** Next:** 2.2 Rational Numbers
**Up:** 2 Numbers
** Previous:** 2 Numbers

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

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.
denotes a set of numbers.
An *n*-ary operator
maps an *n*-tuple of numbers to a single number.
Formally stated,

Addition and multiplication are binary operators.
An *n*-ary function may
be represented as a set *F*, of *n*+1-tuples of numbers:
Boldface is used to indicate vectors.
A set of numbers is closed under an *n*-ary operation
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
is commutative if

it is associative if
it has identity if
and it has as an inverse if
where *i* is the identity for .
A unary operator *g* has an inverse
if
An *n*-ary function
is total if

where states that *g*
is undefined for agument .
A function which is not total is a partial function.
The function *g* is injective (invertible) if
An inverse operator
is a total inverse if it is a total operator,
otherwise it is a partial inverse. A set of numbers
is closed under operator
if and only if is total. The domain
of a function *g* is written formally as .
An *n*-ary function
may be restricted to a set ,
so that is not defined for :
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 denotes exclusive or.
The comparison operator
( )
maps pairs of numbers to booleans:

where , 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:** 2.2 Rational Numbers
**Up:** 2 Numbers
** Previous:** 2 Numbers