Bridge to Abstract Mathematics: Mathematical Proof and Structures

326 PROPERTIES OF NUMBER SYSTEMS Chapter 9

(b) The proof proceeds by induction on n. The case n = 1 is evident.

If the conclusion is true for m, then zm+' = zm. z = (rmeid)(reie) =

rm + e i(m + l)', as required. 0

The special case r = 1 of Theorem 5(b) is known as deMoivre's theorem.
It has an important application; using it we can calculate the n complex
nth roots of any given complex number.

THEOREM 6

If z = rei6, then the n complex numbers r11nei[(6 + 2k")1n1, k = 0, 1,... , n - I, are

the n distinct complex nth roots of z.

such w is an nth root of z. The n values of w are clearly distinct, for if

ei[(e+2kn)ln] = ei[(e+2hr)/n], then (8 + 2kn)ln = (8 + 2hn)ln + 2mn, for some

integer m. Hence 8 + 2kn = 8 + 2hn + 2mnn, so that k - h = mn. But

k - h cannot be an integral multiple of n since both are between 0 and

n - 1, inclusive. Finally, all nth roots of z have this form. For if w =

seiX satisfies wn = z = reie, then wn = sneinx = rei0. Hence s = r'ln and

einx - - eie so that nx = 8 + 2kn and x = (8 + 2kn)ln. There are only n

such roots because if m E N, we can express m, using the division algo-

rithm, in the form m = nq + r, where q, r E N and 0 I r < n. Then

ei[(e + 2mWnI = ei[(e + 2(nq + r)n)lnl = ei[(B/n) + (2nqrrIn) + (2rxln)l = ei[(B + 2rx)/n], where

r is one of the integers 0, 1,2,... , n - 1. Cl

Theorem 6 is particularly useful, and relatively easy to apply, for com-
puting nth roots of complex numbers that are either real or purely imagi-
nary. In particular, if z is real, we may take arg z = 0 if z > 0 and arg z = n
if z < 0.

EXAMPLE 5 Find the four 4th roots of z = 16.

Solution Express z in polar form with r = 16 and 8 = 0. Then r1I4 = 2 and

the four 4th roots have the form 2eir(e+2k")141, k = 0, 1, 2, 3, that is, 2e0,

2ei("12) , 2ei* , and 2ei(3"/2), or in simplified form, 2, 2i, -2, and -2i.

Note from Figure 9.5 that the roots are equally spaced about a circle

of radius 2 in the complex plane.

We conclude this article by stating formally a property of C alluded to

earlier. The proof requires results from the area of complex analysis and

is encountered in any introductory complex variables course.

T H E 0 R E M 7 (Fundamental Theorem of Algebra)

Any polynomial c, + c,z + c2z2 +... + cnf (c, E C, cn # 0) of degree n with

complex coefficients has at least one zero (i.e., root) in C.