy algorithm
Enter a proper fraction (numerator < denominator)
What is an Egyptian Fraction?
An Egyptian fraction expresses a positive rational number as a sum of distinct unit fractions (fractions with numerator 1). This system was used by ancient Egyptians as their primary method for representing fractions.
Example: 2/3 = 1/2 + 1/6
The Greedy Algorithm
This calculator uses the greedy algorithm (also known as Fibonacci's algorithm) to find an Egyptian fraction expansion:
- Find the largest unit fraction 1/n that is less than or equal to a/b
- Subtract 1/n from a/b to get a new fraction
- Repeat until the remainder is zero
For a/b, the smallest n where 1/n ≤ a/b is n = ⌈b/a⌉
Historical Significance
The Rhind Mathematical Papyrus (c. 1650 BCE) contains extensive tables of Egyptian fractions. Ancient Egyptian scribes used these representations for practical calculations involving division of goods, land measurement, and accounting.
Example Expansions
- 2/3 = 1/2 + 1/6
- 3/4 = 1/2 + 1/4
- 4/5 = 1/2 + 1/4 + 1/20
- 5/6 = 1/2 + 1/3
- 2/7 = 1/4 + 1/28
Note: The greedy algorithm always terminates and produces a valid expansion, but it may not always produce the shortest expansion or the one with the smallest denominators.
What is an Egyptian fraction?
An Egyptian fraction is a representation of a positive rational number as a sum of distinct unit fractions (fractions with numerator 1). For example, 2/3 = 1/2 + 1/6. Ancient Egyptians used this system exclusively for representing fractions, as evidenced by the Rhind Mathematical Papyrus from around 1650 BCE.
Why did ancient Egyptians use unit fractions?
Ancient Egyptians found unit fractions practical for dividing resources fairly. If you need to divide 3 loaves among 5 people, expressing 3/5 as 1/2 + 1/10 makes it clear: give each person half a loaf plus a tenth of a loaf. This made physical division of goods more intuitive.
What is the greedy algorithm for Egyptian fractions?
The greedy algorithm (also called Fibonacci's algorithm) repeatedly subtracts the largest possible unit fraction from the remaining value. For a/b, find the smallest n where 1/n <= a/b, subtract 1/n, and repeat with the remainder. This always terminates and produces a valid Egyptian fraction expansion.
Are Egyptian fraction representations unique?
No, most fractions have multiple valid Egyptian fraction representations. For example, 2/3 = 1/2 + 1/6 = 1/3 + 1/4 + 1/12. The greedy algorithm produces one specific representation, but it's not always the shortest or the one with the smallest denominators.