Math
Egyptian Fractions Calculator
Type a fraction such as 4/13 and the calculator writes it as a sum of distinct unit fractions the way Egyptian scribes did: 4/13 = 1/4 + 1/18 + 1/468.
It uses the greedy method (Fibonacci, 1202; Sylvester, 1880): at each step subtract the largest unit fraction 1/⌈q/p⌉ that does not overshoot, and repeat on what is left. Every subtraction is shown, the terms are listed, and the sum is checked exactly. Improper fractions are split into a whole part plus a proper Egyptian fraction.
A positive fraction, decimal, or mixed number. Whole parts are separated off; the fractional part is expanded.
Try an example
Result
Egyptian fraction
—
- Number of unit fractions
- —
- Denominators
- —
- Largest denominator
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- Check
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What the Egyptian fractions calculator solves
Ancient Egyptian scribes wrote every fraction (except 2/3) as a sum of distinct unit fractions, fractions with The top number of a fraction; it counts how many parts you have. 1. The Rhind papyrus tabulates 2/n this way for odd n. An Egyptian fraction for p/q is any such sum: 4/13 = 1/4 + 1/18 + 1/468. This calculator produces one using the greedy algorithm and shows the arithmetic at each step.
The greedy (Fibonacci–Sylvester) algorithm
- Take the largest unit fraction not exceeding p/q: 1/⌈q/p⌉. For 4/13, ⌈13/4⌉ = 4, so 1/4.
- Subtract it: 4/13 - 1/4 = (16 - 13)/52 = 3/52.
- Repeat on the leftover: ⌈52/3⌉ = 18, and 3/52 - 1/18 = (54 - 52)/936 = 2/936 = 1/468.
- Stop when the leftover is itself a unit fraction or zero. The numerators strictly decrease (p⌈q/p⌉ - q < p), which guarantees the process ends within p steps.
| Fraction | Greedy Egyptian fraction | Terms |
|---|---|---|
| 3/4 | 1/2 + 1/4 | 2 |
| 5/6 | 1/2 + 1/3 | 2 |
| 2/3 | 1/2 + 1/6 | 2 |
| 4/13 | 1/4 + 1/18 + 1/468 | 3 |
| 5/121 | 1/25 + 1/757 + 1/763309 + 1/873960180913 + 1/1527612795642093418846225 | 5 |
| 7/15 | 1/3 + 1/8 + 1/120 | 3 |
Greedy is not always shortest
The greedy method always works, but it can produce enormous The bottom number of a fraction; it says how many equal parts make one whole., as 5/121 shows; a smarter split gives 5/121 = 1/33 + 1/121 + 1/363. Finding the expansion with the fewest terms or the smallest largest denominator is a harder problem with no simple formula, and some questions about Egyptian fractions (like the Erdős–Straus conjecture that 4/n always needs at most three terms) are still open.
How to use it
- Type a positive fraction, decimal, or A whole number written next to a fraction, like 2 1/3..
- Read the unit-fraction sum and the list of denominators.
- Open Show the work to see each ceiling division and subtraction, then the exact check that the terms add back to the fraction.
How to read the answer
The result lists distinct unit fractions whose exact sum is the input; a whole part, if any, is written first. The number of terms is at most the numerator of the reduced fraction. Large denominators are a feature of the greedy method, not an error: the sum is still exact, but a shorter expansion may exist.
Common mistakes and edge cases
- Using a unit fraction larger than the fraction itself (1/3 for 4/13 overshoots since 1/3 > 4/13); the ceiling of q/p gives the right denominator.
- Repeating a unit fraction: 2/5 = 1/5 + 1/5 is not an Egyptian fraction; use 1/3 + 1/15.
- Forgetting to reduce first: 6/8 should be treated as 3/4.
- Expecting the greedy answer to be the shortest or nicest one.
- Entering 0 or a negative fraction; Egyptian fractions are for positive values.
Worked examples
4/13
1/4 + 1/18 + 1/468.
Egyptian fraction
1/4 + 1/18 + 1/468
5/6
1/2 + 1/3.
Egyptian fraction
1/2 + 1/3
Decimal input
0.7 = 7/10 = 1/2 + 1/5.
Egyptian fraction
1/2 + 1/5
Mixed number
2 3/8 = 2 + 1/3 + 1/24.
Egyptian fraction
2 + 1/3 + 1/24
Already a unit fraction
1/7 stays as is.
Egyptian fraction
1/7
Whole number
Nothing to expand.
Egyptian fraction
3
Explosive denominators
5/121 takes five terms with a 25-digit denominator.
Egyptian fraction
1/25 + 1/757 + 1/763,309 + 1/873,960,180,913 + 1/1,527,612,795,642,093,418,846,225
Zero
Not allowed.
Egyptian fraction
Error
Frequently asked questions
What is an Egyptian fraction?+
A sum of distinct unit fractions (fractions with numerator 1) that equals a given fraction, such as 4/13 = 1/4 + 1/18 + 1/468.
How does the greedy algorithm work?+
Repeatedly take the largest unit fraction that fits, 1/⌈q/p⌉, subtract it, and continue with the remainder. The numerators shrink each time, so it always finishes.
Does every fraction have an Egyptian fraction expansion?+
Yes, every positive rational number does, and in fact infinitely many. Fibonacci proved the greedy method always terminates.
Why are some denominators so big?+
The greedy step multiplies denominators together, so they can grow doubly exponentially. 5/121 reaches a 25-digit denominator greedily even though 1/33 + 1/121 + 1/363 also works.
Why did the Egyptians write fractions this way?+
Their notation only had symbols for unit fractions (plus 2/3), so any other fraction had to be a sum of them. Tables like the Rhind papyrus 2/n table made the arithmetic practical.
Can a unit fraction be repeated?+
No. By definition the terms are distinct. 1/n can always be split as 1/(n+1) + 1/(n(n+1)) if you need to avoid a repeat.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
References
- Fibonacci, Liber Abaci (1202), Chapter 7 (greedy unit-fraction method); Sylvester, American Journal of Mathematics 3 (1880) — Origin of the greedy algorithm and the proof that it terminates.
- OEIS A050205, Denominators in the greedy Egyptian fraction expansion
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Last updated: September 4, 2026