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ONLINE: MATHEMATICS EXTENSION 2

Topic 2 COMPLEX NUMBERS

EXERCISE p2201

p001

Given the two complex numbers 1 22 30 3 60o oz z

find the real part, imaginary part, modulus and argument for each of the following

complex numbers:

1 1 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2/ /z z z z z z z z z z z z z z z z z z z z

Locate the results on Argand diagrams.

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p002

Given the two complex numbers 1 23 3 / 4 2 5 / 6z z

find the real part, imaginary part, modulus and argument for each of the following

complex numbers:

1 1 2 2 1 2 1 2 1 2 1 2/z z z z z z z z z z z z

Locate the results on Argand diagrams.

p003

Prove that 1

ln 142

ii

p004

Prove that b

ln( ) ln atana

a i b a i b i

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p005

Express the following in rectangular form for k = 0, 2, ,3, .., , 8.

/ 4

e 0,1,3, ,8i k

kz k

Plot the complex numbers on an Argand diagram.

p006

Prove the following using the properties of complex numbers

cos cos cos sin sin

sin sin cos cos sin

p007

Form the product z1 z2 and the quotient z1 / z2 for

1 23 cos / 6 sin / 6 1.5 cos 4 / 3 sin 4 / 3z i z i

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ANSWERS a001

(a)

Given the two complex numbers 1 22 30 3 60o oz z

find the real part, imaginary part, modulus and argument for each of the

following complex numbers:

1 1 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2/ /z z z z z z z z z z z z z z z z z z z z

Locate the results on Argand diagrams.

know

Rectangular form z x i y

Polar form cos sinz R i

Exponential form iz Re

Modulus 2 2z z z R x y

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Argument 1Imtan tan tan

Re

z y y ya

z x x x

Real part cosx R

Imaginary part siny R

1 2 1 2 1 2z z x x i y y

1 2 1 2 1 2z z x x i y y

1 2 1 2 1 2 1 2 2 1z z x x y y i x y x y

1 21

2 2 2

z zz

z z z

1 2

1 2

1 2 1 2 1 2 1 2

1 2 1 2 1 2

1 1 11 2

2 2 2

1 2 1 2 1 2

cos sin

cos sin

i

i

z z R R e R R

R R i

z R Re

z R R

R R i

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o

1 1 1 1 1

o

2 1 2 2 2

2 30 / 6 rad 2 cos / 6 1.7321 2sin / 6 1.0000

3 60 / 3 rad 3 cos / 3 1.5000 3sin / 3 2.5981

z R x y

z R x y

z Re(z) = x Im(z) = y |z| = R Arg(z) =

rad

Arg(z) =

deg

1z 1.7321 1.0000 2.0000 / 6 30

1z 1.7321 -1.0000 2.0000 - / 6 - 30

2z 1.5000 2.5981 3.0000 / 3 60

2z 1.5000 -2.5981 3.0000 - / 3 - 60

1 2z z 3.2321 3.5981 4.8366 0.8389 48.07

1 2z z 3.2321 -1.5981 3.6056 -0.4592 - 26.31

1 2z z 0.2321 -1.5981 1.6148 -1.4266 - 81.74

1 2z z 0.2321 3.5981 3.6056 1.5064 86.31

1 2z z 0 6.0000 6.0000 / 2 90

1 2z z 5.1962 -3.0000 6.0000 - / 6 - 30

1 2/z z 0.5774 -0.3333 0.6667 - / 6 - 30

1 2/z z 0 0.6667 0.6667 / 2 90

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a002

Given the two complex numbers 1 23 3 / 4 2 5 / 6z z

find the real part, imaginary part, modulus and argument for each of the

following complex numbers:

1 1 2 2 1 2 1 2 1 2 1 2/z z z z z z z z z z z z

Locate the results on Argand diagrams.

o

1 1 1

1 1

o

2 1 2

2 2

3 135 3 / 4 rad

3 cos 3 / 4 2.1213 3sin 3 / 4 2.1213

2 150 5 / 6 rad

2 cos 5 / 6 1.7321 2sin 5 / 6 1.0000

z R

x y

z R

x y

z Re(z) = x Im(z) = y |z| = R Arg(z) =

rad

Arg(z) =

deg

1z -2.1213 2.21213 3.0000 - 3 / 4 135.00

1z -2.1213 -2.1213 3.0000 - / 4 - 45.00

2z -1.7321 -1.0000 2.0000 - 5 /6 -150.00

2z -1.7321 -2.5981 1.0000 / 4 45.00

1 2z z -3.8534 1.1213 4.0132 2.8584 163.78

1 2z z -0.3893 3.1455 3.1455 1.6949 97.11

1 2z z 5.7956 -1.5529 6.0000 - 0.2618 -15.00

1 2/z z 0.3882 -1.4489 1.5000 -1.3090 - 75.00

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a003

Prove that 1

ln 142

ii

/ 4

/ 4

11 1 atan 1 / 4

2

1ln 1 ln / 4

2

ii

i

z i z R Arg z

z R e e

i e i

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a004

Prove that b

ln( ) ln atana

a i b a i b i

atan /

atan /

atan /

atan /

ln ln

ln ln

ln atan /

i b ai

i b a

i b a

z a i b z a i b R Arg z b a

z R e a i b e

a i b a i b e

a i b e

a i b i b a

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a005

Express the following in rectangular form for k = 0, 2, ,3, .., , 8.

/ 4

e 0,1,3, ,8i k

kz k

Plot the complex numbers on an Argand diagram.

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/ 4

0

1 1 1

2 2 2

3 3 3

e 0,1,3, ,8

1 / 4 0,1,3, ,8

Re cos / 4 Im sin / 4

cos / 4 sin / 4

1

1cos / 4 sin / 4 1

2

cos 2 / 4 sin 2 / 4

1cos 3 / 4 sin 3 / 4 1

2

i k

k

k k k

k k k k

k k k

z k

z Arg z k k

z x k z y k

z x i y k i k

z

z x i y i i

z x i y i i

z x i y i i

z

4 4 4

5 5 5

6 6 6

7 7 7

8 8 8

cos 4 / 4 sin 4 / 4 1

1cos 5 / 4 sin 5 / 4 1

2

cos 6 / 4 sin 6 / 4

1cos 7 / 4 sin 7 / 4 1

2

cos 8 / 4 sin 8 / 4 1

x i y i

z x i y i i

z x i y i i

z x i y i i

z x i y i

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a006

Prove the following using the properties of complex numbers

cos cos cos sin sin

sin sin cos cos sin

1

2

1 2

cos sin

cos sin

cos sin cos sin

cos cos sin sin cos sin sin cos

cos sin

cos cos cos sin sin

sin cos sin sin cos

i

i

i

z i e

z i e

z z i i e

i

i

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a007

Form the product z1 z2 and the quotient z1 / z2 for

1 23 cos / 6 sin / 6 1.5 cos 4 / 3 sin 4 / 3z i z i

1 2

/ 6 4 / 3

1 2

/ 6 4 / 3 3 / 2

1 2

/ 6 4 / 3 7 / 3

1 2

3 cos / 6 sin / 6 1.5 cos 4 / 3 sin 4 / 3

3 1.5

4.5 4.5

4.5 cos 3 / 2 sin 3 / 2 4.5

/ 2 2

2 cos 7 / 3 sin 7 / 3 1.7321 3

i i

i i

i i

z i z i

z e z e

z z e e

i i

z z e e

i i i