320 PROPERTIES OF NUMBER SYSTEMS Chapter 9
We interject at this point only one discordant note into the rather "tidy"
view, just presented, of the complex numbers as "cooked up," in contrast
to the "more natural" real numbers. We will see in Chapter 10 that the
various numbers systems studied throughout this chapter, including 2, Q,
and R, can all be regarded as invented, in exactly the same mathematical
sense as described earlier in reference to C. Once this fact is accepted,
it becomes possible to regard complex numbers to be just as "real" as
numbers in the system we designate by that name.
BASIC DEFINITIONS AND PROPERTIES
DEFINITION 1
The complex number system (C, + , a) consists of a set C, comprising all ordered
pairs of real numbers (x, y), together with two operations, + and ., where we
specify:(a) Two complex numbers (x,, y,), and (x2, y,) areequal if and onlyif x, = x2and
Y1 = Y2.
(6) Thesum of two complex numbers (x,, y,) and (x,, y,) is the complex number
(x, + x2, Y, + ~2).
(c) The product of two complex numbers (x,, y,) and (x2, y,) is the complex
number (~1x2 - YlY2, XlY2 + ~1~2).Complex numbers are most often denoted by the letters z and w, with
subscripts as necessary. Furthermore, the notation x + yi is customarily
used in place of the ordered pair notation (x, y). Thus - 2 + 3i corresponds
to (-2,3), 5 = 5 + Oi corresponds to (5, O), and i = 0 + li corresponds to
(0, 1). According to (c) of Definition 1, the product of (0, 1) with (0, 1) equals
((O)(O) - (l)(l), (O)(l) + (l)(O)) = (- 1 0) In the alternative notation this
translates to the equation i2 = - 1. This rule, in turn, can be used along
with ordinary high school algebra to provide a practical method of multi-
plying complex numbers.
EXAMPLE 1 Calculate the product z1z2, where z1 = 3 - 7i and 2, =
-4 + 6i.
Solution The product z,z2 = (3 - 7i)(-4 + 6i) equals (3)(-4) +
(-7)(i)(6)(i) + (-7)(i)('--4) + (3)(6)(i) = (- 12) + (-42i2) + (28i) +
(1%) = (-12 + 42) + (28 + 18)i = 30 + 46i. 0The equation i2 = - 1 can be rephrased i = g. We can calculate
easily, using the methods of Example 1, that (ri)2 = r2i2 = - r2 SO that we
have the equation ri = 47 for any real number r. This equation provides
us with the interpretation "complex roots" (no doubt familiar to you) of a
negative value of the quadratic discriminant b2 - 4ac, in applying the
quadratic formula. The following example illustrates this.