On the examples of Egyptian fractions in Liber Abaci

TL;DR

This paper analyzes Fibonacci's methods for expanding fractions into Egyptian fractions in Liber Abaci, clarifying the algorithms, addressing manuscript errors, and providing precise computational descriptions.

Contribution

It offers a detailed, formalized algorithmic interpretation of Fibonacci's Egyptian fraction expansion methods, improving understanding of historical mathematical procedures.

Findings

Clarified Fibonacci's algorithms for Egyptian fractions

Addressed manuscript and translation errors

Provided formal mathematical notation for procedures

Abstract

The focus of this note is to formulate the algorithms and give the examples used by Fibonacci in Liber Abaci to expand any fraction into a sum of unit fractions. The description in Liber Abaci is all verbal and the examples are numbers which may lead to different algorithmic descriptions with the same input and results. An additional complication is that the manuscripts that exist are copies of an older manuscript and in the process new errors are introduced. Additional errors may also be introduced in the transcript and the translation. Fibonacci introduces seven categories each with several numerical examples. We give a precise description of the computational procedure using standard mathematical notation.