The Chemistry Maths Book, Second Edition

8.3 Graphical representation 233

(ii) θ

1

1 − 1 θ

2

1 = 1 − 13 π 212 and, by equation (8.22),

(iii)z

2

2 z

1

1 = 1 (z

1

2 z

2

)

− 1

and, by equation (8.24),

0 Exercises 23, 24

de Moivre’s formula

It follows from the relations (8.20) that the product of three or more complex

numbers has modulus equal to the product of the moduli of the numbers and has

argument equal to the sum of the arguments. For example, for the product of three

numbers,

|z

1

z

2

z

3

| 1 = 1 |z

1

z

2

| 1 × 1 |z

3

| 1 = 1 |z

1

| 1 × 1 |z

2

| 1 × 1 |z

3

|

arg 1 (z

1

z

2

z

3

) 1 = 1 arg 1 (z

1

z

2

) 1 + 1 arg 1 (z

3

) 1 = 1 arg 1 (z

1

) 1 + 1 arg 1 (z

2

) 1 + 1 arg 1 (z

3

)

Therefore

z

1

z

2

z

3

1 = 1 r

1

r

2

r

3

[cos(θ

1

1 + 1 θ

2

1 + 1 θ

3

) 1 + 1 i 1 sin(θ

1

1 + 1 θ

2

1 + 1 θ

3

)]

and, in general,

z

1

z

2

-1z

n

1 = 1 r

1

r

2

-1r

n

[cos(θ

1

1 + 1 θ

2

1 +1-1+ 1 θ

n

) 1 + 1 i 1 sin(θ

1

1 + 1 θ

2

1 +1-1+ 1 θ

n

)] (8.25)

In the special case, when all the numbers are equal toz 1 = 1 r(cos 1 θ 1 + 1 i 1 sin 1 θ),

z

n

1 = 1 r

n

(cos 1 nθ 1 + 1 i 1 sin 1 nθ) (8.26)

For a number on the unit circle in the complex plane(r 1 = 1 1),z 1 = 1 cos 1 θ 1 + 1 i 1 sin 1 θand

(cos 1 θ 1 + 1 i 1 sin 1 θ)

n

1 = 1 cos 1 nθ 1 + 1 i 1 sin 1 nθ (8.27)

z

z

i

2

1

1

2

13

12

13

12

=+

















cos sin

ππ

=−

















2

13

12

13

12

cos sin

ππ

i

=−













+−





























2

13

12

13

12

cos sin

ππ

i

z

z

r

r

i

1

2

1

2

12 12

=−+−













cos(θθ) sin(θθ)

rr

12

=, 2