### 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:
• Two counting numbers can always be added to give a result that is a counting number.
•  You can only subtract a counting number from another counting number and obtain a counting number as the result if the number you subtract is smaller than the number you subtract from.
•  The associative property for addition: (a + b) + c = a + (b + c) for every three counting numbers a, b, and c.
• The commutative property for addition: a + b = b + c for every two counting numbers a and b.
•  The distributive property of multiplication over addition: a(b + c) = ab + ac for every three counting numbers a, b, and c.

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:
• The usual "base ten" or "decimal" system: 1, 2, 3,  … , 10, 11, 12, … 99, 100, ….
• Roman numerals: I, II, III, IV, V, VI, VII, VIII, IX, X, …
• The binary system: 1, 10, 11, 100, 101, …(read as "one", "one, zero", "one, one", "one, zero, zero", etc.) This system used in computer science. We will discuss this and other systems of representing numbers later in this class.

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.