The Chemistry Maths Book, Second Edition

232 Chapter 8Complex numbers

If the two numbers are a complex conjugate pair,z 1 = 1 x 1 + 1 iyandz* 1 = 1 x 1 − 1 iy, then

arg 1 z 1 = 1 −arg 1 z*

and

zz* 1 = 1 |z|

2

1 = 1 x

2

1 + 1 y

2

,arg 1 zz* 1 = 10

In the case of division,

(8.21)

and

(8.22)

For the inverse of a complex number, it follows from (8.22) that

(8.23)

and, becausecos(−θ) 1 = 1 cos 1 θandsin(−θ) 1 = 1 −sin 1 θ,

(8.24)

EXAMPLES 8.6Express each ofz

1

z

2

,z

1

2 z

2

andz

2

2 z

1

as a single complex number

for

(see Examples 8.5)

We have r

2

1 = 11 ,θ

1

1 = 1 π 2 4,andθ

2

1 = 14 π 23. Therefore

(i) θ

1

1 + 1 θ

2

1 = 119 π 212 and, by equation (8.19),

zz rr i

1212 12 12

2

1

=+++













cos(θθ) sin(θθ) = cos

99

12

19

12

ππ

• 
  

















isin

rr

12

=, 2

r

1

=, 2

zizi

12

2

44

4

3

4

3

=+













cos sin ,=+cos sin

ππ π π

11

zr

=−(cosθθisin )

11

zr

=−+−i













cos( ) sin( )θθ

z

z

r

r

i

1

2

1

2

12 12

=−+−













cos(θθ) sin(θθ)

z

z

z

z

z

z

zz

1

2

1

2

1

2

12

=,

















arg =−arg() ()arg

|||*|zz xy== +,

22