Difference between revisions of "Integer"
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An '''integer''' is any whole number, positive, negative, or 0. 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''' is any whole number, positive, negative, or 0. 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 | + | 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 fractional (divided into smaller parts), and various other classifications, such as [[prime]] (only divisible by itself and one). |
| − | Every integer larger than 1 has a unique [[prime]] | + | Every integer larger than 1 has a unique [[prime factorizarion]]. |
Some examples of integers: 1, 10/5, 98058493, -87, -3/3, the square root of 9, and 0. | Some examples of integers: 1, 10/5, 98058493, -87, -3/3, the square root of 9, and 0. | ||
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==Generalizations== | ==Generalizations== | ||
| − | 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 [[ | + | 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]] | ||
Revision as of 20:26, June 13, 2008
An integer is any whole number, positive, negative, or 0. 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 fractional (divided into smaller parts), and various other classifications, such as prime (only divisible by itself and one).
Every integer larger than 1 has a unique prime factorizarion.
Some examples of integers: 1, 10/5, 98058493, -87, -3/3, the square root 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 rational integer.
Generalizations
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 and multiplicative), and inversion (subtraction). Other objects such as matrixes, polynomials, quaternions, and algebraic integers also form rings, and can therefore be viewed as generalizations of the integers.
