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The sum of any two real numbers a and b is again a real number, denoted a + b. The real numbers are closed under the operations of addition, subtraction, multiplication, division, and the extraction of roots.
No matter how terms are grouped in carrying out additions, the sum will always be the same: (a + b) + c = a + (b + c). This is called the associative law of addition.
Given any real number a, there is a real number zero (0) called the additive identity, such that a + 0 = 0 + a = a.
Given any real number a there is a number (-a), called the additive inverse of a, such that (a) + (-a) = 0.
No matter in what order addition is carried out, the sum will always be the same: a + b = b + a. This is called the commutative law of addition.
Any set of numbers obeying laws A-1 through A-4 is said to form a group. If the set also obeys A-5 it is said to be a commutative Abelian group.