42 Chapter^1
( e) Again assuming that z 2 and z 3 are fixed, find the locus of z1 when
). = eit ( t real).
[Gae. Mat., 20 (1968), 192]- Find the locus of the points z such that z, i, and iz are collinear.
1.11 POWERS OF COMPLEX NUMBERSIn Section 1.1 we defined the power zn, n being any integer. For
convenience, it will now be formally stated as follows:De:finiti.on 1.4
- z^0 = 1 (z =/= 0)
- z^1 = z
- Zn = zn-l · z ( n > 1)
4~ z-n = l/zn (z =/= 0)
If zP = w, z is called the base, p the exponent, and w is the power.
From Definition 1.4 the laws of exponents follow easily as in elementary
algebra.Theorem 1.9 If p and q are any two integers, we have:
1. zP · zq = zP+q
- (zP)q = zPq
- zP / zq = zP-q
- (zw)P = zPwP
5. (z/w)P = zP /wP
where the bases are assumed to be different from zero if the corresponding
exponents are 'zero or negative.
Powers of the imaginary unit i have four values, as shown by the formulas
·4k 1
i = '
i4k+l -- ; .,where k is any integer.i ·4k+2 -- -^1 ' i4k+a = -i (1.11-1)
The binomial theorem may be applied to obtain the power (a + bir
since that theorem is based on the distributive and associative laws of
multiplication, which are also valid for complex numbers.
Example
(2 + 3i)^3 = 23 + 3(2)^2 (3i) + 3(2)(3i)^2 + (3i)^3
= -46 + 9i
Given n > 1 nonzero complex numbers in exponential form,
Z 2 -_ .. '2 ei1J2 ' ... '