9.4 PROPERTIES OF THE COMPLEX NUMBER FIELD 319*6. Prove that, in a complete ordered field F, any nonempty subset S of F that is
bounded below in F has a greatest lower bound in F.
- Complete the solution to Example 3 by proving:
(a) If u is a real number such that u2 < 3, then (u + [(3 - ~~)/7])~ < 3.
(b) If u is a real number with u2 > 3, then:
(i) (u2 + 3)/2u < u, and
(ii) [(u2 + 3)/2uI2 > 3. Conclude that (u2 + 3)/2u is an upper bound in Q
for the set S of Example 3. - Suppose a subset I of R is an interval. Prove:
(a) If I is bounded above, but not below, and if b = lub I 4 I, then I = (- oo, b).
(6) If I is bounded, then I has one of the forms (a, b), [a, b), (a, b], or [a, b], for
some a, b E R.
(c) If I is bounded below, but not above, then I has one of the forms [a, a) or (a, oo).
(d) If I is bounded neither above nor below, then I = (-a, a). - Use the intermediate value theorem to prove that if y is a positive real number
and n E N, then there exists a positive real number x such that x" = y. - Verify the lemma following the statement of Theorem 3 for the cases:
(a) x, is the left endpoint of the interval I. [Note: The continuity off on an
interval having x, as its left endpoint means that the limit, as x approaches xo
from the right, of f(x), equals f(x,). In other words, to any E > 0, there corre-
sponds 6 > 0 such that f(xo) - E < f(x) < f (x,) + E whenever x E [x,, xo + a)].
(b) x, is the right endpoint of the interval I.
9.4 Properties of the Complex Number Field
We conclude this chapter with a brief introduction to properties of the field
C of complex numbers. This important mathematical structure is often
cited as an example of an "invented number system." Such a description
is historically accurate. We can trace the origin of complex numbers to
distinguished "inventors," the German mathematician Karl Gauss (1777-
1855) and the Irish mathematician and physicist William Hamilton (1 805-
1865). It is accurate in another sense as well, if we interpret the word "in-
vented" to carry a meaning such as "contrived" or "concocted," as opposed
to "occurring naturally." One way to view the complex numbers is as a
product of the resourcefulness of man, a creation designed to solve a prob-
lem that, in the traditional context, has no solution. The problem we refer
to is the quadratic equation x2 + 1 = 0, and the "traditional context" is
the real number system. As you undoubtedly know, the complex numbers
+i are solutions to this equation, whereas no real number satisfies it.
Indeed, complex numbers enable us to solve easily any quadratic equation
with real coefficients, and, in fact, enable us, in principle, to find a solu-
tion to any polynomial equation with complex coefficients (see Theorem 7).