Geometrical Representation of

Complex Numbers

Geometrical Representation of

Complex Numbers

Complex numbers can be represented on the Argand Diagram as vectors.

Geometrical Representation of

Complex Numbers

Complex numbers can be represented on the Argand Diagram as vectors.

y

x

Geometrical Representation of

Complex Numbers

Complex numbers can be represented on the Argand Diagram as vectors.

y

x

iyxz

Geometrical Representation of

Complex Numbers

Complex numbers can be represented on the Argand Diagram as vectors.

y

x

iyxz

Geometrical Representation of

Complex Numbers

Complex numbers can be represented on the Argand Diagram as vectors.

y

x

iyxz

The advantage of using vectors is that they can be moved around the

Argand Diagram

Geometrical Representation of

Complex Numbers

Complex numbers can be represented on the Argand Diagram as vectors.

y

x

iyxz

The advantage of using vectors is that they can be moved around the

Argand Diagram

Geometrical Representation of

Complex Numbers

Complex numbers can be represented on the Argand Diagram as vectors.

y

x

iyxz

The advantage of using vectors is that they can be moved around the

Argand Diagram

No matter where the vector is placed its length (modulus) and the angle

made with the x axis (argument) is constant

Geometrical Representation of

Complex Numbers

Complex numbers can be represented on the Argand Diagram as vectors.

y

x

iyxz

The advantage of using vectors is that they can be moved around the

Argand Diagram

No matter where the vector is placed its length (modulus) and the angle

made with the x axis (argument) is constant

A vector always

represents

HEAD minus TAIL

Addition / Subtraction

Addition / Subtraction

y

x

Addition / Subtraction

y

x

1z

Addition / Subtraction

y

x

1z

2z

Addition / Subtraction

y

x

1z

2z

To add two complex numbers, place the vectors “head to tail”

Addition / Subtraction

y

x

1z

2z

To add two complex numbers, place the vectors “head to tail”

21 zz

Addition / Subtraction

y

x

1z

2z

To add two complex numbers, place the vectors “head to tail”

21 zz

Addition / Subtraction

y

x

1z

2z

To add two complex numbers, place the vectors “head to tail”

21 zz

To subtract two complex numbers, place the vectors “head to head” (or

add the negative vector)

Addition / Subtraction

y

x

1z

2z

To add two complex numbers, place the vectors “head to tail”

21 zz

To subtract two complex numbers, place the vectors “head to head” (or

add the negative vector)

21 zz

Addition / Subtraction

y

x

1z

2z

To add two complex numbers, place the vectors “head to tail”

21 zz

To subtract two complex numbers, place the vectors “head to head” (or

add the negative vector)

21 zz

Addition / Subtraction

y

x

1z

2z

To add two complex numbers, place the vectors “head to tail”

21 zz

To subtract two complex numbers, place the vectors “head to head” (or

add the negative vector)

21 zz 2121 and

diagonals; twohas vectors

addingby formed ramparallelog the

:

zzzz

NOTE

Addition / Subtraction

y

x

1z

2z

To add two complex numbers, place the vectors “head to tail”

21 zz

To subtract two complex numbers, place the vectors “head to head” (or

add the negative vector)

21 zz 2121 and

diagonals; twohas vectors

addingby formed ramparallelog the

:

zzzz

NOTE

Addition / Subtraction

y

x

1z

2z

To add two complex numbers, place the vectors “head to tail”

21 zz

To subtract two complex numbers, place the vectors “head to head” (or

add the negative vector)

21 zz 2121 and

diagonals; twohas vectors

addingby formed ramparallelog the

:

zzzz

NOTE

Trianglar Inequality

In any triangle a side will be shorter than the sum of the other two sides

Addition / Subtraction

y

x

1z

2z

To add two complex numbers, place the vectors “head to tail”

21 zz

To subtract two complex numbers, place the vectors “head to head” (or

add the negative vector)

21 zz 2121 and

diagonals; twohas vectors

addingby formed ramparallelog the

:

zzzz

NOTE

Trianglar Inequality

In any triangle a side will be shorter than the sum of the other two sides

BCABACABC ;In

Addition / Subtraction

y

x

1z

2z

To add two complex numbers, place the vectors “head to tail”

21 zz

To subtract two complex numbers, place the vectors “head to head” (or

add the negative vector)

21 zz 2121 and

diagonals; twohas vectors

addingby formed ramparallelog the

:

zzzz

NOTE

Trianglar Inequality

In any triangle a side will be shorter than the sum of the other two sides

BCABACABC ;In (equality occurs when AC is a straight line)

Addition / Subtraction

y

x

1z

2z

To add two complex numbers, place the vectors “head to tail”

21 zz

To subtract two complex numbers, place the vectors “head to head” (or

add the negative vector)

21 zz 2121 and

diagonals; twohas vectors

addingby formed ramparallelog the

:

zzzz

NOTE

Trianglar Inequality

In any triangle a side will be shorter than the sum of the other two sides

BCABACABC ;In (equality occurs when AC is a straight line)

2121 zzzz

Addition

If a point A represents and point B

represents then point C representing

is such that the points OACB

form a parallelogram.

1z

2z

21 zz

Addition

If a point A represents and point B

represents then point C representing

is such that the points OACB

form a parallelogram.

1z

2z

21 zz

Subtraction

If a point D represents

and point E represents

then the points ODEB form a

parallelogram.

Note:

1z

12 zz

12 zzAB

12arg zz

1995..ge y

x

P

O

Q

The diagram shows a complex plane with origin O.

The points P and Q represent the complex numbers z and w respectively.

Thus the length of PQ is wz

wzwzi that Show

1995..ge y

x

P

O

Q

The diagram shows a complex plane with origin O.

The points P and Q represent the complex numbers z and w respectively.

Thus the length of PQ is wz

wzwzi that Show

1995..ge y

x

P

O

Q

The diagram shows a complex plane with origin O.

The points P and Q represent the complex numbers z and w respectively.

Thus the length of PQ is wz

wzwzi that Show

zOP is oflength The

1995..ge y

x

P

O

Q

The diagram shows a complex plane with origin O.

The points P and Q represent the complex numbers z and w respectively.

Thus the length of PQ is wz

wzwzi that Show

zOP is oflength The

wOQ is oflength The

1995..ge y

x

P

O

Q

The diagram shows a complex plane with origin O.

The points P and Q represent the complex numbers z and w respectively.

Thus the length of PQ is wz

wzwzi that Show

zOP is oflength The

wOQ is oflength The

wzPQ is oflength The

1995..ge y

x

P

O

Q

The diagram shows a complex plane with origin O.

The points P and Q represent the complex numbers z and w respectively.

Thus the length of PQ is wz

wzwzi that Show

zOP is oflength The

wOQ is oflength The

wzPQ is oflength Thewzwz

OPQ

on inequalityr triangula theUsing

1995..ge y

x

P

O

Q

The diagram shows a complex plane with origin O.

The points P and Q represent the complex numbers z and w respectively.

Thus the length of PQ is wz

wzwzi that Show

zOP is oflength The

wOQ is oflength The

wzPQ is oflength Thewzwz

OPQ

on inequalityr triangula theUsing

(ii) Construct the point R representing z + w, What can be said about the

quadrilateral OPRQ?

1995..ge y

x

P

O

Q

The diagram shows a complex plane with origin O.

The points P and Q represent the complex numbers z and w respectively.

Thus the length of PQ is wz

wzwzi that Show

zOP is oflength The

wOQ is oflength The

wzPQ is oflength Thewzwz

OPQ

on inequalityr triangula theUsing

(ii) Construct the point R representing z + w, What can be said about the

quadrilateral OPRQ?

R

1995..ge y

x

P

O

Q

The diagram shows a complex plane with origin O.

The points P and Q represent the complex numbers z and w respectively.

Thus the length of PQ is wz

wzwzi that Show

zOP is oflength The

wOQ is oflength The

wzPQ is oflength Thewzwz

OPQ

on inequalityr triangula theUsing

(ii) Construct the point R representing z + w, What can be said about the

quadrilateral OPRQ?

R

OPRQ is a parallelogram

?about said becan what , If z

wwzwziii

?about said becan what , If z

wwzwziii

wzwz

?about said becan what , If z

wwzwziii

wzwz i.e. diagonals in OPRQ are =

?about said becan what , If z

wwzwziii

wzwz i.e. diagonals in OPRQ are =

rectangle a is OPRQ

?about said becan what , If z

wwzwziii

wzwz i.e. diagonals in OPRQ are =

rectangle a is OPRQ

2argarg

zw

?about said becan what , If z

wwzwziii

wzwz i.e. diagonals in OPRQ are =

rectangle a is OPRQ

2argarg

zw

2arg

z

w

?about said becan what , If z

wwzwziii

wzwz i.e. diagonals in OPRQ are =

rectangle a is OPRQ

2argarg

zw

2arg

z

wimaginarypurely is

z

w

?about said becan what , If z

wwzwziii

wzwz i.e. diagonals in OPRQ are =

rectangle a is OPRQ

2argarg

zw

2arg

z

wimaginarypurely is

z

w

Multiplication

?about said becan what , If z

wwzwziii

wzwz i.e. diagonals in OPRQ are =

rectangle a is OPRQ

2argarg

zw

2arg

z

wimaginarypurely is

z

w

Multiplication

2121 zzzz

?about said becan what , If z

wwzwziii

wzwz i.e. diagonals in OPRQ are =

rectangle a is OPRQ

2argarg

zw

2arg

z

wimaginarypurely is

z

w

Multiplication

2121 zzzz 2121 argargarg zzzz

?about said becan what , If z

wwzwziii

wzwz i.e. diagonals in OPRQ are =

rectangle a is OPRQ

2argarg

zw

2arg

z

wimaginarypurely is

z

w

Multiplication

2121 zzzz 2121 argargarg zzzz

21212211 cisrrcisrcisr

?about said becan what , If z

wwzwziii

wzwz i.e. diagonals in OPRQ are =

rectangle a is OPRQ

2argarg

zw

2arg

z

wimaginarypurely is

z

w

Multiplication

2121 zzzz 2121 argargarg zzzz

21212211 cisrrcisrcisr

2

2121

by multiplied islength its and

by iseanticlockw rotated is vector the,by multiply weif ..

r

zzzei

If we multiply by the vector

OA will rotate by an angle of in an

anti-clockwise direction. If we

multiply by it will also multiply

the length of OA by a factor of r

1z cis

rcis

If we multiply by the vector

OA will rotate by an angle of in an

anti-clockwise direction. If we

multiply by it will also multiply

the length of OA by a factor of r

1z cis

rcis

ii 2

sin2

cos

Note: 1iz will rotate OA anticlockwise 90 degrees.

If we multiply by the vector

OA will rotate by an angle of in an

anti-clockwise direction. If we

multiply by it will also multiply

the length of OA by a factor of r

1z cis

rcis

ii 2

sin2

cos

Note: 1iz will rotate OA anticlockwise 90 degrees.

2by iseanticlockwrotation a is by tion Multiplica

i

If we multiply by the vector

OA will rotate by an angle of in an

anti-clockwise direction. If we

multiply by it will also multiply

the length of OA by a factor of r

1z cis

rcis

ii 2

sin2

cos

Note: 1iz will rotate OA anticlockwise 90 degrees.

REMEMBER: A vector is HEAD minus TAIL

2by iseanticlockwrotation a is by tion Multiplica

i

y

x

A

O

C

B

)1(D

y

x

A

O

C

B

)1(D

DC DA i

y

x

A

O

C

B

)1(D

DC DA i

1 1C i

y

x

A

O

C

B

)1(D

DC DA i

1 1C i

1 1

1

C i

i i

y

x

A

O

C

B

)1(D

DC DA i

B A DC

1 1C i

1 1

1

C i

i i

y

x

A

O

C

B

)1(D

DC DA i

B A DC

1B C

1 1C i

1 1

1

C i

i i

y

x

A

O

C

B

)1(D

DC DA i

B A DC

1B C

1 1C i

1 1

1

C i

i i

1

1

B i

i i

y

x

A

O

C

B

)1(D

DC DA i

B A DC

1B C

OR

1 1C i

1 1

1

C i

i i

1

1

B i

i i

y

x

A

O

C

B

)1(D

DC DA i

B A DC

1B C

OR

1 1C i

1 1

1

C i

i i

1

1

B i

i i

B C DA

y

x

A

O

C

B

)1(D

DC DA i

B A DC

1B C

OR

1 ( 1)B i i

1 1C i

1 1

1

C i

i i

1

1

B i

i i

B C DA

y

x

A

O

C

B

)1(D

DC DA i

B A DC

1B C

OR

1 ( 1)B i i

1 1C i

1 1

1

C i

i i

1

1

B i

i i

B C DA

1i i

y

x

A

O

C

B

)1(D

DC DA i

B A DC

1B C

OR

1 ( 1)B i i

OR

1 1C i

1 1

1

C i

i i

1

1

B i

i i

B C DA

1i i

y

x

A

O

C

B

)1(D

DC DA i

B A DC

1B C

OR

1 ( 1)B i i

OR

24

DB cis DA

1 1C i

1 1

1

C i

i i

1

1

B i

i i

B C DA

1i i

y

x

A

O

C

B

)1(D

DC DA i

B A DC

1B C

OR

1 ( 1)B i i

OR

24

DB cis DA

1 2 cos sin ( 1)4 4

B i

1 1C i

1 1

1

C i

i i

1

1

B i

i i

B C DA

1i i

y

x

A

O

C

B

)1(D

DC DA i

B A DC

1B C

OR

1 ( 1)B i i

OR

24

DB cis DA

1 2 cos sin ( 1)4 4

B i

1 1C i

1 1

1

C i

i i

1

1

B i

i i

B C DA

1i i

1 ( 1) 1B i

y

x

A

O

C

B

)1(D

DC DA i

B A DC

1B C

OR

1 ( 1)B i i

OR

24

DB cis DA

1 2 cos sin ( 1)4 4

B i

1 1C i

1 1

1

C i

i i

1

1

B i

i i

B C DA

1i i

1 ( 1) 1B i

1 1i i

y

x

A

O

C

B

)1(D

DC DA i

B A DC

1B C

OR

1 ( 1)B i i

OR

24

DB cis DA

1 2 cos sin ( 1)4 4

B i

1 1C i

1 1

1

C i

i i

1

1

B i

i i

B C DA

1i i

1 ( 1) 1B i

1 1i i

1i i

2000..ge y

x

A

O

C

In the Argand Diagram, OABC is a rectangle, where OC = 2OA.

The vertex A corresponds to the complex number

B

2000..ge y

x

A

O

C

In the Argand Diagram, OABC is a rectangle, where OC = 2OA.

The vertex A corresponds to the complex number

B

? toscorrespondnumber complex What Ci

2000..ge y

x

A

O

C

In the Argand Diagram, OABC is a rectangle, where OC = 2OA.

The vertex A corresponds to the complex number

B

? toscorrespondnumber complex What Ci

2OC OA i

2000..ge y

x

A

O

C

In the Argand Diagram, OABC is a rectangle, where OC = 2OA.

The vertex A corresponds to the complex number

B

? toscorrespondnumber complex What Ci

2OC OA i

2C i

2000..ge y

x

A

O

C

In the Argand Diagram, OABC is a rectangle, where OC = 2OA.

The vertex A corresponds to the complex number

B

? toscorrespondnumber complex What Ci

2OC OA i

(ii) What complex number corresponds to the point of intersection D of

the diagonals OB and AC?

2C i

2000..ge y

x

A

O

C

In the Argand Diagram, OABC is a rectangle, where OC = 2OA.

The vertex A corresponds to the complex number

B

? toscorrespondnumber complex What Ci

2OC OA i

(ii) What complex number corresponds to the point of intersection D of

the diagonals OB and AC?

diagonals bisect in a rectangle

2C i

2000..ge y

x

A

O

C

In the Argand Diagram, OABC is a rectangle, where OC = 2OA.

The vertex A corresponds to the complex number

B

? toscorrespondnumber complex What Ci

2OC OA i

(ii) What complex number corresponds to the point of intersection D of

the diagonals OB and AC?

diagonals bisect in a rectangle

2C i

midpoint of D AC

2000..ge y

x

A

O

C

In the Argand Diagram, OABC is a rectangle, where OC = 2OA.

The vertex A corresponds to the complex number

B

? toscorrespondnumber complex What Ci

2OC OA i

(ii) What complex number corresponds to the point of intersection D of

the diagonals OB and AC?

diagonals bisect in a rectangle

2C i

midpoint of D AC

2

A CD

2000..ge y

x

A

O

C

In the Argand Diagram, OABC is a rectangle, where OC = 2OA.

The vertex A corresponds to the complex number

B

? toscorrespondnumber complex What Ci

2OC OA i

(ii) What complex number corresponds to the point of intersection D of

the diagonals OB and AC?

diagonals bisect in a rectangle

2C i

midpoint of D AC

2

A CD

2

2

iD

2000..ge y

x

A

O

C

In the Argand Diagram, OABC is a rectangle, where OC = 2OA.

The vertex A corresponds to the complex number

B

? toscorrespondnumber complex What Ci

2OC OA i

(ii) What complex number corresponds to the point of intersection D of

the diagonals OB and AC?

diagonals bisect in a rectangle

1

2D i

2C i

midpoint of D AC

2

A CD

2

2

iD

Examples

Examples

3OB OA cis

Examples

3OB OA cis

13

B cis

Examples

3OB OA cis

iB2

3

2

1

13

B cis

Examples

3OB OA cis

iB2

3

2

1

13

B cis

OD OB i

Examples

3OB OA cis

iB2

3

2

1

iBD

13

B cis

OD OB i

Examples

3OB OA cis

iB2

3

2

1

iBD

iD2

1

2

3

13

B cis

OD OB i

Examples

3OB OA cis

iB2

3

2

1

iBD

iD2

1

2

3

13

B cis

OD OB i C B OD

Examples

3OB OA cis

iB2

3

2

1

iBD

iD2

1

2

3

1 3 3 1

2 2 2 2C i i

13

B cis

OD OB i C B OD

Examples

3OB OA cis

iB2

3

2

1

iBD

iD2

1

2

3

1 3 3 1

2 2 2 2C i i

iC

2

31

2

31

13

B cis

OD OB i C B OD

( ) i AP AO i

P A i O A

( ) i AP AO i

P A i O A

11 0 zizP

( ) i AP AO i

P A i O A

11 0 zizP

11 izzP

( ) i AP AO i

P A i O A

11 0 zizP

11 izzP

11 ziP

( ) i AP AO i

P A i O A

11 0 zizP

11 izzP

11 ziP

( ) i AP AO i ( ) ii QB BO i

P A i O A

11 0 zizP

11 izzP

11 ziP

Q B i O B

( ) i AP AO i ( ) ii QB BO i

P A i O A

11 0 zizP

11 izzP

11 ziP

Q B i O B

iBBQ

( ) i AP AO i ( ) ii QB BO i

P A i O A

11 0 zizP

11 izzP

11 ziP

Q B i O B

iBBQ

21 ziQ

( ) i AP AO i ( ) ii QB BO i

P A i O A

11 0 zizP

11 izzP

11 ziP

Q B i O B

iBBQ

21 ziQ

2

QPM

( ) i AP AO i ( ) ii QB BO i

P A i O A

11 0 zizP

11 izzP

11 ziP

Q B i O B

iBBQ

21 ziQ

2

QPM

2

11 21 ziziM

( ) i AP AO i ( ) ii QB BO i

P A i O A

11 0 zizP

11 izzP

11 ziP

Q B i O B

iBBQ

21 ziQ

2

QPM

2

11 21 ziziM

2

2121 izzzzM

( ) i AP AO i ( ) ii QB BO i

Exercise 4K; 1 to 8, 11, 12, 13

Exercise 4L; odd

Terry Lee: Exercise 2.6