Egyptian Fraction

Representation as a sum of unit fractions

Egyptian Fraction Calculator

What is calculated?

An Egyptian fraction expresses a fraction as a sum of distinct unit fractions (fractions with numerator 1). All denominators are positive and different.

Historical background

Ancient Egyptians used this notation over 4000 years ago (Rhind Papyrus). They employed unit fractions and the special fraction 2/3 for many computations.

Enter fraction
Example: 3/4
Egyptian fraction
Representation as a sum of distinct unit fractions

Egyptian Fractions Info

Properties

Egyptian fraction rules:

Numerator = 1 Distinct denominators Greedy algorithm Unique representation

Note: Every positive proper fraction (less than 1) can be expressed as a sum of distinct unit fractions.

Examples
Simple: 1/2 = 1/2
Classic: 2/3 = 1/2 + 1/6
Complex: 5/8 = 1/2 + 1/8
Application: 43/48 = 1/2 + 1/3 + 1/16


Greedy algorithm & rules

Principle
\[\frac{a}{b} = \frac{1}{n_1} + \frac{1}{n_2} + \cdots\] Sum of unit fractions
Greedy step
\[n = \lceil \frac{b}{a} \rceil\] Largest unit fraction ≤ a/b
Recursion
\[\frac{a}{b} - \frac{1}{n} = \frac{an - b}{bn}\] Compute remainder fraction
Condition
\[n_1 < n_2 < n_3 < \cdots\] All denominators distinct
Existence
\[\forall \frac{a}{b} : a < b, a > 0\] Every proper fraction decomposable
Uniqueness
\[\text{Greedy} \Rightarrow \text{unique}\] Greedy algorithm yields unique result
Termination
\[\text{finite number of steps}\] Algorithm terminates
Special case
\[\frac{1}{n} = \frac{1}{n}\] Unit fraction remains unchanged

Step-by-step example

Example: Decompose 5/8 into Egyptian fraction
Step 1: Find largest unit fraction
\[n_1 = \lceil \frac{8}{5} \rceil = \lceil 1.6 \rceil = 2\] So: 1/2 ≤ 5/8

The largest unit fraction ≤ 5/8 is 1/2.

Step 2: Compute remainder
\[\frac{5}{8} - \frac{1}{2} = \frac{5}{8} - \frac{4}{8} = \frac{1}{8}\]
Step 3: Check remainder
\[\frac{1}{8}\] is already a unit fraction!
Step 4: Result
\[\frac{5}{8} = \frac{1}{2} + \frac{1}{8}\]
More examples
Decompose 2/3:
\[\frac{2}{3} = \frac{1}{2} + \frac{1}{6}\] (since 2/3 - 1/2 = 1/6)
Decompose 3/4:
\[\frac{3}{4} = \frac{1}{2} + \frac{1}{4}\] (since 3/4 - 1/2 = 1/4)
Decompose 4/5:
\[\frac{4}{5} = \frac{1}{2} + \frac{1}{4} + \frac{1}{20}\] (multiple steps required)
Practical application
Pizza example: 5 pizzas for 8 guests
5/8 = 1/2 + 1/8
→ Each gets half a pizza plus an eighth
Greedy algorithm steps
1. compute n = ⌈b/a⌉
2. subtract 1/n
3. form remainder
4. repeat until 0

The greedy algorithm always finds an optimal representation using the largest possible unit fractions.

Applications of Egyptian fractions

Egyptian fractions have both historical and modern uses:

Historical use
  • Ancient Egyptian mathematics
  • Rhind Papyrus (1650 BC)
  • Practical division tasks
  • Commerce and surveying
Practical sharing
  • Fair distribution of goods
  • Recipes and portions
  • Craft allocation
  • Time management
Mathematical education
  • Develop fraction intuition
  • Algorithmic thinking
  • Understand number theory
  • Historical mathematics
Modern applications
  • Computer algorithms
  • Approximation theory
  • Number theory research
  • Optimization problems

Mathematical context

Description

Egyptian fractions are one of the oldest known representations and illustrate how complex numerical concepts can be built from simple components. The greedy decomposition algorithm is a clear example of recursive problem solving and touches on fundamental number-theoretic ideas. Modern research still investigates optimal representations and related properties.

Summary

Egyptian fractions bridge ancient techniques and modern mathematics, providing insight into historical computation and algorithmic principles. They are useful both pedagogically and in theoretical investigations of rational approximations and combinatorial structures.




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