What this calculator does
Ancient Egyptian arithmetic expressed every fraction as a sum of distinct unit fractions, each with a numerator of 1. Two thirds was the only exception they allowed a special symbol for.
The greedy algorithm always terminates, though the denominators can grow alarmingly. Five sixths splits neatly into 1/2 + 1/3, but three sevenths needs 1/3 + 1/11 + 1/231.
The formula
The largest unit fraction not exceeding the remainder is subtracted repeatedly. Fibonacci proved this greedy approach always terminates for any proper fraction.
| Term | Meaning |
|---|---|
| Unit fraction | A fraction with numerator 1. |
| Greedy algorithm | Repeatedly taking the largest unit fraction that fits. |
| Fibonacci-Sylvester method | The formal name for that greedy approach. |
The inputs explained
| Field | What to enter |
|---|---|
| Numerator | The numerator of the fraction to decompose. |
| Denominator | The denominator. Cannot be zero. |
When to use it
Understanding ancient arithmetic
Egyptian mathematics worked entirely in unit fractions.
Exploring number theory
Egyptian fractions raise questions that remain open today.
Dividing things fairly
The decomposition corresponds to a practical way of sharing loaves.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How long does each decomposition run?
Three fractions decomposed.
| Fraction (numerator 3) | Egyptian fraction | Number of unit fractions |
|---|---|---|
| 3/4 | 1/2 + 1/4 | 2 |
| 3/7 | 1/3 + 1/11 + 1/231 | 3 |
| 3/11 | 1/4 + 1/44 | 2 |
Questions
Why did Egyptians use unit fractions?
Most likely because they map directly onto the practical problem of dividing goods. Splitting five loaves among six people really does mean giving everyone half a loaf plus a third, and that is exactly what the decomposition says.
Does the greedy algorithm always work?
It always terminates, which Fibonacci proved in 1202. It does not always give the shortest decomposition though, and finding the shortest one is a genuinely hard problem.
Why do the denominators grow so fast?
Because each greedy step leaves a remainder with a much larger denominator. For some fractions the growth is explosive, producing denominators with dozens of digits within a handful of terms.
Are there unsolved problems here?
Yes. The Erdős-Straus conjecture, that 4/n can always be written as a sum of exactly three unit fractions, has been open since 1948 and remains unproven despite extensive computational verification.
For continued fraction expansions, see the continued fraction calculator. For rounding to a chosen denominator, see the round to nearest fraction calculator.