The Chemistry Maths Book, Second Edition

230 Chapter 8Complex numbers

EXAMPLES 8.5Expressz 1 = 1 x 1 + 1 iyin polar form.

(i) z 1 = 111 + 1 i

We havex 1 = 11 andy 1 = 11 so that

The principal value oftan

− 1

(1) isπ 24 , andarg 1 z 1 = 1 π 24 because the point lies in

the first quadrant. Therefore

(ii)

We havex 1 = 1 − 122 and so that

The principal value of tan is π 23 and, because x 1 < 10 , it follows that

arg 1 z 1 = 1 tan. Therefore

0 Exercises 16–22

Representation of arithmetic operations

Addition and subtraction

In Figure 8.4, the numbersz

1

1 = 1 x

1

1 + 1 iy

1

andz

2

1 = 1 x

2

1 + 1 iy

2

are represented by points P

and Q, respectively. The representation of the sum

z

1

1 + 1 z

2

1 = 1 (x

1

1 + 1 x

2

) 1 + 1 i(y

1

1 + 1 y

2

)

zi=+cos sin

4

3

4

3

ππ

−

()

+=

1

343 ππ

−

()

1

3

tan tan ( )

−−













=

11

3

y

x

r=













• 
  

















=

2

2

1

2

3

2

1

y=− 32

zi=− −

1

2

3

2

zi=+













2

44

cos sin

ππ

rz x y

y

x

== +=,













=

−−

|| tan tan()

22 1 1

21

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−

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√

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Figure 8.2

Figure 8.3