Integer

The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. The set of all integers is usually denoted in mathematics by Z (or Z in blackboard bold, mathbb{Z}), which stands for Zahlen (German for "numbers"). They are also known as the whole numbers, although that term is also used to refer only to the positive integers (with or without zero). Like the natural numbers, the integers form a countably infinite set.

Order-theoretic properties

Z is a totally ordered set without upper or lower bound. The ordering of Z is given by

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: ... < −2 < −1 < 0 < 1 < 2 < ...

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An integer is positive if it is greater than zero and negative if it is less than zero. Zero is defined as neither negative nor positive.

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The ordering of integers is compatible with the algebraic operations in the following way:

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• if a < b and c < d, then a + c < b + d
• if a < b and 0 < c, then ac < bc. (From this fact, one can show that if c < 0, then ac > bc.)