30 Chapter 1 • The Real Number System
In this sense, every ordered field contains the familiar natural numbers,
integers, and rational numbers.
EXERCISE SET 1.4In each of the following, assume that F is an ordered field. Prove the stated
property.
- Z satisfies all the properties of an ordered field except (M4). [You may
assume (AO) and (MO); see Exercise 13 below.] - Ql is the "smallest" ordered field, in the sense that every ordered field F
contains Ql as a subset. - Let n E Z. If n^2 is divisible by 2, then n is also divisible by 2. [Hint: any
integer is either even (of the form 2k, where k E Z) or odd (of the form
2k + 1, where k E Z).] - There is no element of Ql whose square is 3. [Hint: Use suitable modifi-
cations of Exercise 3 and Theorem 1.4.5.]
In Exercises 5- 10 , assume that F is an ordered field with at least one
irrational element. - If x is rational and y is irrational, then x + y is irrational.
- If x =I- 0 is rational and y is irrational, then xy is irrational.
- If x is irrational, then so are -x and x-^1.
- The set of irrationals is not closed under addition.
- The set of irrationals is not closed under multiplication.
- F contains infinitely many irrational elements.
- The Principle of Mathematical Induction, and its alternative forms, can
use any integer no as the "starting point." (Theorem 1.3.11 establishes
this for natural numbers only.) - Prove that Z satisfies axioms (AO) and (MO).
- (Project) Integer exponents: In Definition 1.3.12 we used mathemat-
ical induction to define an, \:/a in an ordered field F and all n E N. For
a =I- 0 we define a^0 = 1 and a-n = l/an. Prove that with this definition
the familiar "laws of exponents" hold: if a, b are nonzero elements of an
ordered field F, then \:/m, n E Z,
(a) aman = am+n (b) (amt= amn
(c) am /an= am-n (d) (ab)n = anbn
[See Exercises 1.3.20- 1.3.22.]