Egyptian Fractions Calculator

Egyptian fractions are representations of rational numbers as a sum of distinct unit fractions — fractions with 1 in the numerator, like 1/2, 1/3, or 1/7. Ancient Egyptians used this notation over 4,000 years ago, as seen in the Rhind Papyrus. Every positive proper fraction (where the numerator is less than the denominator) can be expressed as a finite sum of distinct unit fractions.

The Greedy Algorithm, attributed to Fibonacci and later Sylvester, works by repeatedly subtracting the largest possible unit fraction at each step. Given a/b, you find the ceiling of b/a to get denominator n, write 1/n, then compute the remainder (a/b − 1/n) and repeat until the remainder is zero. It always terminates and always produces distinct denominators.

Using the Greedy Algorithm: the ceiling of 5/4 is 2, so the first unit fraction is 1/2. The remainder is 4/5 − 1/2 = 3/10. The ceiling of 10/3 is 4, giving 1/4. The remainder is 3/10 − 1/4 = 1/20. So 4/5 = 1/2 + 1/4 + 1/20.

No — Egyptian fraction representations are not unique. Any proper fraction can be expressed as a sum of distinct unit fractions in infinitely many different ways. The Greedy Algorithm produces one specific valid representation, but other algorithms (such as Bleicher/Erdős, binary remainder, or splitting methods) can yield different valid expansions of the same fraction.

Ancient Egyptians preferred unit fractions because their numeral system and arithmetic practices made them easier to work with than general fractions. Using unit fractions simplified tasks like dividing food, land, or wages — for example, distributing 5 loaves among 8 people required expressing 5/8 as a sum of unit fractions that could be physically divided. The only non-unit fraction they regularly used was 2/3.

Yes, the Greedy Algorithm is guaranteed to terminate and produce a valid Egyptian fraction expansion for any positive proper fraction (where numerator < denominator and both are positive integers). Each step strictly reduces the numerator of the remaining fraction, so the process always reaches zero in a finite number of steps.

This calculator handles proper fractions where the numerator is smaller than the denominator (values between 0 and 1). Fractions greater than 1 (improper fractions) can be decomposed by first separating the whole-number part and then applying Egyptian fraction expansion to the remaining proper fraction portion.

Egyptian fractions represent one of humanity's earliest systematic approaches to mathematics, dating back to at least 1650 BC as documented in the Rhind Mathematical Papyrus. The papyrus contains a table of decompositions for fractions of the form 2/n. This notation influenced Greek and later European mathematics, and studying Egyptian fractions remains an active area of number theory research today.