8.2 Algebra of complex numbers 227
EXAMPLES 8.2Multiplication
(i)(2 1 + 13 i)(3 1 + 14 i) 1 = 1 2(3 1 + 14 i) 1 + 13 i(3 1 + 14 i) 1 = 161 + 18 i 1 + 19 i 1 + 112 i
21 = 1 − 61 + 117 i
(ii)i(2 1 − 13 i) 1 = 12 i 1 − 13 i
21 = 131 + 12 i
0 Exercises 4–7
The complex conjugate
Ifz 1 = 1 x 1 + 1 iyis an arbitrary complex number then the number obtained from it by
replacing iby−iis
z* 1 = 1 x 1 − 1 iy (8.9)
and is called the complex conjugateof z(sometimesGis used instead ofz*). zis then
also the complex conjugate ofz. The conjugate pair of complex numbers zandz
has the following properties:
(i) (8.10)
(ii) (8.11)
(iii)zz* 1 = 1 (x 1 + 1 iy)(x 1 − 1 iy) 1 = 1 x
21 + 1 y
2(real and positive) (8.12)
EXAMPLES 8.3Conjugate pairs of complex numbers
(i) Ifz 1 = 121 + 13 ithenz* 1 = 121 − 13 iand
(ii) Ifz 1 = 111 − 1 ithenz* 1 = 111 + 1 iand
(iii) Solve the quadratic equationz
21 − 13 z 1 + 141 = 10 (see Example 2.18).
zi=
±−
=±−
=±
3916
2
1
2
37
1
2
37
1
2
1
1
2
()zz+ =, ()zz i zz− =−, *=+= 112
1
2
2
1
2
32313
22() ()zz+=, zz i zz−=, *=+=
1
2
1
2
(*) ( )( )z z−= +−−x iy x iy iy i z()
==Im
1
2
1
2
(*) ( )( )zz+= ++−xiy xiy x z()
==Re