Egyptian Fraction Calculator

What Are Egyptian Fractions?

An Egyptian fraction expresses a positive rational number as a sum of distinct unit fractions (fractions with numerator 1). Ancient Egyptian scribes used this notation nearly 4,000 years ago, as recorded in the Rhind Mathematical Papyrus.

Greedy Algorithm

$$\frac{n}{d} = \frac{1}{k} + \text{remainder}, \quad k = \left\lceil \frac{d}{n} \right\rceil$$

Repeat on the remainder until zero. This Fibonacci-Sylvester method always terminates for proper fractions.

Example

$$\frac{5}{8} = \frac{1}{2} + \frac{1}{8}$$

This calculator compares greedy, binary, and practical (Rhind-style) expansions side by side with step-by-step breakdowns.

Frequently Asked Questions

Can every fraction be written as an Egyptian fraction?

Yes. Every positive rational number has a finite Egyptian fraction representation. The greedy algorithm always finds one.

Why did Egyptians use unit fractions?

Unit fractions made physical division practical. Splitting loaves equally among workers maps naturally to sums like $\frac{1}{2} + \frac{1}{8}$.

What is the Rhind Papyrus?

The Rhind Mathematical Papyrus (c. 1650 BC) is one of the oldest math texts. It opens with a table decomposing $\frac{2}{n}$ for odd $n$ from 5 to 101.

Why do greedy denominators sometimes get huge?

For fractions like $\frac{5}{121}$, the greedy method produces very large denominators. Alternative algorithms often give shorter, more practical expansions.

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What Are Egyptian Fractions?

Greedy Algorithm

Example

Frequently Asked Questions

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