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Chapter 3
Complex numbers
129
remember
remember
1. The magnitude (or modulus or absolute value) of z = x + yi is the length of the
line segment from (0, 0) to z and is denoted by ⏐z⏐, ⏐x + yi⏐ or mod z.
2. ⏐z⏐ =
3.
4.
5.
6.
x 2 + y 2 and zz = z 2 .
y
arg z = θ where tan θ = -- .
x
n
z × i , n ∈ N produces an anticlockwise rotation of 90n degrees.
z = r cos θ + r sin θ i = r cis θ in polar form.
Arg z is the angle θ in the range −π < θ ≤ π.
3D
Complex numbers
in polar form
In the following exercise give arg z or Arg z correct to three decimal places where it is not
easily expressed as a multiple of π.
Example
xample
17
2 Find the modulus of each of the following.
a z = 5 + 12i
b z = 5 – 2i
d z = −3 − 6i
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Example
xample
18
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Example
xample
19
e z=
3 + 2i
c
z = −4 + 7i
f
z = (2 + i)2
3 If z = 3 + i, w = 4 − 3i and u = −2 + 5i then:
i represent each of the following on an Argand diagram
ii calculate the magnitude in each case.
a z−w
b u+z
c
d w+z
e z+w−u
f
w−u
z2
4 a Show the points z1 = −3 + 0i, z2 = 2 + 5i, z3 = 7 + 5i and z4 = 9 + 0i on the complex
number plane.
b Calculate the area of the shape formed when the four points are connected by
straight line segments in the order z1 to z2 to z3 to z4 and back to z1.
5 a Show the points z = −1 + 3i, u = 3 and w = 3 + 12i on the complex number plane.
b Calculate the area of the triangle produced by joining the three points with straight
line segments.
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Example
xample
20
6 Find the argument of z for each of the following in the interval [0, 2π ]. (Give exact
answers where possible.)
a z = 3 + 2i
b z=
e z = −2 − 2 3i
i z = −6i
f
z = 6 − 10i
j
z = 55
3+i
c
z = 5 − 5i
g z = 3i
cad
Complex 1
1 a Represent z = 4 + 8i on an Argand diagram.
b Calculate the exact distance of z from the origin.
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Math
d z = −4 + 8i
h z= – 7
130
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Example
xample
21
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Example
xample
22
Maths Quest 12 Specialist Mathematics
7 Convert each of the following into Arguments.
3π
11 π
15 π
a -----b − --------c --------2
6
8
19 π
20 π
18 π
e --------f --------g − --------6
7
5
8 Find the modulus and Argument of each of the following complex numbers.
a 3 − 3i
b −5 + 5i
c −1 − 3i
d 4 3 + 4i
e −7 − 10i
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Example
xample
23
f
Example
xample
24
6i − 2
g ( 3 + i )2
9 Express each of the following in polar form z = r cis θ where θ = Arg z.
a z = −1 + i
b z = 6 + 2i
c z = – 5 – 5i d z = 5 – 15i
3
1
e z = − --- – ------- i
2 2
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5π
d − -----4
13 π
h − --------12
f
1 1
z = − --- + --- i
4 4
10 Express each of the following complex numbers in Cartesian form.
2π
a 2 cis -----3
e
π
b 3 cis --4
7π
7 cis ⎛ – ------⎞
⎝ 4⎠
f
π
8 cis --2
c
5π
5 cis -----6
g
3 cis π
π
d 4 cis ⎛ – ---⎞
⎝ 3⎠
11 multiple
ultiple choice
If z = 3 − 50i and w = 5 + 65i the value of ⏐z + w⏐ is:
A 64
B 15
C 17
D 225
E 289
12 multiple
ultiple choice
The perimeter of the triangle formed by the line segments connecting the points
2 − 4i, 14 − 4i and 2 + i is:
A 13
B 30
C 10
D 17
E 25
13 multiple
ultiple choice
The Argument of 4
π
A --B
6
3 – 4i is:
π
--3
14 multiple
ultiple choice
In polar form, 5i is:
π
A cis --B cis 5π
2
5π
C -----6
π
D – --6
π
E – --3
5π
C cis -----2
D 5 cis 5π
π
E 5 cis --2
–3
3
D ------ + ------- i
2
2
– 3 1
E ---------- – --- i
2
2
15 multiple
ultiple choice
Work
ET
SHE
3.1
The Cartesian form of
1
3
A --- + ------- i
2 2
7π
3 cis ⎛ – ------⎞ is:
⎝ 6⎠
–1
3
B ------ + ------- i
2
2
– 3 1
C ---------- + --- i
2
2
Chapter 3
3E
Example
xample
Example
xample
Power of
a complex
number
2 Express the resultant complex numbers in question 1 in Cartesian form.
Power of
a complex
number
program
GC
26
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Example
xample
27
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Example
xample
28
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Example
xample
29
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Example
xample
30
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Example
xample
31
3 Express the following products in polar form.
a (2 + 2i)( 3 + i)
b ( 3 − 3i)(2 3 − 2i)
c
(−4 + 4 3 i)(−1 − i)
4 Express each of the following in the form r cis θ where θ ∈ (−π, π ].
5π
π
3π
π
a 12 cis ------ ÷ 4 cis --b 36 cis ------ ÷ 9 cis ⎛ – ---⎞
⎝ 6⎠
6
3
4
4π
11 π
π
π
d 4 3 cis ------ ÷ 6 cis --------c
20 cis ⎛ – ---⎞ ÷ 5 cis ⎛ – ---⎞
⎝ 2⎠
⎝ 5⎠
7
14
5π
7π
e 3 5 cis ⎛ – ------⎞ ÷ 2 10 cis -----⎝ 12 ⎠
6
3π
π
3 cis ------ and w = 2 cis ⎛ – ---⎞ then express each of the following in:
⎝ 4⎠
4
i polar form
ii Cartesian form.
b w4
c z4
d w5
a z3
5 If z =
6 If z = 1 − i and w = – 3 + i , write the following in Cartesian form.
z3
e -----4a z−4
b w−3
c z−3
d w−5
w
f
z2w3
7 Determine ( 2 + 2i ) 2 ( 1 – 3i ) 4 in Cartesian form.
( 3 – i )6
8 Write ----------------------------3 in the form x + yi.
( 2 – 2 3i )
9 multiple
ultiple choice
π
π
a 5 cis ⎛ – ---⎞ × 8 cis ⎛ – ---⎞ is equal to:
⎝ 3⎠
⎝ 6⎠
B – 2 10i
A 6 2i
C –6 3
b If z =
(
A 1+i
D −6i
E 6 6
6 + 2 ) + ( 6 – 2 ) i then 64 2z –3 is:
B
2i
C 1−i
D
2+i
w4
c If z = –1 – 3i and w = 2 + 2i then -----3- is equal to:
z
A −4 + 4i
C
2
D −4i
B 2 3
E –1 – 2i
E −8
am
progr –C
asio
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1 Express each of the following in the form r cis θ where θ ∈ (−π, π ].
π
π
2π
π
a 2 cis --- × 3 cis --b 5 cis ------ × 4 cis ⎛ – ---⎞
⎝ 3⎠
4
2
3
3π
5π
π
d 6 cis ------ × 5 cis π
c
3 cis ⎛ – ------⎞ × 2 cis ⎛ – ---⎞
⎝ 6⎠
⎝ 2⎠
4
5π
7π
e
7 cis ⎛ – ------⎞ × 2 cis -----⎝ 12 ⎠
12
–TI
25
Basic operations on complex
numbers in polar form
GC
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139
Complex numbers
140
Maths Quest 12 Specialist Mathematics
π
z6
3π
2 cis ------ and w = 3 cis --- , find the modulus and the argument of -----4- .
Example
xample
4
6
w
32
11 If z = 4 + i and w = −3 − 2i, determine (z + w)9.
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10 If z =
12 Find z6 + w4, if z =
13 If z 1 =
2 – 2i and w = 2 − 2i.
3π
2π
5 cis ⎛– ------ ⎞ , z 2 = 2 cis------ and z 3 =
⎝ 5⎠
8
π
10 cis ------ , find the modulus and the
12
z 12 × z 23
argument of ------------------.
4
z3
14 By finding z4 if z = cis θ, show that cos 4θ = cos4θ − 6 cos2θ sin2θ + sin4θ and that
sin 4θ = 4 cos3θ sin θ − 4 cos θ sin3θ.
33
15 Using z = r cis θ, verify that zz = z 2 .
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Example
xample
16 If zn = (1 + i)n, determine the smallest value of n ∈ N so that zn is equal to:
a
( 2 )n
b –( 2 )
n
c
( 2 )n i
n
d –( 2 ) i .
Factorisation of polynomials in C
A polynomial in z is an expression of the form
P(z) = an zn + an − 1 zn − 1 + an − 2 zn − 2 + . . . + a1z + a0,
where n ∈ N is the degree (highest power) of P(z) and an (with an ≠ 0) are the
coefficients.
If an ∈ R, that is, all the coefficients are real, then P(z) is said to be a polynomial
over R. Similarly, if at least one of the an is complex, P(z) is said to be a polynomial
over C.
For example, P(z) = 3z4 − 5z2 + 6 is a polynomial of degree 4 over R and
P(z) = 2iz3 + 3z2 − 8i is a polynomial of degree 3 over C.
The fundamental theorem of algebra
Firstly recall that R ⊂ C and the factor theorem, which states:
If (x − a) is a factor of the polynomial P(x), then P(a) = 0.
In 1799 the German mathematician Carl Friedrich Gauss proved that every polynomial
over C has a solution that is a complex number.
That is, if Pn(z) is a polynomial of degree n over C, then there exists a z0 ∈ C such
that Pn(z0) = 0. This important result can be used to show that a polynomial of degree
n, with n ∈ N, has n solutions.
The proof relies on a repeated application of the fundamental theorem of algebra and
the factor theorem.
Firstly, the fundamental theorem of algebra guarantees that there is a z0 ∈ C such
that Pn(z0) = 0. The factor theorem states that if Pn(z0) = 0 for some z0 then (z − z0) is a
factor of Pn(z) so that Pn(z) = (z − z0)Pn − 1(z), where Pn − 1(z) is a polynomial of degree
n − 1.
Now by applying the fundamental theorem of algebra to Pn − 1(z) there is a z1 ∈ C
such that Pn − 1(z1) = 0 and the factor theorem ensures that Pn − 1(z) = (z − z1)Pn − 2(z).
146
Maths Quest 12 Specialist Mathematics
12 Find the values of a and b (a, b ∈ R) if:
a (z + 1) is a factor of z3 − 2iz2 + aiz + b
39
b (z − i) is a factor of az3 − 3z2 + biz + 12i
c (z + 2i) is a factor of z3 + aiz2 + 2iz + (1 + i)b.
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Example
xample
13 Explain why at least one of the zeros of a polynomial of degree n (where n is an odd
natural number) is a real number.
14 Write down a polynomial of degree 3, whose coefficients are all real, that has 4i and
2 as two of its zeros.
15 Find the values of a, (a ∈ R) for which ai is a solution to:
a P(z) = z3 + 3z2 + 36z + 108
b P(z) = z3 + 6iz2 − 11z − 6i.
16 Factorise z3 + i over C.
17 a Show that P(1) = 0 for P(z) = z4 − (1 + 3i)z3 + 3(i − 1)z2 + (7 + i)z − 4 − i.
b Find the polynomial Q(z) if P(z) = (z − 1)Q(z).
c Determine the values of a ∈ C, b ∈ R if Q(z) is of the form Q(z) = (z − a)3 + b.
18 Factorise z4 + 2z3 + 8z2 + 10z + 15 over C given that z +
5 i is a factor.
19 Factorise P(z) = 9z3 + (9i − 12)z2 + (5 − 12i)z + 5i over C if P(−i) = 0.
a – 11
-.
20 Determine the value of a ∈ R if – 3i is to be a zero of a + z 2 = -------------z2
Graphics Calculator tip! Roots of complex numbers
Casio
tip
removed.
1. To select complex number mode, press MODE and select Radian mode; scroll down
and select a+bi and press ENTER .
2. To find the cube roots of z = –2 – 2i, start by finding one of the roots as follows. Press
MATH , select 4:
3
, enter (–2 – 2i) and press ENTER . So one cube root is z1 = 1 – i.
2π
3. Since cube roots occur at angles of ------ , the second cube root can be found by multi3
2π
plying z1 by cis ------ . Scrolling shows that this root is 0.366 + 1.366i.
3
4π
4. The third cube root is found by multiplying z1 by cis ------ . Scrolling shows that this root
3
is –1.366 – 0.366i. Note that the cube root was recalled using 2nd [ENTRY] twice;
2π
4π
cis ------ was also recalled using 2nd [ENTRY] twice and then edited to make it cis ------ .
3
3
Chapter 3
3G
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Example
xample
40
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Example
xample
41
Complex numbers
153
Solving equations in C
1 Solve the following quadratic equations over C.
a x2 + 2x + 5 = 0
b x2 − 8x + 25 = 0
d 4x2 − 12x + 13 = 0
e 4x2 − 32 x + 4 = 0
c
2 Solve the following equations over C.
a z3 − z2 − z + 10 = 0
b z3 − 2z2 + 3z − 2 = 0
c
3
2
d 3z − 13z + 5z − 4 = 0 e 4z3 − 20z2 + 34z − 20 = 0
x2 − 14x + 149 = 0
2z3 − 7z2 + 10z − 8 = 0
3 For f(z) = z − 4, g(z) = z2 − z + 1 and h(z) = z3 − 5z2 + 5z − 4 show that
f(z) × g(z) = h(z) and hence determine the values of z such that h(z) = 0.
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Example
xample
42
4 Solve these equations over C.
a x4 + 25x2 + 144 = 0
c
9z4 + 35z2 − 4 = 0
b z4 − 3z2 − 4 = 0
d 4x4 + 12x2 + 9 = 0
5 multiple
ultiple choice
The solutions to the equation (z − 3)2 + 4 = 0 are:
A z = 2 + 3i, z = 2 − 3i
B z = 3 − 2i, z = 3 + 2i
C z = 3 + 4i, z = 3 − 4i
D z = 4 − 3i, z = 4 + 3i
E z = 9 + 16i, z = 9 − 16i
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Example
xample
a 1+
7 Find
b 11 + 60i
3i
c
16 + 63i
i in Cartesian form.
program
GC
–TI
43
6 Find the square roots of each of the following in Cartesian form.
Roots of
a complex
number
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Example
xample
44
am
progr –C
asio
π
If one of the square roots of a(1 + i) is a cis --- , the other square root is:
8
7π
7π
7π
B – a cis -----C a cis – -----A a cis -----8
8
8
9π
9π
E – a cis -----D a cis -----8
8
GC
8 multiple
ultiple choice
Roots of
a complex
number
9 Use De Moivre’s theorem to solve the following equations, in polar form.
b z2 = 4 + 4i
c z3 = −4 + 4 3 i
a z2 = 3 – i
3
3
d z =i
e z = −1 − i
f z6 + i = 0
1
---
10 Find ( – 125i ) 3 and determine the value of the sum of the roots.
11 a Find the cube root of 64.
b Show the results on an Argand diagram.
12 Solve the following equations in Cartesian form.
b z4 = 25
c z6 = 64
a z4 = 16
45
13 Find all z satisfying
a z5 = 1
b z8 + 1 = 0.
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Express your answers in polar form.
d z6 = 27
ET
SHE
Work
Example
xample
3.2