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The integers are a very fundamental mathematical set on which arithmetic is based. They are the familiar "whole numbers"; that is, they do not include fractional quantities. They include zero and negative whole numbers—the natural numbers are the set that includes only positive whole numbers. That is, the integers are the set .
Mathematicians denote the set of integers with an ornate capital letter: . They are the 2nd item in this hierarchy of types of numbers:
- The "natural numbers"—, 0, 1, 2, 3, ...
- The "integers"—, positive, negative, and zero
- The "rational numbers"—, or fractions, like 355/113
- The "real numbers"—, including irrational numbers
- The "complex numbers"—, which give solutions to polynomial equations
Definition of the integers
Integer represents a set of signed numbers namely Negative numbers , Zero and Positive numbers . Integer is denoted as I . So Negative number is -I and Positive number is +I
- . Integer
- . Negative integer
- . Positive integer
Negative number is defined as a number has value less than zero
Positive number is defined as a number has value greater zero
Example Integers of octal base
Like positive integer
Like negative integer
For all positive x and y, we extend addition to all integers as follows. The value of x+(-y) depends on which is larger, x or y.
- If ,
- If ,
- Also, = , and
- Finally, for all x, positive, negative, or zero.
This operator, written "-x", has already been defined for positive numbers. For negative numbers, we define
- . Also, .
We define subtraction for all numbers, positive, negative, or zero, in terms of negation and addition
For negative numbers, we define
- Ordering: If x and y are both positive numbers:
- if and only if
- In all cases,
- means x≠y and y > x
From these definitions, we can then prove all the various commutativity, associativity, and distributivity properties, and the trichotomy law. This is left as an exercise.
Group and ring properties
The integers, with their property of addition, constitute a group. Groups are extremely important objects in mathematics. A group requires a set (the integers), an operation (addition), an "inverse" operation (negation—the natural numbers don't have this, so they are not a group), and an "identity" operation (zero). To be a group, the operation must be associative, a property which the integers satisfy. Also, since addition
Furthermore, the integers have a multiplication operator, satisfying the distributive law, and having a multiplicative identity (1). This makes the integers a ring.
For a discussion of the philosophical and historical aspects of the various types of numbers, see Our_Playground:_The_Real_Numbers_and_Their_Development.