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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
Q

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

A

Multiplies their modulii and adds their arguements.

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26
Q

What geometric effect does dividing complex number z by complex number w have?

A

Divides their modulii and subtracts their arguements.

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27
Q

z x z = ?

A

[z]2

28
Q

[z] = ?

A

[z]

29
Q

z + z = ?

A

2Re(z)

30
Q

z - z = ?

A

2Im(z)

31
Q

z + w = ?

A

z + w

32
Q

z x w = ?

A

z x w

33
Q

z⁄w = ?

A

__

(z⁄w)

34
Q

[z][w] = ?

A

[zw]

35
Q

[z]⁄[w] = ?

A

[z⁄w]

36
Q

arg(z) + arg(w) = ?

A

arg(zw)

37
Q

arg(z) - arg(w) = ?

A

arg(z⁄w)

38
Q

If z = rcisΘ,

What is the value of z2?

A

r2cis2Θ

39
Q

If z = rcisΘ,

What is the value of z-1?

A

1⁄rcis-Θ

40
Q

If z = rcis(Θ),

What is the value of zn?

A

rncis(nΘ)

41
Q

When are two vectors equal?

A

When they have the same length and direction.

42
Q

What is a prime vector?

A

A vector that originates from the origin.

43
Q

Why must [z1 + z2] ≤ [z1] + [z2]?

A

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
Q

What is the complex locus equation for x = 3?

What shape is it?

A

Re(z) = 3

Vertical Line

45
Q

What is the complex locus equation for y = -2?

What shape is it?

A

Im(z) = -2

Horizontal Line

46
Q

What is the complex locus equation for:

x2 + y2 = 16

What shape is it?

A

[z] = 4

Circle, radius 4, centre (0,0)

47
Q

What is the complex locus equation for:

(x - 2)2 + (y - 3)2 = 4

What shape is it?

A

[z - (2 + 3i)] = 2

Circle, radius 2, centre (2,3)

48
Q

What is the locus of argz = π⁄6?

A

A line from the origin 30º above the x-axis.

Does not include the origin.

49
Q

What is the equation of a line from the origin 60º above the x-axis, that doesn’t include the origin?

A

What is the locus of argz = π⁄3?

50
Q

What is the locus of arg{z - (1 + 3i)} = π/2?

A

A line from (1, 3), 45º above the x-axis.

Does not include (1, 3).

51
Q

What is the locus of arg{z + 2 + i)} = -2π/3?

A

A line from (-2, -1), 120º below the x-axis.

Does not include (-2, -1).

52
Q

What is the locus of [z - z1] = [z - z2]?

A

A straight line equidistant from z1 and z2.

53
Q

What shape does this equation form if Θ = 0?

arg(z - z1 ⁄z - z2) = Θ

A

A straight line passing through z1 and z2, with a gap between z1 and z2, and not including z1 and z2.

54
Q

What shape does this equation form if 0 < Θ < π⁄2?

arg(z - z1 ⁄z - z2) = Θ

A

The major arc of a circle, going anticlockwise from z1 to z2, not including z1 and z2.

55
Q

What shape does this equation form if Θ = π⁄2?

arg(z - z1 ⁄z - z2) = Θ

A

A semicircle, going anticlockwise from z1 to z2, not including z1 and z2.

56
Q

What shape does this equation form if π⁄2 < Θ < π?

arg(z - z1 ⁄z - z2) = Θ

A

The minor arc of a circle, going anticlockwise from z1 to z2, not including z1 and z2.

57
Q

What shape does this equation form if Θ = π?

arg(z - z1 ⁄z - z2) = Θ

A

A staright line from z1 to z2, not including z1 and z2.

58
Q

Which direction should you always sketch when drawing a locus with the form:

arg(z - z1 ⁄z - z2) = Θ

A

Anticlockwise from z1 to z2.

59
Q

State De Moivre’s theorem.

A

(cisΘ)n = cis(nΘ)

60
Q

sin(-Θ) = ?

A

-sin(Θ)

61
Q

cos(-Θ) = ?

A

cos(Θ)

62
Q

cos3Θ = ?

A

1⁄4cos3Θ + 3⁄4cosΘ

63
Q

sin3Θ = ?

A

3⁄4sinΘ - 1⁄4sin3Θ

64
Q

Describe a method for finding z if:

zn = 1

A

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
Q

Describe a method for z if:

zn = -1

A

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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