Difference between revisions of "Integer"

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(with the basics having been so poorly defined, there would be no reason to mix in stuff which is way over the head of most readers)
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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.
 
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.
  
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 [[composite]] (divisible into other integers), and various other classifications, such as [[prime]] (only divisible by itself and one).
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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 composite (divisible into other integers), and various other classifications, such as [[prime]] (only divisible by itself and one).
  
 
Every integer larger than 1 has a unique [[prime factorization]].
 
Every integer larger than 1 has a unique [[prime factorization]].
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Likewise, the following numbers are not integers: 5/10, the square root of -9, 8.75, and [[pi]].  
 
Likewise, the following numbers are not integers: 5/10, the square root of -9, 8.75, and [[pi]].  
  
See also: [[algebraic numbers]]
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See also:  
 
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*[[algebraic numbers]]
==Generalizations==
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*[[abstract algebra]]
 
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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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[[category:mathematics]]
 
[[category:mathematics]]

Revision as of 17:48, November 19, 2008

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".

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.

An integer may be even (divisible by two) or odd (not divisible by two), positive (more than zero) or negative (less than zero), whole (undivided) or composite (divisible into other integers), and various other classifications, such as prime (only divisible by itself and one).

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.

See also: