Egyptian Fractions

The Papyrus of Scribe Ahmes: Mathematics from a Tomb

In **1858**, the Scottish antiquarian **Henry Rhind** purchased an ancient papyrus in Luxor. It was nearly 3,500 years old. The text began: "Rules for enquiring into nature, and for knowing all that exists, every mystery, every secret." The author - scribe **Ahmes** - had copied it from an even older original (~1850 BCE).

Rules for enquiring into nature, and for knowing all that exists, every mystery, every secret. - Opening words of the Rhind Papyrus

The **Erdős - Straus conjecture** (1948): any 4/n can be expressed as a sum of three unit fractions. Verified for n up to 10^17. But there is no proof! Egyptian fractions are not a museum exhibit - they are living mathematics with open problems.

4,000 years ago, Egyptians wrote fractions in an entirely different way: only 1/2, 1/3, 1/4... How to express 3/5? As a sum: 1/2 + 1/10. This seems strange, but it works. And it still generates unsolved mathematical problems today.

• **History of mathematics:** among the oldest mathematical texts
• **Algorithms:** greedy algorithms and their limitations
• **Number theory:** open problems (Erdős - Straus conjecture)

Unit Fractions

A **unit fraction** is a fraction with numerator 1. The ancient Egyptians used only such fractions (plus 2/3). This seems like a restriction, but it works surprisingly well.

**Unit fractions:** 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, ... **Egyptian notation:** An oval (mouth) was placed above the number: 𓂋 over III = 1/3 𓂋 over IIII = 1/4

Any fraction can be expressed as a sum of distinct unit fractions. This is called the Egyptian representation.

Egyptian Representation

The **Egyptian representation** of a fraction is a sum of distinct unit fractions. Each unit fraction is used at most once.

**Theorem:** Any fraction p/q (0 < p/q < 1) can be expressed as a sum of distinct unit fractions. Proof: the greedy algorithm (next section).

Finding the optimal (shortest) representation is an NP-complete problem. For some fractions the difference is enormous.

The Greedy Algorithm

**Fibonacci's greedy algorithm** (13th century) is guaranteed to find an Egyptian representation. At each step, take the largest unit fraction not exceeding the remainder.

**Why the algorithm terminates:** After subtracting 1/n from p/q, the numerator of the result is strictly less than p. Proof: pn - q < p (because n > q/p, so pn > q). The numerator decreases → the algorithm is finite.

The greedy algorithm is simple but not optimal. Finding the shortest Egyptian representation is an open algorithmic problem.

The Rhind Papyrus

The **Rhind Mathematical Papyrus** is an Egyptian mathematical text dating to ~1650 BCE. It contains tables decomposing fractions 2/n for odd n from 3 to 101.

**The Rhind Papyrus:** • Purchased by Henry Rhind in Luxor in 1858 • Written by the scribe Ahmes around 1650 BCE • A copy of an even older text (~1850 BCE) • Contains 87 mathematical problems • Held in the British Museum

Egyptian fractions are not merely a historical curiosity. They continue to generate unsolved mathematical problems to this day.

Key Ideas

• Unit fraction: numerator = 1
• Any fraction can be represented as a sum of distinct unit fractions
• The greedy algorithm works but is not optimal
• The Rhind Papyrus - a table of 2/n decompositions

Related Topics

Egyptian fractions are connected to fractions and algorithms:

• Fractions — Alternative representation
• GCD — Simplifying results
• History — Ancient Egyptian mathematics

Вопросы для размышления

• Why did Egyptians choose unit fractions?
• How do you find the shortest Egyptian representation?
• What modern applications might Egyptian fractions have?

Связанные уроки

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