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Key Features
Instant Calculation
Get accurate results in real time with our optimized algorithm.
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Ancient Egyptian mathematics represented all fractions (except 2/3) as sums of distinct unit fractions — fractions with a numerator of 1. This tool decomposes any proper fraction into such a sum using the greedy (Fibonacci-Sylvester) algorithm.
How it works
At each step, the algorithm takes the largest unit fraction that doesn't exceed the remaining fraction (the ceiling of denominator/numerator gives that unit fraction's denominator), subtracts it, and repeats on what's left until nothing remains.
- Enter numerator.
- Enter denominator.
- Click Calculate to see your results.
Examples
A simple fraction
2/3 decomposes as 1/2 + 1/6, since 1/2 is the largest unit fraction not exceeding 2/3, and the remainder 2/3 − 1/2 = 1/6 is itself already a unit fraction.
Who should use it
- Studying ancient Egyptian mathematics and number theory history.
- Recreational mathematics and puzzle-solving.
Industry applications
- Mathematics education
- Number theory research and recreational mathematics
Advantages
- Always produces a valid decomposition for any proper fraction.
- Shows every step of the greedy subtraction process.
Limitations
- Only accepts proper fractions (numerator smaller than denominator after reduction).
Common mistakes to avoid
- Assuming the greedy algorithm gives the shortest possible decomposition — it doesn't always, though it's a reliable, simple method that always works.
Best practices
- Use the greedy method when you need any valid unit-fraction decomposition quickly; for a minimal number of terms, more specialized algorithms exist but are far more complex.
Tips
- If the resulting denominators grow very large, that's a known characteristic of the greedy algorithm for certain fractions — it doesn't indicate an error.