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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).

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]].

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]]

See also:  

 

*[[algebraic numbers]]

*[[abstract algebra]]

 

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.

[[category:mathematics]]

[[category:mathematics]]