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© 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