Egyptian Fractions
Convert fractions to and from Egyptian fractions (sums of unit fractions like 1/2 + 1/3 + 1/6).
Input Fraction
Egyptian Fractions
Common Examples
What are Egyptian Fractions?
Egyptian fractions are a way of representing fractions as sums of distinct unit fractions (fractions with 1 as the numerator). Ancient Egyptians used this system because they only had symbols for unit fractions and the fraction 2/3.
How Egyptian Fractions Work
Any positive fraction can be expressed as a sum of distinct unit fractions. For example:
- 2/3 = 1/2 + 1/6
- 5/6 = 1/2 + 1/3
- 3/7 = 1/3 + 1/11 + 1/231
The Greedy Algorithm
The most common method to find Egyptian fractions is the greedy algorithm:
- Find the largest unit fraction less than or equal to your fraction
- Subtract it from your fraction
- Repeat until the remainder is zero
This always works, though it doesn't always produce the shortest representation.
Egyptian Fractions in Geocaching
Egyptian fractions appear in geocaching puzzles due to their:
- Historical appeal: Ancient Egyptian themes
- Mathematical challenge: Converting between forms
- Multiple representations: Same fraction, different sums
- Hieroglyphic connection: Often paired with Egyptian symbols
Historical Context
The Rhind Mathematical Papyrus (c. 1650 BCE) contains extensive tables of Egyptian fractions. Egyptian scribes needed these for dividing goods, measuring land, and calculating with fractions in everyday commerce.
Mathematical Properties
- Every positive rational number has an Egyptian fraction representation
- Representations are not unique (multiple ways to express the same fraction)
- The greedy algorithm always terminates
- Finding the shortest representation is computationally hard
Eye of Horus Fractions
The ancient Egyptians also used parts of the Eye of Horus symbol to represent the fractions 1/2, 1/4, 1/8, 1/16, 1/32, and 1/64 (powers of 1/2), used in measuring grain.