ABSTRACT
For the details of this “bottom up” approach, the interested reader is referred to [7] or [10]. We shall instead take a “top down” approach, describing the real number system axiomatically.
1.2 Algebraic Properties of R In this section we list the axioms that govern the use of addition (+) and
The operations of addition and multiplication satisfy the following field axioms, where a, b, c denote arbitrary members of R:
• Closure under addition: a+ b ∈ R. • Associative law of addition: (a+ b) + c = a+ (b+ c). • Commutative law of addition: a+ b = b+ a. • Existence of an additive identity: There exists a member 0 of R such
that a+ 0 = a for all a ∈ R. • Existence of additive inverses: For each a ∈ R there exists a member −a
of R such that a+ (−a) = 0. • Closure under multiplication: a · b ∈ R. • Associative law of multiplication: (a · b) · c = a · (b · c). • Commutative law of multiplication: a · b = b · a. • Existence of a multiplicative identity: There exists a real number 1 6= 0
such that a · 1 = a for all a ∈ R. • Existence of multiplicative inverses: For each a 6= 0 there exists a member a−1 of R such that a · a−1 = 1.
