Egyptian Fraction Calculator

Work out egyptian fraction instantly with clear inputs, formula shown and shareable results.

About Egyptian Fraction Calculator

An Egyptian fraction writes a proper fraction as a sum of distinct unit fractions. Fibonacci's greedy algorithm repeatedly subtracts the largest unit fraction that fits, 1/⌈q/p⌉, and always terminates.

Formula

Greedy step

p/q = 1/⌈q/p⌉ + (p·⌈q/p⌉ − q)/(q·⌈q/p⌉)

Worked example

With the default values below, Egyptian Fraction Calculator returns Egyptian fraction expansion of 1/25 + 1/757 + 1/763309 + 1/873960180913 + 1/1.5276127956420934e+24. Change any input above and every figure here is recalculated in your browser.

Inputs used

Numeratorminimum 1: 5
Denominatorminimum 2: 121

Results

Egyptian fraction expansion: 1/25 + 1/757 + 1/763309 + 1/873960180913 + 1/1.5276127956420934e+24
Number of unit fractions: 5
Sum of the unit fractions: 0.0413223140
Largest denominator used: 873,960,180,913

Step by step

1Values used
2Greedy step
3Egyptian fraction expansion= 1/25 + 1/757 + 1/763309 + 1/873960180913 + 1/1.5276127956420934e+24
4Number of unit fractions= 5
5Sum of the unit fractions= 0.0413223140
6Largest denominator used= 873,960,180,913

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 10 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

What does 5/121 expand to?

The greedy algorithm gives 1/25 + 1/757 + 1/763309 + …, converging quickly to the exact value.

Why did Egyptians use these?

Their notation only had symbols for unit fractions (plus 2/3), so every quantity had to be expressed as a sum of them.