Egyptian Fractions Calculator
Convert proper fractions to Egyptian fraction expansions using greedy, pairing, or splitting algorithms
Convert Fraction to Egyptian Form
Top number of the fraction
Bottom number of the fraction
Method for decomposition
Example: Converting 6/7
Greedy Algorithm Example
Step 1: 6/7 = 1/⌈7/6⌉ + remainder
Step 2: ⌈7/6⌉ = ⌈1.167⌉ = 2
Step 3: 6/7 = 1/2 + (6×2 - 7)/(7×2) = 1/2 + 5/14
Step 4: 5/14 = 1/⌈14/5⌉ + remainder = 1/3 + 1/42
Result: 6/7 = 1/2 + 1/3 + 1/42
Algorithm Comparison
Greedy Algorithm
Most efficient, shortest expansions
✓ Recommended for most cases
Pairing Algorithm
Resolves conflicts by pairing identical fractions
✓ Educational value
Splitting Algorithm
Creates many terms quickly
⚠️ Limited to small numerators
Historical Context
Used in ancient Egypt (3200 BC+)
Found in the Rhind Papyrus (81 of 87 problems)
Used for dividing bread among workers
Fibonacci proved every fraction has an expansion
Understanding Egyptian Fractions
What are Egyptian Fractions?
An Egyptian fraction is a representation of a proper fraction (numerator < denominator) as a sum of distinct unit fractions (fractions with numerator = 1). For example: 3/4 = 1/2 + 1/4.
Key Properties
- •All unit fractions must be distinct (no repetition)
- •Every proper fraction has an Egyptian expansion
- •Expansions are not unique (multiple representations exist)
- •Used for practical division problems in ancient times
Greedy Algorithm Formula
x/y = 1/⌈y/x⌉ + (remainder)
- ⌈y/x⌉: Ceiling function (round up)
- Remainder: (x×⌈y/x⌉ - y) / (y×⌈y/x⌉)
- Repeat: Apply to remainder until numerator = 1
Example: 4/5 = 1/2 + 1/4 + 1/20
Modern Applications
- • Equal division problems
- • Mathematical education
- • Computer science algorithms
- • Number theory research
Special Cases
- • 2/3 had its own Egyptian symbol
- • 3/4 also had a special symbol
- • 1/2 was written with unique notation
- • These didn't need unit fraction expansion
Mathematical Interest
- • Proof of finite termination (Fibonacci)
- • Optimization problems
- • Continued fraction connections
- • Algorithmic complexity studies