Egyptian Fractions Calculator
Our Egyptian fractions calculator will teach you how to write fractions in the Egyptian style. It's not hard — well, the math isn't, at least! Type in any proper fraction and the tool instantly rewrites it as a sum of distinct unit fractions, shows every step of the greedy algorithm, verifies the result, and compares three different expansion methods side by side.
Here you will learn:
- How Egyptians did math;
- What an Egyptian fraction is;
- How to use Egyptian fractions;
- How to calculate Egyptian fractions — the greedy algorithm; and
- How to use our Egyptian fractions calculator.
Discover the ancient Egyptian fractions with us. 1/3, 1/2, 1, go!
Egyptian math
Egyptians were good at math! Their millennial civilization saw the development of many innovative concepts, spanning from arithmetic to geometry. Look at the pyramids — we even call them pyramids!
The first evidence of the use of mathematics dates back to 3200 BC. Let this sink in: we are closer to Rome than Romans were to those first additions! In many papyruses, historians found traces of mathematical problems and solutions. Thanks to them, we know that Egyptians knew how to calculate surfaces and volumes of many shapes and had knowledge of mathematical constants like π.
Egyptians had a particular way to write fractions — today, we call them Egyptian fractions — and they made extensive use of it. Let's take a look!
What are Egyptian fractions?
Our knowledge of the Egyptian fraction expansion comes mainly from a papyrus called the Rhind papyrus, where, out of 87 mathematical problems, as many as 81 involve fractions. Historians think that the Egyptians' interest in fractions stemmed from the particular labor force they used in their construction projects. When managing hundreds of workers, you don't want conflict over an unevenly split piece of bread!
An Egyptian fraction is a way to represent a proper fraction (a fraction where the numerator is smaller than the denominator) as a sum of distinct unit fractions. A unit fraction is simply a fraction with 1 on top: 1/2, 1/3, 1/17, and so on.
a/b = 1/n1 + 1/n2 + … + 1/nk, where n1 < n2 < … < nk
Like this!
5/8 = 1/2 + 1/8
Two rules matter here. First, every denominator must be a whole number. Second — and this is the part that makes the problem interesting — all denominators must be different. Writing 5/8 as 1/8 + 1/8 + 1/8 + 1/8 + 1/8 is perfectly true arithmetic, but it is not an Egyptian fraction: the Egyptians would have refused to repeat a term.
💡 The ancient Egyptians made two exceptions to the "unit numerator" rule: they had special symbols for 2/3 and (more rarely) 3/4. Everything else was built out of unit fractions.
A remarkable fact: every proper fraction has an Egyptian fraction expansion — and in fact infinitely many of them. There is no fraction you can type into our calculator that will defeat it.
Applications of the Egyptian fractions
Why bother splitting a fraction into a sum of unit fractions? Because unit fractions are easy to share fairly, and that was the whole point.
The classic Rhind papyrus problem asks: how do you divide 5 loaves of bread among 8 workers? Each worker is owed 5/8 of a loaf. The naive approach is to cut every loaf into 8 slices and hand 5 slices to each person — that's 40 tiny cuts and 40 crumbly pieces.
The Egyptian approach uses the expansion 5/8 = 1/2 + 1/8:
- Cut 4 loaves in half → 8 halves → one half per worker (that's the 1/2).
- Cut the last loaf into 8 pieces → one eighth per worker (that's the 1/8).
Twelve cuts instead of forty, and — crucially — everyone visibly gets the same pieces. Nobody can complain that their neighbor's share looks bigger. Set the amount to divide field in the calculator to 5 loaves and you will see exactly this breakdown.
The same logic still applies today whenever a quantity must be split into visibly equal, easy-to-measure portions — money, land, ingredients, work shifts. That's why our calculator lets you attach a currency (US dollar, Russian ruble, euro, and 17 more), a metric unit (kg, m, L…), or a US customary unit (lb, ft, gal…) to the quantity being divided.
Beyond the practical side, Egyptian fractions are alive and well in modern mathematics. They connect to continued fractions, to number theory problems such as the famous unsolved Erdős–Straus conjecture (that 4/n = 1/a + 1/b + 1/c for every n > 1), and to greedy algorithms in computer science.
Greedy algorithms: how to calculate an Egyptian fraction expansion
The most famous method is the greedy algorithm, described by Fibonacci in 1202 and later analyzed by Sylvester — hence its other name, the Fibonacci–Sylvester algorithm.
A greedy algorithm is one that, at every step, grabs the biggest bite it can and never looks back. Here, the "biggest bite" is the largest unit fraction that still fits inside what's left of our fraction. Given a remainder x/y, that unit fraction is:
1 / ⌈y / x⌉
where ⌈ ⌉ is the ceiling function — round up to the next whole number. Subtract that unit fraction, simplify what's left, and repeat until nothing remains:
x/y − 1/c = (x·c − y) / (y·c), with c = ⌈y/x⌉
Does this ever end? Yes — and that's the beauty of it. At each step the new numerator x·c − y is strictly smaller than the old numerator x. A sequence of positive whole numbers can't decrease forever, so the algorithm must hit a numerator of 1 — a unit fraction — and stop. Fibonacci's greedy algorithm always terminates, for every proper fraction, and the denominators it produces always increase, so they are automatically distinct.
The greedy algorithm in action
Let's expand 5/8 by hand, the way the calculator does it.
Step 1
Our remainder is 5/8. Compute the ceiling: ⌈8/5⌉ = ⌈1.6⌉ = 2. So the biggest unit fraction that fits is 1/2. Subtract it:
5/8 − 1/2 = (5·2 − 8) / (8·2) = 2/16 = 1/8
Step 2
The remainder is 1/8 — already a unit fraction! We take it as it is and we're done:
5/8 = 1/2 + 1/8
A longer example
Now try 7/15:
- ⌈15/7⌉ = ⌈2.14…⌉ = 3 → take 1/3; remainder 7/15 − 1/3 = 2/15.
- ⌈15/2⌉ = ⌈7.5⌉ = 8 → take 1/8; remainder 2/15 − 1/8 = 1/120.
- Remainder is a unit fraction → take 1/120.
7/15 = 1/3 + 1/8 + 1/120
⚠️ Greedy is simple, but it isn't always graceful. Feed it 5/121 and it produces denominators in the millions, while a smarter method finds 5/121 = 1/33 + 1/121 + 1/363. Being greedy is fast, not wise — which is exactly why other algorithms exist.
Other Egyptian fractions algorithms
Every proper fraction has infinitely many Egyptian expansions, so there is plenty of room for other strategies. Our calculator runs these alongside the greedy one so you can compare the number of terms and the size of the largest denominator:
The divisor (practical number) method
This is closest to what the scribes actually did. Look for a multiplier k such that the numerator k·x can be written as a sum of distinct divisors of the denominator k·y. For 5/8: the divisors of 8 are 8, 4, 2, 1, and 5 = 4 + 1, so
5/8 = 4/8 + 1/8 = 1/2 + 1/8
Denominators like 8, 12, 20 and 24 — the practical numbers — are especially rich in divisors, which is one reason they show up so often in ancient tables.
The Golomb–Farey algorithm
A clever number-theory trick. For x/y in lowest terms, find z with x·z ≡ 1 (mod y). Then x/y = 1/(y·z) + k/z with k = (x·z − 1)/y, and since z < y, the problem shrinks at every step. It guarantees denominators no larger than y·(y − 1) — a hard ceiling that greedy simply doesn't offer.
The odd greedy algorithm
A variant for fractions with an odd denominator: at each step round up to the next odd number, so every denominator in the expansion is odd. It's known to work in practice, but the denominators can explode spectacularly, and whether it terminates for every such fraction is still an open question. Our calculator reports it as unavailable when the numbers run away.
The splitting method
Start with x copies of 1/y and repeatedly apply the identity 1/n = 1/(n+1) + 1/(n·(n+1)) to break up repeats until all denominators differ. Elegant to state, but it produces long expansions in a hurry.
How to use our Egyptian fraction calculator
It takes seconds:
- Enter your fraction. Type the numerator on top and the denominator below. It must be a proper fraction — numerator smaller than denominator. Not in lowest terms? No problem, the calculator reduces it for you and tells you so. In a hurry? Tap one of the quick example buttons.
- Read the expansion. The headline result gives your fraction as a sum of distinct unit fractions, along with the number of terms, the largest denominator, and the decimal value.
- Follow the steps. The step-by-step table shows each remainder, each ⌈y/x⌉ ceiling, the unit fraction taken, and what's left — the whole greedy algorithm laid bare. The bar chart shows how the unit fractions stack up against the whole.
- Optional: divide a real quantity. Pick a unit or currency and enter an amount, and the calculator turns each unit fraction into an actual share — 5 loaves, $120, 3.5 kg, whatever you need. This is the Rhind papyrus bread problem, live.
- Optional: compare methods. The last table pits the greedy expansion against the divisor, Golomb–Farey, and odd greedy methods. Sometimes greedy wins; often it doesn't.
Ancient math
It's worth pausing on how strange, and how sophisticated, this all is. The Egyptians had no algebra, no equals sign, and no zero as a number — yet they built a fraction system flexible enough to handle any division a foreman could throw at it, and they compiled reference tables (like the 2/n table on the Rhind papyrus, covering every odd n from 5 to 101) so scribes wouldn't have to redo the work.
Egyptian fractions stayed in use for a very long time. Greek mathematicians adopted them, and they remained the standard way to write fractions in Europe throughout the Middle Ages — which is precisely why Fibonacci devoted a chapter of his Liber Abaci (1202) to expanding fractions into unit fractions, giving us the greedy algorithm along the way. Our modern a/b notation only won out later.
FAQs
What is an Egyptian fraction?
An Egyptian fraction is a proper fraction written as a sum of distinct unit fractions — fractions with numerator 1 and different whole-number denominators. For example, 5/8 = 1/2 + 1/8.
Does every fraction have an Egyptian fraction expansion?
Yes. Every proper fraction can be expanded, and in infinitely many ways. Fibonacci's greedy algorithm proves it constructively: it terminates for every proper fraction, because the numerator of the remainder strictly decreases at each step.
How do I write 5/8 as an Egyptian fraction?
Take the largest unit fraction that fits: ⌈8/5⌉ = 2, so take 1/2. The remainder is 5/8 − 1/2 = 1/8, which is already a unit fraction. Therefore 5/8 = 1/2 + 1/8.
Why can't denominators repeat?
That's the defining rule of the notation — and it had a practical purpose. Repeating 1/8 five times to make 5/8 is just "cut everything into eighths," which gives no advantage. Requiring distinct denominators forces you to find the genuinely efficient split, with the fewest cuts and the most obviously fair shares.
Is the greedy algorithm always the best method?
No. Greedy is guaranteed to finish, but it can produce huge denominators — 5/121 is the classic cautionary tale. Other methods often find shorter or tidier expansions, which is why our calculator shows several of them at once.
What was the 2/n table?
A lookup table on the Rhind papyrus listing Egyptian expansions of 2/n for every odd n from 5 to 101 (for instance 2/5 = 1/3 + 1/15). Since doubling was central to Egyptian multiplication, having these expansions pre-computed saved scribes an enormous amount of work.