There is a number 0, called zero, such that the sum of any number a and zero is a.

Corresponding to each number a, there is a number (-a), called the opposite of a, such that the sum of a and (-a) is zero.

Also, for any two numbers a, b there is a unique number x such that a + x = b, namely, x = b + (-a), more commonly written x = b - a.

A set with an addition law satisfying the above properties, and such that all sums lie in the set, is called an additive group.

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