# Egyptian multiplication and some of its ramifications

## TL;DR

This paper explores Egyptian multiplication and exponentiation, highlighting their historical significance and deriving algorithms for division, fractional powers, and logarithms based on these ancient methods.

## Contribution

It presents a unified framework for understanding Egyptian multiplication and exponentiation, deriving classical algorithms and proposing new ones for fractional powers.

## Key findings

- Egyptian multiplication can be used for efficient division algorithms.
- The method yields a new approach to fractional power computation.
- It provides historical context for algorithms like logarithms and division.

## Abstract

Multiplication and exponentiation can be defined by equations in which one of the operands is written as the sum of powers of two. When these powers are non-negative integers, the operand is integer; without this restriction it is a fraction. The defining equation can be used in evaluation mode or in solving mode. In the former case we obtain "Egyptian" multiplication, dating from the 17th century BC. In solving mode we obtain an efficient algorithm for division by repeated subtraction, dating from the 20th century AD. In the exponentiation case we also distinguish between evaluation mode and solving mode. Evaluation mode yields a possibly new algorithm for raising to a fractional power. Solving mode yields the algorithm for logarithms invented by Briggs in the 17th century AD.