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ORDERED FIELDS The set R of real numbers can be described as a “complete ordered field.” In this section we present the axioms of an ordered field and in the next section we give the completeness axiom. The purpose of this development is to identify the basic properties that characterize the real numbers. After stating the axioms of an ordered field, we derive some of the basic algebraic properties that the reader no doubt has used for years without question. It is not our intent to derive all these properties, but simply to illustrate how this might be done by giving a few examples. Other properties are left for the reader to prove as exercises. Having done this, we shall subsequently assume familiarity with all the basic algebraic properties (whether we have proved them specifically or not). We begin by assuming the existence of a set R, called the set of real numbers, and two operations“+” and “.”, called addition and multiplication, such that the following properties apply: A1 For all x, y € R, x+y e R and if x = w and y = z, then x + y = w + z. A2. For all x, ye R, x+y = y + x. SO A3 For all x, y, ze R, X+ (y + z) = (x + y) + z. A4. There is a unique real number 0 such that x + 0 = x, for all x e R. AS, For each x € R there is a unique real number -x such that x + (-x) = 0.
The Real Numbers MI For all x, ye Rrye Rand if x w and y, then xy=w M2 For all X, YER X-y-yor. M3. For all x, y, Te Rr (2) - (x+y).I. M4 There is a unique real number 1 such that I 0 and r1= x for all TER M5. For each re R with #0there is a unique real number 1/x such that r. (1/x) = 1 We also writer in place of 1/x DL For all x,y,ze R. x.ly+2) = y + x3 or These first 11 axioms are called the field axioms because they describe a syster known as a field in the study of abstract algebra Axioms A2 and M2 are called the commutative laws and axioms A3 and M3 are the associative laws Axiom DL is the distributive law that shows how addition and multiplication relate to each other. Because of axioms Al and MI, we can think of addition and multiplication functions that map R XR into R When writing multiplication we often omit the raised dot and write xy instead of ry The other basic operations are defined as follows: The difference x - y is defined to be the sum x + (-) The quotient is defined to be the product And we may let 2 represent 1 + 1 atid xrepresent x - x, etc. 2.1 PRACTICE Show that the set of irrational numbers with addition and multiplication is not a field. In particular, show that axioms Al and MI do not apply. In addition to the field axioms, the real numbers also satisfy four order axioms. These axioms identify the properties of the relation "<" As is common practice, we may write y > instead of x <y, and Sy is an abbreviation for " <yor xy" The notation ">is defined analogously A real number is called nonnegative if x 20 and positive if x > 0. A pair of simultaneous inequalities such as " <y and y <:" is often written in the shorter form "x<y. The relation"<"satisfies the following properties 01. For all x, y e R. exactly one of the relations - y, x >y, or <y holds (trichotomy law). 02. For all x,y,z € R, if x < y and y<s, then x<: 03. For all x,y,ze Rifr<y, then x + : <y+z. 04. For all x, y, ze R, if x <y and :> 0, then xz <y: The following theorem illustrates how the axioms may be used to derive some familiar algebraic properties.