Complex Numbers

Question 1

What are

• *two other names** for the
• *complex plane**?

Answer 1

z-plane

Argand plane

Question 2

(a + bi ) (a − bi ) = _____

Answer 2

a² + b²
&
|a + bi |²

Question 3

For complex number
z = a + bi,

|z| = _____

Answer 3

√(a² + b²)

In other words:
√( Re(z)² + Im(z)²)

Absolute value means
distance from zero, whether the number is real or imaginary.

The absolute value of a complex number can be obtained by applying the Pythagorean theorem.

Question 4

If you
don’t know the
real part and imaginary part of a
complex number,
what
other information
may allow you to
uniquely describe that number?

Answer 4

Absolute value

&

Angle
(or argument)

(or modulus)

Question 5

If you
don’t know the
absolute value and angle of a
complex number,
what
other information
may allow you to
uniquely describe that number?

Answer 5

Real part

&

Imaginary part

Question 6

In complex numbers,
what does
absolute value
refer to?

Answer 6

The

• *absolute value**, or
• *modulus**, gives the
• *distance** of the number from the
• *origin** in the complex plane.

Question 7

In complex numbers,
what does
angle
refer to?

Answer 7

The

• *angle**, or
• *argument**, is the angle the number forms with the
• *positive real axis**.

Question 8

How do you find the

• *angle** of complex number
• a + bi ***?

Θ = _____?

Answer 8

Θ = tan⁻¹ (b/a_)_

• The slope of the line that intersects the number and the origin.*
• Just as with real numbers on a Cartesian plane.*

Question 9

What are

• *three forms** of a
• *complex number**?

Answer 9

Rectangular form:
a + bi

Polar form:
r (cosΘ + i sinΘ))

Exponential form:
r•​e^{<em>i</em> Θ}

Question 10

What
form is the complex number

2√(3) + 2i

written in?

Answer 10

Rectangular form

Question 11

What is the
rectangular form of a
complex number
useful for?

Answer 11

Adding and subtracting
complex numbers.
It’s the sum of two terms: the real part and the imaginary part.

Plotting
in the complex plane.
It gives you the two coordinates.

Question 12

What is the
polar form of a
complex number
useful for?

Answer 12

Multiplying and dividing
complex numbers.

The product of two numbers with absolute values
r₁ and r₂ and angles Θ₁ and Θ₂ will have an absolute value r₁ • r₂ and angle Θ₁ + Θ₂.

Example:

• z₁ has absolute value r₁ and angle Θ₁.
• z₂ has absolute value r₂ and angle Θ₂.
• z₁ • z₂ = (r₁ • r₂) (cos(Θ₁ + Θ₂) + i sin(Θ₁ + Θ₂)).

Question 13

How does the
exponential form of a complex number
relate to the
polar form?

Answer 13

It
uses the same attributes
as polar form (absolute value and angle), but
displays them in a
more compact way.

Example:

• z₁ has absolute value r₁ and angle Θ₁.
• z₂ has absolute value r₂ and angle Θ₂.
• z₁ • z₂ = (r₁ • e^{<em>i</em> Θ₁}) • (r₂ • e^{<em>i</em> Θ₂}) = (r₁ • r₂) • e^{<em>i</em> (Θ₁+Θ₂)}

Question 14

What
form is the complex number

4 (cos (π/6) +i sin (π/6))

written in?

Answer 14

Polar form

Question 15

What
form is the complex number

4e^{(π/6)<em>i</em>}

written in?

Answer 15

Exponential form

Question 16

How do you state

• *complex number** z in
• *rectangular form**?

z = _____

Answer 16

z = a + bi

a: real part
b: imaginary part
* i:* imaginary unit

Question 17

How do you state

• *complex number** z in
• *polar form**?

z = _____

Answer 17

z = r (cosΘ + i sinΘ)

r: absolute value or modulus
Θ: angle or argument
i: imaginary unit

Question 18

How do you state

• *complex number** z in
• *exponential form**?

z = _____

Answer 18

z = r•​e^{<em>i</em> Θ}

r: absolute value or modulus
Θ: angle or argument
i: imaginary unit

Question 19

In what quadrant of the complex plane does

3 + 4i

lie?

Answer 19

Quadrant I

Question 20

In what quadrant of the complex plane does

−3 + 4i

lie?

Answer 20

Quadrant II

Question 21

In what quadrant of the complex plane does

−3 − 4i

lie?

Answer 21

Quadrant III

Question 22

In what quadrant of the complex plane does

3 − 4i

lie?

Answer 22

Quadrant IV

Question 23

Why is the
quadrant of a
complex number
important?

Answer 23

Because the calculator may give the
opposite angle,
so you may need to
add 180° or π to the result.

Example:

• −3 + 4i
  (located in Quadrant II)
• tan⁻¹ (4/−3) ≈ −53°
  (located in Quadrant IV)
• Must add 180° to result from calculator to obtain the opposite angle:
  −53° + 180° = 127°

Question 24

How do you restate

z = 4 (cos(π/6) + i sin(π/6))

in
rectangular form?

Answer 24

Expand
the parentheses in the
polar representation.

Question 25

What are the steps to restating z = **2√(3) + 2*i*** in **polar form**?

Answer 25

1. **Find the angle** Θ = tan⁻¹ (2 / (2√(3))) Θ = π/6 2. **Find the absolute value** r = √( (2√(3))² + 2²) r = √( 12 + 4) r = 4 3. **Plug in** z = 4 (cos(π/6) + *i* sin(π/6))

Question 26

Given complex number z's **absolute value *r*** and **angle Θ**, how do you obtain its **rectangular form**?

Answer 26

z = ***r* cos(Θ) + *r* sin(Θ) *i*** ## Footnote *This results from using trigonometry in the right triangle formed by the number and the Real axis.*

Question 27

How do you obtain the **real part** of a complex number given its **absolute value *r*** and **angle Θ**?

Answer 27

***r* cos(Θ)** *This results from using trigonometry in the right triangle formed by the number and the Real axis.*

Question 28

How do you obtain the **imaginary part** of a complex number given its **absolute value *r*** and **angle Θ**?

Answer 28

***r* sin(Θ)** *This results from using trigonometry in the right triangle formed by the number and the Real axis.*

Question 29

Given ​z₁ = r₁ (cosΘ₁ + *i* sinΘ₁) & ​z₂ = r₂ (cosΘ₂ + *i* sinΘ₂), **z₁ • z₂ = \_\_\_\_\_**

Answer 29

z₁ • z₂ = **(r₁ • r₂) (cos(Θ₁ + Θ₂) + *i* sin(Θ₁ + Θ₂))** _Example_: * z₁ = 5 (cos 15° + *i* sin 15°) z₂ = 3 (cos 45° + *i* sin 45°) * z₁ • z₂ = (5 • 3) (cos (15° + 45°) + *i* sin (15° + 45°)) * = 15 (cos 60° + *i* sin 60°)

Question 30

Given ​z₁ = r₁ (cosΘ₁ + *i* sinΘ₁) & ​z₂ = r₂ (cosΘ₂ + *i* sinΘ₂), **_z₁_ = \_\_\_\_\_ z₂**

Answer 30

_z₁_ = **(r₁ / r₂) (cos(Θ₁ − Θ₂) + *i* sin(Θ₁ − Θ₂))** z₂ _Example_: * z₁ = 75 (cos 289° + *i* sin 289°) z₂ = 25 (cos 83° + *i* sin 83°) * z₁ / z₂ = (75 / 25) (cos (289° − 83°) + *i* sin (289° − 83°)) * = 3 (cos 206° + *i* sin 206°)

Question 31

Given ​z₁ = r₁ e^{i Θ₁} & ​z₂ = r₂ e^{i Θ₂}, **z₁ • z₂ = \_\_\_\_\_**

Answer 31

z₁ • z₂ = **(r₁ • r₂) e^{i (Θ₁ + Θ₂)}** _Example_: * z₁ = 9 e^{i π/4} z₂ = 3 e^{i π/12} * z₁ • z₂ = r₁ e^{i Θ₁} • r₂ e^{i Θ₂} * = (r₁ • r₂) e^{i (Θ₁ + Θ₂)} * = (9 • 3) e^{i (π/4 + π/12)} * = 27 e^{i (3π/12 + π/12)} * = 27 ​e^{i (4π/12)} * = 27 e^{i π/3}

Question 32

Given ​z₁ = r₁ e^{i Θ₁} & ​z₂ = r₂ e^{i Θ₂}, **_z₁_ = \_\_\_\_\_ z₂**

Answer 32

z₁ / z₂ = **(r₁ / r₂) e^{i (Θ₁ − Θ₂)}** _Example_: * z₁ = 9 e^{i π/4} z₂ = 3 e^{i π/12} * _z₁_ = z₁ • z₂⁻¹ z₂ * = 9 e^{i π/4} (3 e^{i π/12})⁻¹ * = 9 (1/3) e^{i π/4 − i π/12} * = 3 e^{i (π/4 − π/12)} * = 3 e^{i (3π/12 − π/12)} * = 3 ​e^{i (2π/12)} * = 3 e^{i π/6}

Question 33

If * z₁ = 1.5 (cos45° + *i* sin45°), * z₂ = 3 (cos15° + *i* sin15°), and * z₃ = z₁ z₂ what will the * *graph of z₃** look like * *relative to that of z₁**?

Answer 33

**Scaled by a factor of 3** **Rotated by +15° (counterclockwise)** (add angles) * z₃ = z₁ z₂ * = 1.5 (cos45° + *i* sin45°) • 3 (cos15° + *i* sin15°) * = (1.5 • 3) (cos(45° + 15°) *i* sin(45° + 15°)) * = 4.5 (cos60° + *i* sin60°) | (multiply absolute values)

Question 34

If * z₁ = 5 (cos75° + *i* sin75°), * z₂ = 2 (cos30° + *i* sin30°), and * z₃ = _z₁_ z₂ what will the * *graph of z₃** look like * *relative to that of z₁**?

Answer 34

**Scaled by a factor of 1/2** **Rotated by −30° (clockwise)** (subtract angles) * z₃ = _z₁_ z₂ * = _5 (cos75° + *i* sin75°)_ 2 (cos30° + *i* sin30°) * = (5/2) _(cos(75°−30°) + *i* sin(75°− 0°))_ * = 2.5 (cos45° + *i* sin45°) | (divide absolute values)

Question 35

If z is a complex number, u is a complex number, ū is the complex conjugate of u, and _q = z•u•ū_, then how will the * *angle of q** appear relative to the * *angle of z**?

Answer 35

They will have the **same angle**. ## Footnote *If the _angle of u is Θ_, then the _angle of ū is −Θ_.* *So the successive multiplications have _no total rotation_.*

Question 36

If z is a complex number, u is a complex number, ū is the complex conjugate of u, and _q = z•u•ū_, then how will the * *magnitude of q** appear relative to the * *magnitude of z**?

Answer 36

The magnitude will be **z | u |²**. ## Footnote *Both u and ū have the _same absolute value_.* *So the total effect of mutiplying by u and ū is to stretch everything by a factor of | u | • | ū | = | u |².*

Question 37

**_z_ = _a + b*i*_ = \_\_\_\_\_ u c + d*i***

Answer 37

**_z • ū_ |u|²** * _z_ = _a + b*i*_ u c + d*i* * = _a + b*i*_ • _(c − d*i* )_ c + d*i* (c − d*i* ) * = _(__a + b*i* )_ _(c − d*i* )_ c² + d² * = **_z • ū_** **|u|²**

Question 38

If z₁ is a complex number, then how will the * *graph of z₁ • *i*** appear * *relative to that of z₁**?

Answer 38

**Rotated +90° about the origin**. _Note_: * multiplication by *i* ² is a +180° rotation (*i* ² = −1) * multiplication by *i* ³ is a +270° rotation (*i* ³​ = −*i *) * multiplication by *i* ⁴ is a +360° rotation (*i* ⁴ = 1)

Question 39

Given ​z = r (cosΘ + *i* sinΘ) **zⁿ = \_\_\_\_\_**

Answer 39

​zⁿ = **rⁿ (cos(n • Θ) + *i* sin(n • Θ))**

Question 40

Given z = 2(cos60° + *i* sin60°), what is * *absolute value** of * *z³**?

Answer 40

**8** |z³| = r³ = 2³ = 8 ## Footnote *Stretched by a factor of 2 three times, so ultimately stretched by a factor of 2³ = 8.*

Question 41

Given z = 2(cos60° + *i* sin60°), what is * *angle** of * *z³**?

Answer 41

**180°** 60° + 60°+ 60° = 3 • 60° = 180° ## Footnote *Rotated by +60° three times, so ultimately rotated by 180°.*

Question 42

Visualize **z = √(2) (cos 45° + *i* sin 45°)** on the complex plane.

Answer 42

Question 43

The plot of z = √(2) (cos 45° + *i* sin 45°) is below. Disregarding any change affecting the angle, what would **z⁴** look like?

Answer 43

Question 44

The plot of z = √(2) (cos 45° + *i* sin 45°) is below. Disregarding any change affecting the modulus, what would **z⁴** look like?

Answer 44

Question 45

What are the steps to solving **(−2 + 2√(3) *i* )³**?

Answer 45

* **Find the absolute value** r = √(a² + b²) = √((−2)² + (2√(3))²) = √(4 + 12) = 4 * **Find the angle** NOTE: Θ is in Quadrant II Θ = tan⁻¹ (b/a) = tan⁻¹ ((2√(3)/−2) = −60° + 180° (must add 180° to obtain opposite angle) = 120° * **Restate in polar form** z = r (cosΘ + *i* sinΘ) −2 + 2√(3) *i* = 4 (cos120° + *i* sin120°) * **Apply exponentiation** ​zⁿ = rⁿ (cos(n • Θ) + *i* sin(n • Θ)) (−2 + 2√(3) *i* )³ = 4³ (cos(120°•3) + *i* sin(120°•3)) = 64 (cos360° + *i* sin360°) = 64 (1) + 64*i* (0) = 64

Question 46

If z⁵ = 32, **what are the steps** to calculating **z = \_\_\_\_\_**?

Answer 46

* **Restate in polar form** z⁵ = 32 = r⁵ (cos5Θ + *i* sin5Θ) * **Find the absolute value** z⁵ = 32 r⁵ = 32 r = (32)^(1/5) r = 2 * **Find the angle** cos5Θ = 1 5Θ = 0° + 360°k (must add 360°k to include all possible angles) Θ = 0° + 72°k = 72° (where 0° _\<_ Θ _\<_ 90°) * **Plug in** z = r (cosΘ + *i* sinΘ) = 2 (cos72° + *i* sin72°) = 2cos72° + 2*i* sin72°) ≈ 0.618 + 1.902*i*