19 1.4 Ordered Field Axioms In this section and the next we present an axiomatic description of the set of real numbers. That is, we will assume that there exists a set, denoted byR, satisfying the ordered field axioms, stated below, together with the completeness axiom, presented in the next section. In this way we identify the basic properties that characterize the real numbers. After listing the ordered field axioms we derive from them additional familiar properties of the real numbers. We conclude the section with the definition of absolute value of a real number and with several results about it that will be used often later in the text. We assume the existence of a set R(the set of real numbers) and two operations +and· (addition and multiplication) assigning to each pair of real numbers x,y, unique real numbers x+y andx· y and satisfying the following properties: (A1) (x+y)+z =x+(y+z) for all x,y,z ∈R. (A2) x+y =y+x for all x,y ∈R. (A3) There exists a unique element 0∈Rsuch that x+0=x for all x ∈R. (A4) For eachx ∈R, there exists a unique element −x ∈Rsuch that x+(−x) =0. (M1) (x· y)· z =x· (y· z) for all x,y,z ∈R. (M2) x· y =y· x for all x,y ∈R. (M3) There exists a unique element 1∈Rsuch that 1̸=0 and x· 1=x for all x ∈R. (M4) For each x ∈R\ {0}, there exists a unique element x−1 ∈Rsuch that x· (x−1) =1. (We also write 1/x instead of x−1.) (D1) x· (y+z) =x· y+x· z for all x,y,z ∈R. We often write xy instead of x· y. In addition to the algebraic axioms above, there is a relation <on Rthat satisfies the order axioms below: (O1) For all x,y ∈R, exactly one of the three relations holds: x =y, y <x, or x <y. (O2) For all x,y,z ∈R, if x <y andy <z, then x <z. (O3) For all x,y,z ∈R, if x <y, then x+z <y+z. (O4) For all x,y,z ∈R, if x <y and 0<z, then xz <yz. We will use the notation x ≤y to mean x <y or x =y. We may also use the notation x >y to represent y <x and the notationx ≥y to mean x >y or x =y. A set Ftogether with two operations +and · and a relation<satifying the 13 axioms above is called anordered field. Thus, the real numbers are an example of an ordered field. Another example of an ordered field is the set of rational numbers Qwith the familiar operations and order. The integers Zdo not form a field since for an integer mother than 1 or −1, its reciprocal 1/mis not an integer and, thus, axiom (M4) above does not hold. In particular, the set of positive integers Ndoes not form a field either. As mentioned above the real numbers Rwill be defined as the ordered field which satisfies one additional property described in the next section: the completeness axiom. From these axioms, many familiar properties of Rcan be derived. Some examples are given in the next proposition. The proof illustrates how the given axioms are used at each step of the derivation.
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