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HSC Extension 2 Maths > Complex Numbers > Flashcards

Complex Numbers Flashcards

1
Q

Define a rational number.

A

p⁄q where p and q are integers and q != 0

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2
Q

Define an integer.

A

A whole number.

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3
Q

Define a real number.

A

A rational or irrational numbers.

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4
Q

If z = a + ib

Re(z) = ?

A

a

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5
Q

If z = a + ib

Im(z) = ?

A

b

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6
Q

What two conditions must be fulfilled for two complex numbers to be equal?

A

Their real parts are equal.

Their imaginary parts are equal.

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7
Q

If you have (a + ib)(a - ib), what is it equal to and what type of number is it?

A

a^2 + b^2

A real number

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8
Q

Define a natural number.

A

A positive, whole number.

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9
Q

If z = a + ib,

what does z equal?

A

a - ib

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10
Q

The conjugate of a sum of complex numbers…

A

is equal to the sum of the conjuagtes of the complex numbers.

___________ __ __ __

So z1 + z2 + … + zn = z1 + z2 + … + zn

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11
Q

The conjugate of a product of complex numbers…

A

is equal to the product of the conjuagtes of the complex numbers.

___________ __ __ __

So z1 x z2 x … x zn = z1 x z2 x … x zn

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12
Q

What equation gives you the square roots of a complex number?

A

___________ __________

_____ / _____ / _____

√ a + ib = √ √a2 + b2 + a ± √ √a2 + b2 - a

2 2

Where ± is the same sign as that of ib.

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13
Q

What equation do you simultaneously solve to find the square root of a complex number?

A

Let (a + ib)2 = z

∴ a2 - b2 + 2abi = z

(1) a2 - b2 = Re(z)
(2) 2ab = Im(z)

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14
Q

How do you factorise the sum of two squares?

x2 + y2 = ?

A

Use i2 to turn it into a difference of two squares, then factorise.

x2 + y2 = x2 + i2y2

= (x + iy)(x - iy)

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15
Q

What is √ i equal to?

A

√ i = ± ( 1⁄√<span> 2 </span> - 1⁄√<span> 2 </span> i )

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16
Q

What are the two primary forms of coordinates on the complex number plane?

A

Rectangular Form - (x, iy)

Mod-Arg Form - r(cosΘ + isinΘ)

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17
Q

What is the complex number product rule?

A

z1z2 = r1r2[cis(Θ1 + Θ2)]

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18
Q

What is the complex number quotient rule?

A

z1 ⁄z2 = r1 ⁄r2 [cis(Θ1 - Θ2)]

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19
Q

What is an Argand Diagram?

A

The complex number plane.

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20
Q

What is z on an Argand diagram?

A

A reflection of z in the x-axis.

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21
Q

What geometric effect does multiplying complex number z by real number c have?

A

Multiplies the distance from the origin (modulus) by c.

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22
Q

What geometric effect does multiplying complex number z by i have?

A

Rotates (the arguement of) z around the origin by 90º anticlockwise.

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23
Q

What geometric effect does multiplying complex number z by cisπ⁄2 have?

A

Rotates (the arguement of) z around the origin by 90º anticlockwise.

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24
Q

What geometric effect does multiplying complex number z by cisΘ have?

A

Rotates (the arguement of) z around the origin by Θ anticlockwise.

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25
What geometric effect does multiplying complex number z by complex number w have?
Multiplies their modulii and adds their arguements.
26
What geometric effect does dividing complex number z by complex number w have?
Divides their modulii and subtracts their arguements.
27
z x z = ?
[z]2
28
[z] = ?
[z]
29
z + z = ?
2Re(z)
30
z - z = ?
2Im(z)
31
z + w = ?
z + w
32
z x w = ?
z x w
33
z⁄w = ?
\_\_ | (z⁄w)
34
[z][w] = ?
[zw]
35
[z]⁄[w] = ?
[z⁄w]
36
arg(z) + arg(w) = ?
arg(zw)
37
arg(z) - arg(w) = ?
arg(z⁄w)
38
If z = rcisΘ, What is the value of z2?
r2cis2Θ
39
If z = rcisΘ, What is the value of z-1?
1⁄rcis-Θ
40
If z = rcis(Θ), What is the value of zn?
rncis(nΘ)
41
When are two vectors equal?
When they have the same length and direction.
42
What is a prime vector?
A vector that originates from the origin.
43
Why must [z1 + z2] ≤ [z1] + [z2]?
As vectors, they form the three sides of a triangle. The longest side of a triangle is always smaller than the other two sides added together.
44
What is the complex locus equation for x = 3? What shape is it?
Re(z) = 3 Vertical Line
45
What is the complex locus equation for y = -2? What shape is it?
Im(z) = -2 Horizontal Line
46
What is the complex locus equation for: x2 + y2 = 16 What shape is it?
[z] = 4 Circle, radius 4, centre (0,0)
47
What is the complex locus equation for: (x - 2)2 + (y - 3)2 = 4 What shape is it?
[z - (2 + 3i)] = 2 Circle, radius 2, centre (2,3)
48
What is the locus of argz = π⁄6?
A line from the origin 30º above the *x*-axis. Does not include the origin.
49
What is the equation of a line from the origin 60º above the *x*-axis, that doesn't include the origin?
What is the locus of argz = π⁄3?
50
What is the locus of arg{z - (1 + 3i)} = π/2?
A line from (1, 3), 45º above the *x*-axis. Does not include (1, 3).
51
What is the locus of arg{z + 2 + i)} = -2π/3?
A line from (-2, -1), 120º below the x-axis. Does not include (-2, -1).
52
What is the locus of [z - z1] = [z - z2]?
A straight line equidistant from z1 and z2.
53
What shape does this equation form if Θ = 0? arg(z - z1 ⁄z - z2) = Θ
A straight line passing through z1 and z2, with a gap between z1 and z2, and not including z1 and z2.
54
What shape does this equation form if 0 \< Θ \< π⁄2? arg(z - z1 ⁄z - z2) = Θ
The major arc of a circle, going anticlockwise from z1 to z2, not including z1 and z2.
55
What shape does this equation form if Θ = π⁄2? arg(z - z1 ⁄z - z2) = Θ
A semicircle, going anticlockwise from z1 to z2, not including z1 and z2.
56
What shape does this equation form if π⁄2 \< Θ \< π? arg(z - z1 ⁄z - z2) = Θ
The minor arc of a circle, going anticlockwise from z1 to z2, not including z1 and z2.
57
What shape does this equation form if Θ = π? arg(z - z1 ⁄z - z2) = Θ
A staright line from z1 to z2, not including z1 and z2.
58
Which direction should you always sketch when drawing a locus with the form: arg(z - z1 ⁄z - z2) = Θ
Anticlockwise from z1 to z2.
59
State De Moivre's theorem.
(cisΘ)n = cis(nΘ)
60
sin(-Θ) = ?
-sin(Θ)
61
cos(-Θ) = ?
cos(Θ)
62
cos3Θ = ?
1⁄4cos3Θ + 3⁄4cosΘ
63
sin3Θ = ?
3⁄4sinΘ - 1⁄4sin3Θ
64
Describe a method for finding z if: zn = 1
zn = 1 zn = cis0 z = (cis0)1⁄n z = (cis(0, ±2π, ±4π...))1⁄n *number of terms equal to* *(n - 1)⁄2* z = cis(0), cis(±2π⁄n), cis(±4π⁄n)... Simplify
65
Describe a method for z if: zn = -1
zn = -1 zn = cisπ z = (cisπ)1⁄n z = (cis(π, ±3π, ±5π...))1⁄n *number of terms equal to* *(n - 1)⁄2* z = cis(0), cis(±3π⁄n), cis(±5π⁄n)... Simplify

HSC Extension 2 Maths (3 decks)

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