8 1. From the Real Numbers to the Complex Numbers

this case that the set P of positive rational numbers is closed under addition and

multiplication.

Once the set P of positive elements in a ﬁeld has been speciﬁed, it is easier to

work with inequalities than with P. We write x y if and only if x - y ∈ P. We

also use the symbols x ≥ y, x ≤ y, x y as usual. The order axioms then can be

written as follows:

1) If x 0 and y 0, then x + y 0 and xy 0.

2) Given x ∈ F, one and only one of three things holds: x = 0, x 0, x 0.

Henceforth we will use inequalities throughout; we mention that these inequal-

ities will compare real numbers. The complex numbers cannot be made into an

ordered ﬁeld. The following lemma about ordered ﬁelds does play an important

role in our development of the complex number ﬁeld C.

Lemma 3.1. Let F be an ordered ﬁeld. For each x ∈ F, we have

x2

= x * x ≥ 0.

If x 6 = 0, then x2 0. In particular, 1 0.

Proof. If x = 0, then x2 = 0 by Proposition 2.1, and the conclusion holdis. If

x 0, then x2 0 by axiom 1) for an ordered ﬁeld. If x 0, then -x 0, and

hence (-x)2 0. By 3) and 4) of Proposition 2.1 we get

(8)

x2

=

(-1)(-1)x2

= (-x)(-x) =

(-x)2

0.

Thus, if x 6 = 0, then

x2

0.

By deﬁnition (see Section 3.1), the real number system R is an ordered ﬁeld.

The following simple corollary motivates the introduction of the complex number

ﬁeld C.

Corollary 3.1. There is no real number x such that x2 = -1.

3.1. The completeness axiom for the real numbers. In order to ﬁnally deﬁne

the real number system R, we require the notion of completeness. This notion is

considerably more advanced than our discussion has been so far. The ﬁeld axioms

allow for algebraic laws, the order axioms allow for inequalities, and the complete-

ness axiom allows for a good theory of limits. To introduce this axiom, we recall

some basic notions from elementary real analysis. Let F be an ordered ﬁeld. Let

S ⊂ F be a subset. The set S is called bounded if there are elements m and M in

F such that

m ≤ x ≤ M

for all x in S. The set S is called bounded above if there is an element M in F such

that x ≤ M for all x in S, and it is called bounded below if there is an element m

in F such that m ≤ x for all x in S. When these numbers exist, M is called an

upper bound for S and m is called a lower bound for S. Thus S is bounded if and

only if it is both bounded above and bounded below. The numbers m and M are

not generally in S. For example, the set of negative rational numbers is bounded

above, but the only upper bounds are 0 or positive numbers.

We can now introduce the completeness axiom for the real numbers. The

fundamental notion is that of least upper bound. If M is an upper bound for S,

then any number larger than M is of course also an upper bound. The term least