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© John Wiley & Sons Australia, Ltd 2009 1
WorkSHEET 2.2 Number systems: complex numbers
Name: ___________________________ 1
Write yixii
ii
+--
-++ form in the
3235
23 .
( ) ( )
i
i
ii
ii
iii
iiii
iiiii
ii
ii
ii
ii
ii
6558
6546558465
4595139165
919571313919
57
94961510
4236
3232
3235
22
23
3235
23
2
2
2
2
--=
--=
---=
+--=
+-
-=
---+
--
-+-=
++
´--
---
´++
=
--
-++
4
2 Express
45cis2 p in General / Cartesian form.
i
i
i
i
--=
÷ø
öçè
æ--=
÷øö
çèæ --=
÷øö
çèæ +=
121
212
4sin
4cos2
45sin
45cos2
45cis2
pp
pp
p
3
Maths Quest Maths C Year 11 for Queensland Chapter 2: Number systems: complex numbers WorkSHEET 2.2
© John Wiley & Sons Australia, Ltd 2009 2
3 Find the modulus and Argument of z = -7 + 7i.
( )
43Argument
27 Modulus4
34
quadrant) (2nd1tan17
7tan
27
249
98
4949
7)7(
1
22
p
p
=
=
=
-=
-=
-=-
=
=
´=
=
+=
+-=
-
π
π
πθ
θ
z
3
4 (a) Sketch z = 5 – 5i. (b) Express z = 5 – 5i in polar form.
(a)
(b)
4
55tan)Arg(
25
50
2525
)5(5
1
22
p-=
÷øö
çèæ -=
=
=
+=
-+=
-z
z
z = 5 2 cis(-4p )
3
Maths Quest Maths C Year 11 for Queensland Chapter 2: Number systems: complex numbers WorkSHEET 2.2
© John Wiley & Sons Australia, Ltd 2009 3
5 Sketch the following complex numbers and express them in polar form: (a) z = 2 + 3i (b) w = –4 – 2i
(a)
0.98 cis 13
98.023tan)Arg(
13
94
32
1
22
=
=
÷øö
çèæ=
=
+=
+=
-
z
z
z
(b)
)68.2( cis 52
68.246.0
42tan)Arg(
52
20
416
)2()4(
1
22
-=
-=-=
÷øö
çèæ--
=
=
=
+=
-+-=
-
w
w
w
p
6
Maths Quest Maths C Year 11 for Queensland Chapter 2: Number systems: complex numbers WorkSHEET 2.2
© John Wiley & Sons Australia, Ltd 2009 4
6 Use Pascal’s Triangle to expand (–1 – i)5.
Fifth row is 1 5 10 10 5 1 (–1 – i)5
= (–1)5 + 5(–1)4(–i) + 10(–1)3(–i)2 + 10(–1)2(–i)3 + 5(–1)(–i)4 + (–i)5
= –1 + 5(1)(–i) + 10(-1)(-1) + 10(1)i + 5(-1)(1) – i = –1 – 5i + 10 + 10i – 5 – i
= 4 + 4i
4
7 Convert z = 4 cis
3p and w = 6 cis
32p into
standard form.
z = 4 cis3p
z = 4(cos3p + isin
3p )
z = 4(21 + i
23 )
z = 2 + 2 3 i
w = 6 cis32p
w = 6(cos32p + isin
32p )
w = 6(-21 + i
23 )
w = -3 + 3 3 i
5
Maths Quest Maths C Year 11 for Queensland Chapter 2: Number systems: complex numbers WorkSHEET 2.2
© John Wiley & Sons Australia, Ltd 2009 5
8 Perform the following operations using
z = 4 cis3p and w = 6 cis
32p .
Convert into standard form: (a) z + w (b) w – z (c) z × w
(d) wz
(e) zw
Using the answers from the previous question:
i
iiwz
351
333322 (a)
+-=
+-+=+
i
iizw
35
)322(333 (b)
+-=
+-+-=-
24)01(24
)sin(cos24 cis 24
32 cis 6
3 cis 4 (c)
-=+-=+=
=
´=´
ppp
pp
i
wz
i
i
wz
33
31
23
21
32
3 cis
32
32
3 cis
32
cis 6 cis 4
(d)3
23
-=
÷÷ø
öççè
æ-=
÷øö
çèæ-=
÷øö
çèæ -=
=
p
pp
p
p
i
i
i
zw
433
43
23
21
23
3sin
3cos
23
3 cis
23
332 cis
23
cis 4 cis 6
(e)3
32
+=
÷÷ø
öççè
æ+=
÷øö
çèæ +=
=
÷øö
çèæ -=
=
pp
p
pp
p
p
7
Maths Quest Maths C Year 11 for Queensland Chapter 2: Number systems: complex numbers WorkSHEET 2.2
© John Wiley & Sons Australia, Ltd 2009 6
9 Sketch the following complex numbers and express them in standard form. Use common trigonometric ratios and special triangles.
(a) z = 2 cis43p
(b) z = 4 cis6p
(c) z = 2 cis ÷øö
çèæ -32p
(a) z = 2 cis43p
z = 2(cos43p + isin
43p )
z = 2(-21 +
21 i)
z = - 2 + 2 i
(b) z = 4 cis6p
z = 4(cos6p + isin
6p )
z = 4(23 +
21 i)
z = 2 3 + 2i
(c) z = 2 cis ÷øö
çèæ -32p
z = 2(cos ÷øö
çèæ -32p + isin ÷
øö
çèæ -32p )
z = 2(-21-23 i)
z = -1 - 3 i
9
Maths Quest Maths C Year 11 for Queensland Chapter 2: Number systems: complex numbers WorkSHEET 2.2
© John Wiley & Sons Australia, Ltd 2009 7
10 Express i52 + in standard and polar form.
Let i52 + = a + bi 2 + 5i = a2 + 2abi – b2) Therefore 2 = a2 – b2 (1) and 5 = 2ab (2)
From (2): a = b25
Substituting a into (1):
2 = 2
25÷øö
çèæb
– b2
2 = 24
25b
– b2
8b2 = 25 – 4b4 0 = 4b4 + 8b2 – 25
b2 = 8
)25)(4(488 2 --±-
= 84648 ±-
» –3.7 or 1.7 (Ignore b2 = –3.7 because b must be real) Therefore, b2 = 1.7 and b » ± 1.3. For b = -1.3, a = -1.9. For b = 1.3, a = 1.9. Therefore i52 + = ±(1.9 + 1.3i) In polar form: (always sketch the number first) | i52 + | = 22 3.19.1 + = 2.3
q = tan–1 9.13.1
= 0.6 Argument = 0.6 or -2.5
i52 + = 2.3 cis 0.6 or 2.3 cis (–2.5)
7
