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2.1 Integers

The integer number system is a basic system of numbers. The set of all   integers is denoted by tex2html_wrap_inline32159 :


This number system is particularly simple and forms the basis for all of the other number systems presented here. Although the reader is assumed to be familiar with the integers, some semi-formal discussion follows, which serves to refresh the reader's memory and to illustrate common features of all number systems. The integers can be constructed from the   natural numbers; purists construct the naturals using set theory [41, 17, 59].

Integers can be combined through addition and multiplication. Operators abstract the notion of combining numbers, by allowing for unary and 0-ary operators. The terms function and operator are interchangable.   tex2html_wrap_inline32169 denotes a set of numbers. An n-ary operator tex2html_wrap_inline32173 maps an n-tuple of numbers to a single number. Formally stated,


Addition and multiplication are binary operators. An n-ary function   tex2html_wrap_inline32179 may be represented as a set F, of n+1-tuples of numbers:  


Boldface is used to indicate vectors.

  A set of numbers tex2html_wrap_inline32169 is closed under an n-ary operation tex2html_wrap_inline32173 if


Integers are closed under addition and multiplication: the sum or product of any two integers is another integer.

Since binary operators are so prevalent several properties of binary operators will be relevant. A binary operator tex2html_wrap_inline32173 is commutative if


it is associative if


it has identity tex2html_wrap_inline32193 if


and it has tex2html_wrap_inline32195 as an inverse if


where i is the identity for tex2html_wrap_inline32173 . A unary operator g has an inverse tex2html_wrap_inline32203 if


An n-ary function tex2html_wrap_inline32207 is total if


where tex2html_wrap_inline32209 states that g   is undefined for agument tex2html_wrap_inline32213 . A function which is not total is a partial function. The function g is injective (invertible) if


An inverse operator tex2html_wrap_inline32217 is a total inverse if it is a total operator, otherwise it is a partial inverse. A set of numbers tex2html_wrap_inline32169 is closed under operator tex2html_wrap_inline32221 if and only if tex2html_wrap_inline32173 is total. The domain   of a function g is written formally as tex2html_wrap_inline32227 .


An n-ary function tex2html_wrap_inline32207 may be restricted to a set tex2html_wrap_inline32233 , so that tex2html_wrap_inline32235 is not defined for tex2html_wrap_inline32237 :  


Negation is the total inverse of addition. Subtraction is defined as the sum of a number with another number's additive inverse:


A serious limitation of the integers is the lack of a total inverse of multiplication. Division is defined as the product of a number with another number's multiplicative inverse:


It follows that the integers are not closed under division.

Addition and multiplication over the integers jointly satisfy the following distributive law:


multiplication is said to distibute over addition. Addition and multiplication over the integers do not satisfy the following, alternative, distributive law:


The first distributive law will be hereafter referred to as ``the'' distributive law.

Another nice property of the integers is that comparing any pair of integers will always result in exactly one of three orderings. Equivalently, every pair of distinct integers contains a larger member:


where tex2html_wrap_inline32239 denotes exclusive or.

The comparison operator tex2html_wrap_inline32241 ( tex2html_wrap_inline32243 ) maps pairs of numbers to booleans:


where   tex2html_wrap_inline32245 , the set of booleans.

Common Practice

Almost all computers have hardware dedicated to performing very quick operations on integers. Many systems strictly limit the magnitude of the integers to guarantee certain limits on computational resource requirements, while some do not. Although the manipulations of integers by computers is a fascinating and vitally important research area we will envision integers as a basic data type with rudimentary operations.

next up previous notation contents
Next: 2.2 Rational Numbers Up: 2 Numbers Previous: 2 Numbers
Jeff TupperMarch 1996