Bridge to Abstract Mathematics: Mathematical Proof and Structures

322 PROPERTIES OF NUMBER SYSTEMS Chapter 9

Because C satisfies the multiplicative inverse axiom, it is possible to de-
fine division by a nonzero complex number, namely, if z, and z2 E C with
z2 # 0, we define zl/z2 to be z,z;'. For example, if z, = 2 - 3i and z2 =
5 + i, then zl/z2 = (2 - 3i)/(5 + i) = (2 - 3i)[(5 - i)/26] = (7 - 17i)/26 =
& - (3)i. As a further example, if z, = 1 and z2 = i, then zl/z2 = l/i =
(l)(i) = (1-1 = i. Note that i has the unusual property that its
multiplicative and additive inverses are identical. Verification of various
properties involving division of complex numbers is part of Exercise 7(b).

COMPLEX CONJUGATE AND MODULUS

The formula z- = (x - yi)/(x2 + y2), for the multiplicative inverse of a
nonzero complex number z = x + yi, contains two important quantities
related to z. If z = x + yi is represented by one arrow in Figure 9.4(a),
the quantity x - yi is represented by the other. Note the symmetry with
respect to the y (or imaginary) axis. The real number (x2 + y2) represents
the square of the common length of both arrows [see Theorem 2(f), and
Figure 9.4(b)]. These two quantities, which may be calculated for any com-
plex number z, are of sufficient importance to warrant formal designation.

DEFINITION 3

Let z = x + yi be a complex number. We define:

(a) The complex conjugate of z, denoted z*, by the rule z* = x - yi

(b) The modulus of z, denoted Izl, by the rule Izl = (x2 + y2)lI2

Figure 9.4 Complex conjugate and modulus of a complex number z.

y (imaginary axis)

y (imaginary axis)