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Integers include natural numbers, their negatives, and zero, represented as {..., -3, -2, -1, 0, +1, +2, +3,...}. They are used in various applications, such as accounting and measuring quantities below zero, and can be added, subtracted, multiplied, and divided with specific rules for signs and absolute values. The historical acceptance of integers in mathematics evolved over time, and they are fundamental in operations involving gains and losses.

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