16 Chapter 1 • The Real Number System- Prove the second claim of Theorem 1.2.10 (c).
- Prove Theorem 1.2.10 (d).
- Prove Theorem 1.2.11 (b). [Hint: for x E P, proceed as in the proof of
x
Part (a), using
2
in place of x + l.] - For the field F ={a+ b.J2: a, b E Q} defined in Exercise 1.1-A.8, let
P' = {a+ b.J2: a > b.J2}, where the ">" symbol here refers to the
ordinary ">" relation in R Prove that F is an ordered field with respect
to P'. - Give an example of a field F that can be ordered with respect to two
different subsets, P and P'. - Prove that the field C of complex numbers^3 cannot be ordered; that is ,
it is impossible to find a subset P of C satisfying the axioms (01)- (03).
[Hint: Find one of the properties established in the theorems above that
cannot hold in C regardless of the choice of subset P ~ C.] - Prove that the field defined in Exercise 1.1-A.9 cannot be ordered.
ABSOLUTE VALUE
Definition 1.2.12 Suppose F is an ordered field. For each x E F, we define
the absolute value of x to belxl = { x _if x ~ 0
- x if x < 0.
Theorem 1.2.13 (Basic Properties of Absolute Value) Let F be an or-
dered field. Then, "Ix, y E F,(a) lxl ~ O;(b) I - xi= lxl;
(d) -lxl :::; x:::; lxl;
(d) lx-yl=ly-xl;
(e) lxyl = lxllYI·
- See Exercise l.l-A.11