8. Complex Numbers and Polar Coordinates
8.1
8.2
8.3
8.4
8.5
8.6
Complex Numbers
Trigonometric Form for Complex Numbers
Products and Quotients in Trigonometric Form
Roots of a Complex Number
Polar Coordinates
Equations in Polar Coordinates and Their Graphs
1
8.1 Complex Numbers
1) Definitions of complex numbers
2) Equality for Complex Numbers
3) Addition and subtraction of complex numbers
4) Multiplication of complex numbers
5) Division of complex numbers
6) The power of i
7) Problems
2
8.1 Complex Numbers
1) Definitions of Complex Numbers
Definition. A complex number is any number that
can be written in the form
a + bi
where a and b are real numbers and i2 = –1
•
•
•
•
•
i is called the imaginary unit, whose square is –1
a is called real part of the complex number
b is called imaginary part of the complex number
If b = 0, then a + bi = a, which is a real number
If a = 0, then a + bi = bi, which is called an imaginary number
3
8.1 Complex Numbers
2) Equality for Complex Numbers
Definition. Two complex numbers are equal if their
real parts are equal and their imaginary parts are
equal. i.e. for real numbers a, b, c, and d,
a + bi = c + di if and only if a = b and c = d
4
8.1 Complex Numbers
3) Addition and Subtraction of Complex Numbers
Definition. If z1 = a1 + b1i and z2 = a2 + b2i are two
complex numbers, then
z1 + z2 = (a1 + a2) + (b1 + b2)i
z1 − z2 = (a1 − a2) + (b1 − b2)i
5
8.1 Complex Numbers
4) Multiplication of Complex Numbers
Definition
If z1 = a1 + b1i and z2 = a2 + b2i are two complex
numbers, then
z1 + z2 = (a1 + b1i) (a2 + b2i)
= (a1a2 − b1b2) + (a1b2 − a2b1)i
6
8.1 Complex Numbers
5) Division of Complex Numbers
Definition (conjugate of a complex number)
if a + bi and a − bi are called complex conjugates.
Their product is the real number a2 + b2:
(a + bi) (a − bi) = a2 – (bi)2
= a2 – b2i2
= a2 – b2(–1)
= a2 + b2
This property if used for dividing complex numbers.
7
8.1 Complex Numbers
5) Division of Complex Numbers
To divide two complex numbers, multiply both the numerator and
denominator by the conjugate of the denominator.
e.g.8
Sol.
5i
Divide
.
2 − 3i
5i
5i
2 + 3i
=
⋅
2 − 3i
2 − 3i 2 + 3i
10 i + 15 i 2
=
13
15
10
= −
+
i
13
13
8
8.1 Complex Numbers
6) The Power of i
i =i
i2 = –1
i3 = –i
i4 = 1
i5 = i
Period is 4.
9
8.1 Complex Numbers
7) Problems
(1) Write each expression in terms of i.
(a) − 49
[2]
(b) − 20
Ans. 7i
[8]
Ans. i 20 = 2i 5
(2) Write in terms of i and then simplify
Ans. –8i
− 16 ⋅ − 4
[12]
(3) Find x and y so that each equation is true.
(7x – 1) + 4i = 2 + (5y + 2)i
[16]
Ans. x = 3/7, y = 2/5
10
8.1 Complex Numbers
7) Problems
(4) Combine the complex numbers.
[(4 – 5i) – (2 + i)] + (2 + 5i)
(5) Simplify each power of i.
(a) i13
[36]
Ans. i
[32]
Ans. 4 – i
(b) i35
[42]
Ans. –i
(6) Find the following products.
(a) (3 – 2i)2
[48]
(b) 3i(1 + 2i)(3 + i) [54]
Ans. 5 + 12i
Ans. –15 + 3i
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8.1 Complex Numbers
7) Problems
(7) Find the quotient. Write the answer in standard form
for complex numbers.
3i
2 + i
Ans.
3
6
+
i
5
5
12
8.2 Trigonometric Form for Complex Numbers
1) The Graph of a Complex Number
2) The Absolute Value or Modulus of a Complex Number
3) The Argument of a Complex number
4) The Trigonometric Form of complex numbers
5) Conversion between standard form and trig form for a
complex numbers
6) Problems
13
8.2 Trigonometric Form for Complex Numbers
1) The Graph of a Complex Number
Definition
The graph of the complex number x + yi is a vector
(arrow) that extends from the origin out to point (x, y).
y
x+yi
r
θ
x
y
x
14
8.2 Trigonometric Form for Complex Numbers
2) The Absolute Value or Modulus of a Complex Number
Definition
The absolute value or modulus of the complex number
x + yi is the distance from the origin to the point (x, y).
If this distance is denoted by r, then
r = | z |= | x + yi |=
y
x2 + y2
x+yi
r
θ
x
y
x
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8.2 Trigonometric Form for Complex Numbers
3) The Argument of a Complex number
Definition
The argument of the complex number is the smallest
positive angle θ from the positive real axis to the
graph of z.
y
x+yi
r
θ
x
y
x
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8.2 Trigonometric Form for Complex Numbers
4) The Trigonometric Form of complex numbers
y
x+yi
r
θ
x
y
x
Definition. If x + yi is a complex number in standard form,
then the trigonometric form for z is given by:
z = r(cosθ + i sin θ) = r cisθ,
Where
r = x2 + y2
and θ is determined by
x
y
cos θ =
and sin θ = .
r
r
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8.2 Trigonometric Form for Complex Numbers
5) Conversion between standard form and trig form for a
complex numbers
From standard form to trig form: x + yi to (r, θ)
r =
x
y
and sin θ = .
x + y , cos θ =
r
r
2
2
From trig form to standard form: (r, θ) to x + yi
x = r cos(θ), y = r sin(θ).
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8.2 Trigonometric Form for Complex Numbers
6) Problems
(1) Graph each complex number; give the absolute value
of the number.
(a) –4
[8]
(b) –3 – 4i
[10]
Ans. modulus = 4
Ans. modulus = 5
(2) Graph each complex number along with its conjugate.
(a) –3i
[14]
(b)
–2 – 5i
[18]
(3) Write in standard form.
(a) 4(cos30° + i sin30°)
2
3 + 2i
(b)
1 cis(240°)
−
1
2
−
3
2
i
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8.2 Trigonometric Form for Complex Numbers
6) Problems
(c) 2 cis( 74π )
[26]
1–i
(4) Write in standard form. Round the numbers to the
nearest hundredth.
(a) 1 cis 261° [32]
(b) 10 cis5.5
[34]
–0.16 – 0.99i
7.09 – 7.06i
(5) Write in trigonometric form. Round all angles to the
nearest hundredth of degree.
(a) 3 – 4i
[48]
(b) –11 + 2i
[54]
5 cis(306.87°)
5 5 cis (169.70o )
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8.3 Products and Quotients in Trigonometric Form
1) Multiplication of complex numbers
2) De Moivre’s Theorem
3) Division of complex numbers
4) Problems
21
8.3 Products and Quotients in Trigonometric Form
1) Multiplication of Complex Numbers
Theorem (Multiplication)
If z1 = r1 cisθ1 and
z2 = r2 cisθ2
are two complex numbers in trigonometric form, then their
products, z1z2, is
z1z2 = r1r2 cis (θ1 + θ2)
In words, to multiply two complex numbers in
trigonometric form, multiply absolute values and add angles.
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8.3 Products and Quotients in Trigonometric Form
2) De Moivre’s Theorem
DeMoivre’s Theorem
If
z = r cisθ
is a complex number in trigonometric form, then the nth
power of z is
zn = rn cis(nθ )
In words, to raise a complex number to nth power, simply
raise its modulus to the nth power and multiply the
argument by the power.
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8.3 Products and Quotients in Trigonometric Form
3) Division of complex numbers
Theorem (Division)
If z1 = r1cisθ1 and z2 = r2 cisθ2
are two complex numbers in trigonometric form, then their
products, z1/z2, is
z1
r
= 1 cis ( θ 1 − θ 2 )
z2
r2
In words, to divide two complex numbers, divide their
absolute values and subtract angles.
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8.3 Products and Quotients in Trigonometric Form
4) Problems
(1) Multiply, leave answer in trig form.
[4]
9(cos115° + i sin115°)⋅4(cos51° + i sin51°)
Ans.
36 cis(166°)
(2) Use De Voivre’s Theorem to simplify, write the
answer in standard form. [30]
(
3+i
)
4
Ans.
− 8 + 8i 3
(3) Divide, leave answer in trig form.
21(cos 63 o + i sin 63 o )
14 (cos 44 o + i sin 44 o )
Ans.
[38]
3
2
cis (19 o )
25
8.4 Roots of a Complex Number
1) nth roots of a complex number
2) Problems
26
8.4 Roots of a Complex Number
1) Nth roots of a complex number
Suppose we want to solve a equation: x4 = 3.
From intermediate algebra, we know that it has at least one
solution:
x= 4 3
(it is called 4th root of 3)
Actually, it has 4 solutions. In general, for any complex
number z, xn = z has n solutions, called nth roots of
complex number z.
27
8.4 Roots of a Complex Number
1) nth roots of a complex number
Theorem (Roots). The nth roots of the complex
number,
z = r(cosθ + i sin θ )
Are given by
wk = r
=r
1/ n
1/ n
[cos( n +
θ
cis
(
θ
n
+
360 o
n
360o
n
k
k ) + i sin( n +
θ
)
360 o
n
k )]
Where k = 0, 1, …, n – 1
28
8.4 Roots of a Complex Number
2) Problems
(1) Find the two square roots of 9(cos310° + i sin310°).
Graph two roots.
[4]
Ans.
3cis(155°) and 3cis(335°)
(2) Find the two square roots of z = –4i.
[10]
–4i = 4cis(270°)
2 − i 2 or − 2 + i 2
(3) Find 3 cube roots of 27(cos303° + i sin303°).
Ans.
[16]
3cis(101°), 3cis(221°), and 3cis(341°)
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8.4 Roots of a Complex Number
2) Problems
(4) Solve equation:
Ans.
x4 + 81 = 0
[26]
3cis(45°), 3cis(135°), 3cis(225°), 3cis(315°)
± 3 22 ± i 3 22
(5) Find the four 4th roots of z = cos 43π + i sin 43π .
[28]
cis ( π3 ), cis( 56π ), cis ( 43π ), cis ( 116π )
30
8.5 Polar Coordinates
1) Polar coordinates
2) Problems
(I don’t like the idea that r can be negative)
31
8.5 Polar Coordinates
1) Polar coordinates
y
x+yi
r
θ
x
•
•
y
x
(x, y) is called rectangular coordinate
(r, θ ) is called polar coordinate
32
8.5 Polar Coordinates
1) Graph in polar coordinates.
(2, 135°)
(1, –135°)
(–2.5, 45°)
2) What other ordered pairs name the same points
as (2, 60°)
3) Convert ( 2 , –135°) to rectangular coordinates.
4) Convert (− 2 3 , 2) to polar coordinates.
33
8.6 Inverse Trigonometric Functions
Graph in polar coordinates
1) r = 3.
2) θ = 45°.
3) r = 3sinθ.
4) r = 2 + 2sinθ.
34