© John Wiley & Sons Australia, Ltd 2009 1

WorkSHEET 2.1 Number systems: complex numbers

Name: ___________________________ 1 Using the imaginary number i, write

expressions for: (a) 81- (b) 5-

(a) 28181 i=-

i9±= (b) 255 i=-

i5±=

2

2 Simplify 2352 7653 +-+- iiii in the form of iyx + .

303

23522)1)(1(3)1)(1(5)1(212))((3)(5)(2)(

2352242442

7653

-=+-=

++---=+---+--=+-+-=

+-+-

iiii

iiiiiiiiiiii

iiii

3

3 Plot the following complex numbers on the same fully labelled Argand diagram. (a) A = 2 + 3i (b) B = –5 – i (c) C = 1 – 4i (d) D = –4 + 2i (e) E = 6i (f) F = –4

6

Maths Quest Maths C Year 11 for Queensland Chapter 2: Number systems: complex numbers WorkSHEET 2.1

© John Wiley & Sons Australia, Ltd 2009 2

4 Perform the following operations using z = 2 – 6i w = –1 + 2i x = 4 + i

(a) 2z (b) w – z (c) 3x + 2w (d) x × w (e) z × w

(a) 2z = 2(2 – 6i) = 4 – 12i

(b) w – z = –1 + 2i – (2 – 6i)

= –1 + 2i – 2 + 6i = –3 + 8i

(c) 3x + 2w = 3(4 + i) + 2(–1 + 2i)

= 12 + 3i – 2 + 4i = 10 + 7i

(d) x × w = (4 + i)(–1 + 2i)

= –4 + 8i – i – 2 = –6 + 7i

(e) z × w = (2 – 6i)(–1 + 2i)

= –2 + 4i + 6i + 12 = 10 + 10i

5

5 Plot the answers from the previous question on the same Argand diagram.

5

Maths Quest Maths C Year 11 for Queensland Chapter 2: Number systems: complex numbers WorkSHEET 2.1

© John Wiley & Sons Australia, Ltd 2009 3

6 Find the modulus of each of the following complex numbers: (a) z = 2 – 5i (b) w = –3 + 6i (c) y = –6 + i (d) x = 4

29

254

)5(2 (a) 22

=

+=

-+=z

53

45

369

6)3( (b) 22

=

=

+=

+-=w

37

136

1)6( (c) 22

=

+=

+-=y

(d) |x| = 4

4

7 Find the values of x and y if ( )( ) iyixi 22332 +=++ .

( )( )

( ) ( ) iiyxyxiyiixiyxiiyxi

2232332223332222332

2

+=++-+=+++

+=++

( )( )

( ) ( )

( )

.5,4 isSolution 5102

22122243

:(2) into4 ngSubstituti45213

:43)4(669:32)3(4664:21

)2(223)1(2332

-==-=-=

=+=+===

+=+´=-´

=+=-

yxyyyyxxx

yxyx

yxyx

4

Maths Quest Maths C Year 11 for Queensland Chapter 2: Number systems: complex numbers WorkSHEET 2.1

© John Wiley & Sons Australia, Ltd 2009 4

8 Multiply each of the following numbers by their complex conjugates. (a) 2 + 5i (b) –1 + 7i (c) –3i

(a) (2 + 5i)(2 – 5i)

= 4 – 25i2 = 4 + 25 = 29

(b) (–1 + 7i)(–1 – 7i)

= 1 – 49i2 = 1 + 49 = 50

(c) (–3i)(3i)

= –9i2 = 9

3

Maths Quest Maths C Year 11 for Queensland Chapter 2: Number systems: complex numbers WorkSHEET 2.1

© John Wiley & Sons Australia, Ltd 2009 5

9 (a) Sketch z = –4 + 3i and .

_z

(b) Perform the following operations for

z = –4 + 3i. Give the real, Re (w), and imaginary, Im (w), parts of the answer.

(i) w = z + _z

(ii) w = z – _z

(iii) w = z _z

(a)

(b)

(i) w = z + _z

w = –4 + 3i + (–4 – 3i) w = – 8 Re (w) = –8, Im (w) = 0

(ii) w = z – _z

w = –4 + 3i – (–4 – 3i) w = –4 + 3i + 4 + 3i w = 6i Re (w) = 0, Im (w) = 6

(iii) w = z _z

w = (–4 + 3i)(–4 – 3i) w = 16 + 12i - 12i – 9i² w = 16 + 9 w = 25 Re (w) = 25, Im (w) = 0

7

Maths Quest Maths C Year 11 for Queensland Chapter 2: Number systems: complex numbers WorkSHEET 2.1

© John Wiley & Sons Australia, Ltd 2009 6

10 Perform the following operations using z = –1 + 4i and w = 2 – i.

(a) zz

(b) zw

(c) wz

17815

178161

1611681

)41()41(

)41()41( (a)

2

i

i

iiii

ii

zz

--=

--=

++-

=

+-+-

´--+-

=

1776

17742

161482

)41()41(

)41()2( (b)

2

i

i

iiiii

ii

zw

--=

---=

+++--

=

----

´+--

=

5765

74214

482)2()2(

)2()41( (c)

2

i

i

iiiii

ii

wz

+-=

+--=

+++--

=

++

´-+-

=

6