TABLE OF CONTENTS In this chapter we introduce real numbers on an axiomatic basis, solve inequalities, introduce the absolute value and discuss the least upper bound axiom. In the concluding section we outline an alternative development of the real number system.
Real numbers satisfy three groups of axioms field axioms, order axioms and the least upper bound axiom. We discuss each group separately.
A set F together with two functions ( x, y) 
x+ y, (x, y) xy from F F into F is called a field if the axioms in Table 4.1 are satisfied for all x, y, z in F.
| A 1 | : x+ y= y+ x | M 1 | : xy= yx |
| A 2 | : x+( y+ z)=( x+ y)+ z | M 2 | : x( yz)=( xy) z |
| A 3 | : There is an element 0 ? F such that 0+ x= x for all x in F | M 3 | : There is an element 1 ? F, 1 ?0, such that 1 x= x for all x in F; |
| A 4 | : For every element x ? F there exists an element ( ? x) ? F such that ( ?x)+ x=0. | M 4 | : For every element x ? F, x ?0, there exists... |
