Decompose any fraction into a sum of distinct unit fractions using the ancient Egyptian method. Perfect for mathematical history, number theory, and algorithm exploration.
Last updated: April 2026 | By Patchworkr Team
Egyptian fractions represent one of humanity's earliest mathematical innovations, dating back over 4,000 years to ancient Egypt. Rather than writing fractions with arbitrary numerators like modern mathematics (e.g., 3/4 or 5/12), ancient Egyptians expressed all fractions (except 2/3, which had a special symbol) as sums of unit fractions—fractions where the numerator is always 1. For example, 3/4 would be written as 1/2 + 1/4. This system was practical for dividing goods like bread, grain, and land, where splitting a resource into distinct unit portions was more intuitive than working with complex fractions.
Every positive rational number can be expressed as a finite sum of distinct unit fractions—a remarkable property discovered by ancient mathematicians through the greedy algorithm. This algorithm, later formalized by medieval mathematicians like Fibonacci, works by repeatedly subtracting the largest possible unit fraction until nothing remains. While modern mathematics moved away from this system for everyday calculations, Egyptian fractions remain mathematically fascinating because they involve problems that are still unsolved today. For instance, it remains unknown whether all fractions can be expressed with fewer than a certain number of terms, and the Erdős-Straus conjecture concerns the minimum number of unit fractions needed to express fractions of the form 4/n.
Type a positive whole number in the numerator field. This is the top part of your fraction. For example, enter 5 to start working with 5/12.
Enter a positive whole number larger than or equal to the numerator in the denominator field. The numerator must be less than or equal to the denominator for a proper or improper fraction.
The calculator applies the greedy algorithm to find an Egyptian fraction representation. The result shows the sum of unit fractions, the original fraction, decimal equivalent, and the number of terms.
Study how the fraction breaks down into unit fractions. Each denominator in the Egyptian representation is unique. The number of terms shows the complexity of the decomposition.
Experiment with various numerators and denominators to observe different decomposition patterns. Notice how some fractions decompose into few terms while others require many.
Using Egyptian fractions for fair resource division
A unit fraction has a numerator of 1 and a positive integer denominator. Examples: 1/2, 1/3, 1/100. All Egyptian fractions are sums of distinct unit fractions.
Yes! Every positive rational number can be expressed as a sum of distinct unit fractions. This is guaranteed mathematically, though the number of terms varies.
No. Multiple valid Egyptian fraction representations exist for the same fraction. The greedy algorithm finds just one specific solution.
Ancient Egyptians had a unique symbol for 2/3, unlike other fractions. Its special treatment suggests it was commonly used in practical calculations.
The greedy algorithm finds the largest unit fraction that fits into the target fraction, subtracts it, and repeats with the remainder until nothing is left.
No. The greedy algorithm is guaranteed to terminate, but it doesn't always produce the representation with the fewest terms. That's a harder problem.
They're primarily studied in number theory and combinatorics. Modern research includes the Erdős-Straus conjecture and related unsolved problems.
Yes, improper fractions work too. For example, 7/5 can be expressed as 1 + 1/5 or decomposed further using the greedy algorithm.
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