Egyptian Fractions: Why Ancient Scribes Wrote 5/6 as 1/2 + 1/3

Ask a modern student to write five sixths and you get 5/6. Ask an Egyptian scribe four thousand years ago and you get 1/2 + 1/3. Same value, completely different way of recording it. The Egyptians almost never used a fraction with a numerator other than 1. Every quantity below a whole was built up from pieces that each represented one share: a half, a third, a seventh, a forty-second. We call these unit fractions, and a sum of distinct ones is an Egyptian fraction.

This sounds like a quirk, but it shaped how an entire civilization did arithmetic for over a millennium. The decomposition rules behind it are still a clean teaching example for greedy algorithms today. Let's walk through what they are, how to build one by hand, and where the math gets surprisingly deep.

What Counts as an Egyptian Fraction

The rule has two parts. Every term must have 1 on top, and no two terms may repeat. So 1/2 + 1/3 is a legal Egyptian fraction for 5/6, but 1/3 + 1/3 is not, even though it also sums to 2/3, because the two thirds are identical.

The "distinct" requirement is what makes the problem interesting. Repeating allowed, you could write anything trivially: 3/7 is just 1/7 + 1/7 + 1/7. Forbidding repeats forces you to find genuinely different denominators, and that is where the algorithms and the history live.

The Egyptians kept exactly one exception to the unit-fraction rule. They had a dedicated symbol for 2/3 and used it constantly, because two-thirds came up so often in dividing goods and grain that giving it its own glyph saved effort. Everything else was a sum of distinct unit fractions, looked up or computed on the spot.

The Rhind Papyrus and the 2/n Table

Most of what we know about Egyptian fraction practice comes from the Rhind Mathematical Papyrus, copied around 1550 BCE by a scribe named Ahmes from an even older source. It opens with a reference table: the Egyptian fraction expansion of 2/n for every odd n from 3 to 101. This was the lookup table a working scribe needed, because doubling a unit fraction is exactly the operation that produces a 2/n value, and Egyptian arithmetic was built on doubling and halving.

The table is not random. 2/5 is recorded as 1/3 + 1/15, 2/7 as 1/4 + 1/28, and the choices consistently favor small, manageable denominators over what a naive method would produce. Scholars still debate the exact rules the table-makers followed, which tells you something: there is no single "correct" Egyptian fraction for a given value, only many valid ones, and choosing a good one is a craft.

The Greedy Algorithm: Always Take the Biggest Piece That Fits

In 1202, Fibonacci described a method in his Liber Abaci that mechanically produces an Egyptian fraction for any proper fraction. It is now usually called the Fibonacci-Sylvester greedy algorithm, and the rule is a single sentence:

At each step, subtract the largest unit fraction that does not exceed what remains, then repeat on the leftover.

That is the whole algorithm. To find the largest unit fraction 1/k that fits under a value a/b, you take the ceiling of b/a. Subtract, reduce, and run again on the remainder. Fibonacci proved this always terminates, because the numerator of the remainder strictly shrinks at every step, so it must reach zero.

Here is the worked example everyone starts with. Take 2/3:

The largest unit fraction not exceeding 2/3 is 1/2, because ceiling of 3/2 is 2. (1/1 is too big; 1/2 fits.)
Subtract: 2/3 - 1/2 = 1/6.
1/6 is already a unit fraction, so we stop.
Result: 2/3 = 1/2 + 1/6.

Now try 5/6 to see two clean steps:

Largest unit fraction under 5/6 is 1/2 (ceiling of 6/5 is 2).
Subtract: 5/6 - 1/2 = 1/3.
1/3 is already a unit fraction, so we stop.
Result: 5/6 = 1/2 + 1/3.

Run the same machine on 4/5 and you get 1/2 + 1/4 + 1/20. The greedy method never fails, but notice that 4/5 also equals 1/2 + 1/5 + 1/10, which uses smaller denominators. Greedy gives you a valid answer, not always the prettiest one. That tension is the single most important thing to understand about the algorithm.

Where the Greedy Method Goes Wrong (and Right)

The greedy algorithm has two faces. On the one hand it is guaranteed: feed it any proper fraction and it always halts with a finite, distinct-denominator sum. That guarantee is exactly why it is the algorithm a calculator should ship, because it is deterministic and provably terminating.

On the other hand, "always works" is not "always best." Greedy can produce alarmingly large denominators. The textbook horror case is 5/121, where the greedy expansion ends with a denominator in the tens of thousands, while a hand-chosen decomposition stays tiny. And 3/7 greedily becomes 1/3 + 1/11 + 1/231 — correct, but 231 is a big jump from a fraction with denominator 7.

I went down this rabbit hole expecting a tidy little party trick and came out genuinely impressed. What surprised me was how fast the denominators can explode: I fed the calculator a few innocent-looking fractions and watched a four-term expansion end in five-digit denominators. Greedy is honest but blunt, and once you see that, the Rhind Papyrus stops looking like a primitive curiosity and starts looking like the work of people who had figured out, by hand, how to choose good decompositions where a naive rule would give ugly ones.

Compute It Yourself, Step by Step

You can grind the greedy algorithm by hand, but the ceiling arithmetic gets tedious past two or three steps, and that is exactly where mistakes creep in. The Egyptian Fraction Calculator runs the Fibonacci-Sylvester method and shows every subtraction, so you see the largest unit fraction chosen at each stage, the remainder, and the final sum rather than just an answer dropped on you. It reduces the input first (so 10/12 and 5/6 give the same 1/2 + 1/3), splits any whole part off an improper fraction, and copies the result as plain text like 1/2 + 1/3 for pasting into a worksheet or script.

A few practical reminders that the step view makes obvious:

Reduce before you expand. 10/12 decomposed unreduced can give a different, longer answer than 5/6. Always simplify first.
Improper fractions have a whole part. 7/3 is 2 + 1/3, not a single sum below 1. The classical problem only covers the part under one.
Greedy is not minimal. If you need the shortest or smallest-denominator form, greedy is a starting point, not the last word.

If you want to verify the reductions and common denominators by hand, the Fraction Calculator handles ordinary add, subtract, and simplify operations, which pairs naturally with checking each greedy subtraction step.

Why This Still Matters

Egyptian fractions are not just history. They are a rare case where a real algorithm — greedy, with a clean termination proof — sits on top of a question people actually faced: how do you split five loaves fairly among six people? The scribe's answer, give everyone 1/2 + 1/3 of a loaf, is both the math and the dinner. That blend of a provable method and a concrete sharing problem is what makes the topic such good teaching material, and why a step-by-step tool beats a black box that only prints the final sum.

Type a fraction, watch the greedy choice unfold, and you are doing the same arithmetic a scribe did with a reed pen four thousand years ago — just faster.

Made by Toolora · Updated 2026-06-13