Bridge to Abstract Mathematics: Mathematical Proof and Structures

324 PROPERTIES OF NUMBER SYSTEMS Chapter 9

Partial proof (e) If z2 = x + yi # 0, then z;' = (x - yi)/(x2 + y2), so

that (z, ')* = (x + yi)/(x2 + y2). On the other hand, z2 = x - yi, so that

(2:)-' = (x + yi)/(x2 + y2), as well. Hence we have (zT1)* = (x + yi)/

(x2 + y2) = (~2)- ', as desired.

(f) If z2 # 0, then (zl/z2)* = (z, z;')* = [by (d)] (z:)(zil)* =

[by (e)l (z:)(z:) - ' = z:/z:

(h) If z, = x + yi # 0, then z;' = (x - yi)/(x2 + y2), so that P;'l =

{[x2/(x2 + y2)2] + [y2/(x2 + y2)2])112 = [(x2 + y2)/(x2 + y2) Ill2 =

1/(x2 + y2)lI2 = 1/1z21.

The remaining portions of the proof are left to you in Exercise 9.^0

POLAR FORM AND DEMOIVRE'S THEOREM

Let r and 8 be polar coordinates of the ordered pair (x, y), where x # 0,

y # 0, and r > 0. Then we may represent the complex number x + yi in the

form r(cos 8 + i sin 8). This representation is called a polar form of x + yi.

Note that r is uniquely determined by x and y, and the stipulation that r

be positive. Specifically, if z = r(cos 8 + i sin 8), then lzl = (r2 cos2 8 +

r2 sin2 8)'12 = [r2(cos2 8 + sin2 8)]'12 = = r; that is, r is simply the

modulus of z. Unfortunately, the situation is not so simple for 8, as the

following example indicates.

EXAMPLE 4 Describe all possible polar representations of z = 2 + 2ai.

Solution Note first that r = IzI = [(2)' + (2fi)']'12 = fi = 4. Thus we

may express z as z = 4(4 + i&2) = 4(cos n/3 + i sin rr/3). Hence r = 4,

8 = 7113 provides one polar representation of z. The periodicity of cos

and sin, both of period 2n, dictates that r = 4 and 8, = 43 + 2kn, where

k is any integer, provide infinitely many additional (and, in fact, all possi-

ble) polar representations of z. 0

Given a nonzero complex number z = x + yi, any number 8 such that

z = r cos 8 + ir sin 8 is called an argument of z, denoted arg z. Any nonzero

complex number has infinitely many values of arg z. The unique value of

arg z such that - n < arg z I n is called the principal value of arg z. Hence

the principal value of arg z in Example 4 is n/3; some other values are

• 543, 743, and 131r/3.