Egyptian Fraction — Definition, Formula & Examples

An Egyptian fraction is a way of writing a fraction as the sum of different unit fractions (fractions with 1 in the numerator). For example,

\frac{3}{4}

can be written as

\frac{1}{2} + \frac{1}{4}

An Egyptian fraction representation of a positive rational number $\frac{a}{b}$ is a finite sum of distinct unit fractions $\frac{1}{n_1} + \frac{1}{n_2} + \cdots + \frac{1}{n_k}$ where $n_1 < n_2 < \cdots < n_k$ are positive integers.

How It Works

To convert a fraction into an Egyptian fraction, find the largest unit fraction that fits inside it, subtract it, and repeat with the remainder. This approach is called the Greedy Algorithm. At each step, you choose

\frac{1}{n}

where

n

is the smallest integer making

\frac{1}{n} \leq

your remaining fraction. You continue until the remainder is itself a unit fraction. Every positive fraction has at least one Egyptian fraction representation.

Worked Example

Problem: Write 5/7 as an Egyptian fraction.

Step 1: Find the largest unit fraction less than or equal to 5/7. Since 7 ÷ 5 = 1.4, round up to 2, so the first unit fraction is 1/2.

\frac{1}{2} \leq \frac{5}{7}

Step 2: Subtract 1/2 from 5/7 to find the remainder.

\frac{5}{7} - \frac{1}{2} = \frac{10}{14} - \frac{7}{14} = \frac{3}{14}

Step 3: Find the largest unit fraction less than or equal to 3/14. Since 14 ÷ 3 ≈ 4.67, round up to 5, so the next unit fraction is 1/5.

\frac{1}{5} \leq \frac{3}{14}

Step 4: Subtract 1/5 from 3/14.

\frac{3}{14} - \frac{1}{5} = \frac{15}{70} - \frac{14}{70} = \frac{1}{70}

Step 5: The remainder is already a unit fraction, so we are done.

\frac{5}{7} = \frac{1}{2} + \frac{1}{5} + \frac{1}{70}

Answer:

\frac{5}{7} = \frac{1}{2} + \frac{1}{5} + \frac{1}{70}

Why It Matters

Ancient Egyptians used this system over 3,500 years ago because their notation only allowed unit fractions. Studying Egyptian fractions today builds strong skills in fraction subtraction and finding common denominators, and it appears in number theory competitions and recreational math puzzles.

Common Mistakes

Mistake: Using the same unit fraction twice, such as writing 2/5 as 1/5 + 1/5.

Correction: Every unit fraction in an Egyptian fraction must be distinct. A valid representation of 2/5 is 1/3 + 1/15.