Laws of Addition

A1. 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; this means that applying any of these operations to real numbers yields a quantity that also is a real number.

A2. 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.

A3. Given any real number a, there is a real number zero (0) called the additive identity, such that a + 0 = 0 + a = a.

A4. Given any real number a, there is a number (-a), called the additive inverse of a, such that (a) + (-a) = 0.

A5. 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 A1 to A4 is said to form a group. If the set also obeys A5, it is said to be an Abelian, or commutative, group.