WHAT IS A NUMBER SYSTEM?                M 326K, Spring 2004

Like many words and phrases, the phrase "number system" has more than one meaning.

Meaning I: A collection of things (usually called numbers) together with operations on those numbers and the properties that the operations satisfy.

Example: The counting numbers (1, 2, 3, …) together with the operations of addition, subtraction, multiplication, and division and the properties they satisfy. These properties include:

Meaning II: A system for representing (that is expressing or writing) numbers of a certain type.

Example: There are several systems for representing the counting numbers. These include:

Meaning III: A combination of Meanings I and II. In other words, a collection of numbers together with operations, properties of the operations, and a system of representing these numbers.

Example: If we consider the counting numbers as a number system using Meaning I, it wouldn't matter whether we expressed 4 as 4 (decimal), IV (Roman numeral), or 100 (binary), but using Meaning III, it would matter.