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.
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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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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 fractional (divided into smaller parts), 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 fractional (divided into smaller parts), and various other classifications, such as [[prime]] (only divisible by itself and one).

Revision as of 17:24, 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 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.