Egyptian Fractions Calculator
Write any fraction as a sum of distinct unit fractions, the way ancient Egyptian scribes did. Use the greedy method or find the fewest terms with the smallest pieces, and see the pieces cut from a whole.
How to Use
- Type a fraction such as 4/13 (a decimal such as 0.3 works too).
- Choose the greedy method, which always takes the largest piece that fits, or fewest terms, which searches for the shortest sum with the smallest largest denominator.
- Read the sum of unit fractions, the number of terms and the smallest piece.
- The table shows what the other method gives, for comparison.
- The bar shows the pieces cut from one whole.
Worked Example
4/13 by the greedy method. 13 ÷ 4 = 3.25, rounded up to 4, so the largest unit fraction that fits is 1/4. 4/13 − 1/4 = 3/52. 52 ÷ 3 = 17.3, rounded up to 18: take 1/18, and 3/52 − 1/18 = 1/468, a unit fraction. So 4/13 = 1/4 + 1/18 + 1/468.
4/13 with smaller pieces. Searching all three-term sums finds 1/4 + 1/26 + 1/52: over 52, that is 13 + 2 + 1 = 16, and 16/52 = 4/13. Three terms either way, but the smallest piece is 1/52 instead of 1/468.
The common mistake: repeating a fraction. 2/5 = 1/5 + 1/5 is true but not an Egyptian fraction, because the unit fractions must all be different. The scribes wrote 2/5 = 1/3 + 1/15.
Show Work
Formulas
The Scribes’ Fractions
Egyptian arithmetic, known from the Rhind and Moscow mathematical papyri, worked with unit fractions and a special sign for 2/3. The Rhind papyrus, copied by the scribe Ahmes around 1550 BC, starts with a table expressing 2/n as unit fractions for every odd n from 5 to 101, the working tool for division by doubling and halving.
Unit fractions stayed in use in Greek and medieval European mathematics; Fibonacci’s Liber Abaci (1202) gives the greedy method among several others. Modern number theory took them up again in the 20th century, with problems such as the Erdős–Straus conjecture still unsolved.
About This Calculator
This calculator writes a positive fraction, or the fractional part of a larger number, as a sum of distinct unit fractions, either by the greedy Fibonacci–Sylvester method or by searching for the fewest terms with the smallest largest denominator (up to four terms). It checks the sum exactly, compares the two methods and draws the pieces on a bar.
Everything runs in your browser; nothing is sent anywhere. The greedy method works on any fraction; the fewest-terms search is for denominators up to 100,000.
Related calculators: Continued Fraction Converter, Stern–Brocot Tree, and Decimal to Fraction Converter.
Frequently Asked Questions
What is an Egyptian fraction?
A sum of different unit fractions, fractions with 1 on top: 3/4 = 1/2 + 1/4. Ancient Egyptian scribes wrote every fraction this way (apart from 2/3), so 2/5 was written 1/3 + 1/15. Every positive fraction can be written as an Egyptian fraction, in fact in infinitely many ways.
How does the greedy method work?
Take the largest unit fraction that is not bigger than what is left, subtract it, and repeat. For 4/13: 13/4 = 3.25, rounded up to 4, so take 1/4, leaving 3/52; 52/3 rounded up is 18, take 1/18, leaving 1/468, done: 4/13 = 1/4 + 1/18 + 1/468. Fibonacci described it in 1202 and Sylvester proved in 1880 that it always ends.
Why can the greedy method give huge denominators?
Because it never looks ahead. 5/121 by the greedy method needs five terms, the last with a denominator of 25 digits, while 1/33 + 1/121 + 1/363 does it in three small ones. The fewest-terms search here finds expansions like that.
How did the Egyptians use them?
For sharing. Dividing 4 loaves among 13 people as 1/4 + 1/18 + 1/468 of a loaf each is awkward, but dividing 2 loaves among 5 as 1/3 + 1/15 is practical: cut every loaf into thirds, give each person one third, and share the leftover third in five. The Rhind papyrus, about 1550 BC, opens with a table of 2/n for odd n from 5 to 101.
Are there open problems about them?
Yes. The Erdős–Straus conjecture says 4/n can always be written with three unit fractions for n ≥ 2; it has been checked for n up to 10¹⁷ but never proved. Egyptian fractions are a classic source of easy-to-state, hard-to-solve number theory.
How do I use the Egyptian Fractions Calculator?
Simply type your numbers and read the result, which refreshes the instant you change something. There is nothing to submit and nothing to wait for.
Is it free? Does it work without internet?
Yes to both. It is free with no sign-up, and once the page has loaded it keeps working even with no internet.
Where does my data go?
Nowhere — every calculation runs on your own device. Nothing you enter is uploaded, logged, or stored.
Common Use Cases
Greedy
7/15 = 1/3 + 1/8 + 1/120.
Smaller pieces
4/13 = 1/4 + 1/26 + 1/52 instead of the greedy 1/468.
Rhind papyrus
2/5 = 1/3 + 1/15 and 2/7 = 1/4 + 1/28.
A famous case
5/121 = 1/33 + 1/121 + 1/363, where greedy needs a 25-digit denominator.
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