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
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