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Precalculus > Complex Numbers > Flashcards

Complex Numbers Flashcards

1
Q

What are

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

z-plane

Argand plane

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2
Q

(a + bi ) (a − bi ) = _____

A

a2 + b2
&
|a + bi |2

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3
Q

For complex number
z = a + bi,

|z| = _____

A

√(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.

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4
Q

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?

A

Absolute value

&

Angle
(or argument)

(or modulus)

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5
Q

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?

A

Real part

&

Imaginary part

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6
Q

In complex numbers,
what does
absolute value
refer to?

A

The

  • *absolute value**, or
  • *modulus**, gives the
  • *distance** of the number from the
  • *origin** in the complex plane.
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7
Q

In complex numbers,
what does
angle
refer to?

A

The

  • *angle**, or
  • *argument**, is the angle the number forms with the
  • *positive real axis**.
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8
Q

How do you find the

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

Θ = _____?

A

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.*
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9
Q

What are

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

Rectangular form:
a + bi

Polar form:
r (cosΘ + i sinΘ))

Exponential form:
r•​e<em>i</em> Θ

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10
Q

What
form is the complex number

2√(3) + 2i

written in?

A

Rectangular form

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11
Q

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

A

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.

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12
Q

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

A

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)).
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13
Q

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

A

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)
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14
Q

What
form is the complex number

4 (cos (π/6) +i sin (π/6))

written in?

A

Polar form

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15
Q

What
form is the complex number

4e(π/6)<em>i</em>

written in?

A

Exponential form

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16
Q

How do you state

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

z = _____

A

z = a + bi

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

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17
Q

How do you state

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

z = _____

A

z = r (cosΘ + i sinΘ)

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

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18
Q

How do you state

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

z = _____

A

z = r•​e<em>i</em> Θ

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

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19
Q

In what quadrant of the complex plane does

3 + 4i

lie?

A

Quadrant I

20
Q

In what quadrant of the complex plane does

−3 + 4i

lie?

A

Quadrant II

21
Q

In what quadrant of the complex plane does

−3 − 4i

lie?

A

Quadrant III

22
Q

In what quadrant of the complex plane does

3 − 4i

lie?

A

Quadrant IV

23
Q

Why is the
quadrant of a
complex number
important?

A

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°
24
Q

How do you restate

z = 4 (cos(π/6) + i sin(π/6))

in
rectangular form?

A

Expand
the parentheses in the
polar representation.

25
Q

What are the steps to restating

z = 2√(3) + 2i

in
polar form?

A
  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))
26
Q

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

A

z = r cos(Θ) + r sin(Θ) i

This results from using trigonometry in the right triangle formed by the number and the Real axis.

27
Q

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

A

r cos(Θ)

This results from using trigonometry in the right triangle formed by the number and the Real axis.

28
Q

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

A

r sin(Θ)

This results from using trigonometry in the right triangle formed by the number and the Real axis.

29
Q

Given

​z1 = r1 (cosΘ1 + i sinΘ1)
&
​z2 = r2 (cosΘ2 + i sinΘ2),

z1 • z2 = _____

A

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°)
30
Q

Given

​z1 = r1 (cosΘ1 + i sinΘ1)
&
​z2 = r2 (cosΘ2 + i sinΘ2),

z1 = _____
z2

A

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°)
31
Q

Given

​z1 = r1 e<em>i</em> Θ1
&
​z2 = r2 e<em>i</em> Θ2,

z1 • z2 = _____

A

z1 • z2 = (r1 • r2) e<em>i</em> (Θ1 + Θ2)

Example:

  • z1 = 9 e<em>i</em> π/4
    z2 = 3 e<em>i</em> π/12
  • z1 • z2 = r1 e<em>i</em> Θ1 • r2 e<em>i</em> Θ2
  • = (r1 • r2) e<em>i</em> (Θ1 + Θ2)
  • = (9 • 3) e<em>i</em> (π/4 + π/12)
  • = 27 e<em>i</em> (3π/12 + π/12)
  • = 27 ​e<em>i</em> (4π/12)
  • = 27 e<em>i</em> π/3
32
Q

Given

​z1 = r1 e<em>i</em> Θ1
&
​z2 = r2 e<em>i</em> Θ2,

z1 = _____
z2

A

z1 / z2 = (r1 / r2) e<em>i</em> (Θ1 − Θ2)

Example:

  • z1 = 9 e<em>i</em> π/4
    z2 = 3 e<em>i</em> π/12
  • z1 = z1 • z2−1
    z2
  • = 9 e<em>i</em> π/4 (3 e<em>i</em> π/12)−1
  • = 9 (1/3) e<em>i</em> π/4 − <em>i</em> π/12
  • = 3 e<em>i</em> (π/4 − π/12)
  • = 3 e<em>i</em> (3π/12 − π/12)
  • = 3 ​e<em>i</em> (2π/12)
  • = 3 e<em>i</em> π/6
33
Q

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**?
A

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)

34
Q

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**?
A

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)

35
Q

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**?
A

They will have the
same angle.

If the
angle of u is Θ, then the
angle of ū is −Θ.

So the successive multiplications have
no total rotation.

36
Q

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**?
A

The magnitude will be
z | u |2.

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.

37
Q

z = a + bi = _____
u c + di

A

z • ū
|u|2

  • z = a + bi
    u c + di
  • = a + bi • (c − di )
    c + di (c − di )
  • = (__a + bi ) (c − di )
    c2 + d2
  • = z • ū
    |u|2
38
Q

If

z1 is a complex number,

then how will the

  • *graph of z1 • i** appear
  • *relative to that of z1**?
A

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)
39
Q

Given

​z = r (cosΘ + i sinΘ)

zn = _____

A

​zn = rn (cos(n • Θ) + i sin(n • Θ))

40
Q

Given

z = 2(cos60° + i sin60°),

what is

  • *absolute value** of
  • *z3**?
A

8
|z3| = r3 = 23 = 8

Stretched by a factor of 2 three times, so ultimately stretched by a factor of 23 = 8.

41
Q

Given

z = 2(cos60° + i sin60°),

what is

  • *angle** of
  • *z3**?
A

180°
60° + 60°+ 60° = 3 • 60° = 180°

Rotated by +60° three times, so ultimately rotated by 180°.

42
Q

Visualize

z = √(2) (cos 45° + i sin 45°)

on the
complex plane.

A
43
Q

The plot of
z = √(2) (cos 45° + i sin 45°)
is below.

Disregarding any change affecting the angle,
what would
z4
look like?

A
44
Q

The plot of
z = √(2) (cos 45° + i sin 45°)
is below.

Disregarding any change affecting the modulus,
what would
z4
look like?

A
45
Q

What are the steps to solving

(−2 + 2√(3) i )3?

A
  • 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) + 64i (0)
    = 64
46
Q

If

z5 = 32,

what are the steps to calculating

z = _____?

A
  • 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° + 2i sin72°)
    ≈ 0.618 + 1.902i

Precalculus (9 decks)

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