Difference between revisions of "Integer"

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An '''integer''' is any whole number, positive, negative, or 0. Starting at 1 and going up are the [[counting numbers]] {1, 2, 3, 4, ...}, sometimes called "natural numbers".  
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An '''integer''' is any number that does not have any fractional or decimal components. Numbers such as 1,2 and -3 are integers, but 2½, √5 are not. The mathematical symbol for this set is <math>\mathbb{Z}</math>. More precisely, the set of all integers consists of all natural numbers {1, 2, 3, 4, ...}, their negatives {-1, -2, -3, -4, ...} and 0.  A formal definition is that it is the only [[integral domain]] whose positive elements are well ordered and in which order is preserved by addition. Integers are often colloquially referred to as "whole numbers," though this terminology is not well defined and is often used to refer to only a subset of integers, such as the positive integers.
  
More precisely, the set of all integers consists of all natural numbers {1, 2, 3, 4, ...}, their negatives {-1, -2, -3, -4, ...} and 0.  A formal definition is that it is the only [[integral domain]] whose positive elements are well ordered and in which order is preserved by addition.
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An integer may be:
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* [[even number|even]] (divisible by two)
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* [[odd number|odd]] (not divisible by two)
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* [[positive number|positive]] (more than [[zero]])
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* [[negative number|negative]] (less than zero)
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* [[whole number|whole]] (undivided)
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* [[composite]] (divisible into other integers) or [[prime]] (only divisible by itself and one)
  
An integer may be [[even number|even]] (divisible by two) or [[odd number|odd]] (not divisible by two), [[positive number|positive]] (more than [[zero]]) or [[negative number|negative]] (less than zero), [[whole number|whole]] (undivided) or fractional (divided into smaller parts), and various other classifications, such as [[prime]] (only divisible by itself and one).
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Every integer larger than 1 has a unique [[prime factorization]].
 
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Every integer larger than 1 has a unique [[prime factorizarion]].
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Some examples of integers: 1, 10/5, 98058493, -87, -3/3, both square roots of 9, and 0.
 
Some examples of integers: 1, 10/5, 98058493, -87, -3/3, both square roots of 9, and 0.
  
Likewise, the following numbers are not integers: 5/10, the square root of -9, 8.75, and [[pi]]. However, in larger systems of [[algebraic numbers]], the squareroot of -9 is considered to be an integer, but not a <b>rational integer</b>.
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Likewise, the following numbers are not integers: 5/10, the square root of -9, 8.75, and [[pi]].  
  
==Generalizations==
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Some [[subset]]s of the integers are often used. They have their own symbols:
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{| class="wikitable"
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|-
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! set
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! name
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! symbol
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|-
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| ..., -2, -1, 0, 1, 2, ...
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| integers
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| <math>\mathbb{Z}</math>
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|-
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| 1, 2, 3, 4, ...
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| positive integers
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| <math>\mathbb{Z}^+</math>
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|-
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| 0, 1, 2, 3, 4, ...
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| nonnegative integers
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| <math>\mathbb{Z}^*</math>
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|-
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| 0, -1, -2, -3, -4, ...
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| nonpositive integers
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|-
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| -1, -2, -3, -4, ...
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| negative integers
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| <math>\mathbb{Z}^-</math>
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|}
  
The set of integers form what is called in [[abstract algebra]] a [[ring]]. A ring is a set equipped with operations + and x with the usual properties learned in high-school algebra ([[commutativity]], [[distributivity]], [[linearity]], and [[associativity]]), identities for both operations ([[Additive identity of addition|additive]] and [[Multiplicative identity|multiplicative]]), and inversion ([[subtraction]]). Other objects such as [[matrix]]es, [[polynomial]]s, [[quaternion]]s, and [[algebraic integer]]s also form rings, and can therefore be viewed as generalizations of the integers.
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==See also==
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*[[algebraic numbers]]
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*[[abstract algebra]]
  
[[category:mathematics]]
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[[Category:Mathematics]]

Latest revision as of 14:22, July 13, 2016

An integer is any number that does not have any fractional or decimal components. Numbers such as 1,2 and -3 are integers, but 2½, √5 are not. The mathematical symbol for this set is . More precisely, the set of all integers consists of all natural numbers {1, 2, 3, 4, ...}, their negatives {-1, -2, -3, -4, ...} and 0. A formal definition is that it is the only integral domain whose positive elements are well ordered and in which order is preserved by addition. Integers are often colloquially referred to as "whole numbers," though this terminology is not well defined and is often used to refer to only a subset of integers, such as the positive integers.

An integer may be:

Every integer larger than 1 has a unique prime factorization.

Some examples of integers: 1, 10/5, 98058493, -87, -3/3, both square roots of 9, and 0.

Likewise, the following numbers are not integers: 5/10, the square root of -9, 8.75, and pi.

Some subsets of the integers are often used. They have their own symbols:

set name symbol
..., -2, -1, 0, 1, 2, ... integers
1, 2, 3, 4, ... positive integers
0, 1, 2, 3, 4, ... nonnegative integers
0, -1, -2, -3, -4, ... nonpositive integers
-1, -2, -3, -4, ... negative integers

See also