26 - Two-Way Numbers and an Alternate Technique for Multiplying Two Numbers

Summary

Introduction

In 1726, John Colson (1680–1759), a British mathematician and member of the Royal Society of London, devised an ingenious way to represent positive integers using what he called negativo-affirmative figures.[2] With his scheme positive and negative digits are intermingled and the basic arithmetic operations of addition, subtraction, and multiplication are as straightforward as in decimal arithmetic. The figures can be used to encrypt integers and have been rediscovered on several occasions. One version makes unnecessary the use of the digits 6, 7, 8,and 9, another rotates the digits 180°. Colson referred to his method as a “promiscuous scheme” to simplify the basic operations of arithmetic. In the process, he discovered a more compact and efficient way to multiply two numbers. This article is appropriate for an advanced elementary or secondary school mathematics class and represents a block of mathematical-historical material.

Historical Background

There are several ways to represent positive integers other than using the standard decimal system. For example, the internal operations of computers are executed using the binary system which is translated into the hexadecimal system making it easier for humans to understand it. Colson's negativo-affirmative figures offer students an introduction to ciphering and a different perspective on the basic arithmetic operations. For example, consider the negativo-affirmative expression 3 5 7 8 4 which represents the positive integer 2 5 6 2 4. To understand why this is true, replace every digit in 3 5 7 8 4 with a bar over it with a zero to obtain 3 0 7 0 4.