X2 t01 03 argand diagram (2012)

The Argand Diagram

The Argand DiagramComplex numbers can be represented geometrically on an Argand Diagram.

The Argand DiagramComplex numbers can be represented geometrically on an Argand Diagram.

1 2 3 4-1-2-3-4

123

-3-2-1

x

y

The Argand DiagramComplex numbers can be represented geometrically on an Argand Diagram.

1 2 3 4-1-2-3-4

123

-3-2-1

x

y

(real axis)

The Argand DiagramComplex numbers can be represented geometrically on an Argand Diagram.

1 2 3 4-1-2-3-4

123

-3-2-1

x

y

(real axis)

(imaginary axis)

The Argand DiagramComplex numbers can be represented geometrically on an Argand Diagram.

1 2 3 4-1-2-3-4

123

-3-2-1

x

y

(real axis)

(imaginary axis)

A = 2

The Argand DiagramComplex numbers can be represented geometrically on an Argand Diagram.

1 2 3 4-1-2-3-4

123

-3-2-1

x

y

(real axis)

(imaginary axis)

A = 2

A

The Argand DiagramComplex numbers can be represented geometrically on an Argand Diagram.

1 2 3 4-1-2-3-4

123

-3-2-1

x

y

(real axis)

(imaginary axis)

A = 2B = -3i

A

The Argand DiagramComplex numbers can be represented geometrically on an Argand Diagram.

1 2 3 4-1-2-3-4

123

-3-2-1

x

y

(real axis)

(imaginary axis)

A = 2B = -3i

A

B

The Argand DiagramComplex numbers can be represented geometrically on an Argand Diagram.

1 2 3 4-1-2-3-4

123

-3-2-1

x

y

(real axis)

(imaginary axis)

A = 2B = -3iC = -2 + i

A

B

The Argand DiagramComplex numbers can be represented geometrically on an Argand Diagram.

1 2 3 4-1-2-3-4

123

-3-2-1

x

y

(real axis)

(imaginary axis)

A = 2B = -3iC = -2 + i

A

B

C

The Argand DiagramComplex numbers can be represented geometrically on an Argand Diagram.

1 2 3 4-1-2-3-4

123

-3-2-1

x

y

(real axis)

(imaginary axis)

A = 2B = -3iC = -2 + iD = 4 - i

A

B

C

The Argand DiagramComplex numbers can be represented geometrically on an Argand Diagram.

1 2 3 4-1-2-3-4

123

-3-2-1

x

y

(real axis)

(imaginary axis)

A = 2B = -3iC = -2 + iD = 4 - i

A

B

C

D

The Argand DiagramComplex numbers can be represented geometrically on an Argand Diagram.

1 2 3 4-1-2-3-4

123

-3-2-1

x

y

(real axis)

(imaginary axis)

A = 2B = -3iC = -2 + iD = 4 - iE = 4 + i

A

B

C

D

The Argand DiagramComplex numbers can be represented geometrically on an Argand Diagram.

1 2 3 4-1-2-3-4

123

-3-2-1

x

y

(real axis)

(imaginary axis)

A = 2B = -3iC = -2 + iD = 4 - iE = 4 + i

A

B

C

D

E

The Argand DiagramComplex numbers can be represented geometrically on an Argand Diagram.

1 2 3 4-1-2-3-4

123

-3-2-1

x

y

(real axis)

(imaginary axis)

A = 2B = -3iC = -2 + iD = 4 - iE = 4 + i

A

B

C

D

E

NOTE: Conjugates are reflected in the real (x) axis

The Argand DiagramComplex numbers can be represented geometrically on an Argand Diagram.

1 2 3 4-1-2-3-4

123

-3-2-1

x

y

(real axis)

(imaginary axis)

A = 2B = -3iC = -2 + iD = 4 - iE = 4 + i

A

B

C

D

E

NOTE: Conjugates are reflected in the real (x) axis

Every complex number can be represented by a unique point on the Argand Diagram.

Mod-Arg Form

y

xO

Mod-Arg FormModulusThe modulus of a complex number is the length of the vector OZy

xO

Mod-Arg Form

y

xO

ModulusThe modulus of a complex number is the length of the vector OZz = x + iy

Mod-Arg Form

y

xO

ModulusThe modulus of a complex number is the length of the vector OZz = x + iy

Mod-Arg Form

y

xO

ModulusThe modulus of a complex number is the length of the vector OZz = x + iy

x

y

Mod-Arg Form

y

xO

ModulusThe modulus of a complex number is the length of the vector OZz = x + iy

x

y 22

222

yxr

yxr

Mod-Arg Form

y

xO

ModulusThe modulus of a complex number is the length of the vector OZz = x + iy

x

y 22

222

yxr

yxr

zr

Mod-Arg Form

y

xO

ModulusThe modulus of a complex number is the length of the vector OZz = x + iy

x

y 22

222

yxr

yxr

zr

22 yxz

Mod-Arg Form

y

xO

ModulusThe modulus of a complex number is the length of the vector OZz = x + iy

x

y 22

222

yxr

yxr

zr

22 yxz

ArgumentThe argument of a complex number is the angle the vector OZ makes with the positive real (x) axis

Mod-Arg Form

y

xO

ModulusThe modulus of a complex number is the length of the vector OZz = x + iy

x

y 22

222

yxr

yxr

zr

22 yxz

ArgumentThe argument of a complex number is the angle the vector OZ makes with the positive real (x) axis

zarg

Mod-Arg Form

y

xO

ModulusThe modulus of a complex number is the length of the vector OZz = x + iy

x

y 22

222

yxr

yxr

zr

22 yxz

ArgumentThe argument of a complex number is the angle the vector OZ makes with the positive real (x) axis

zarg

xyz 1tanarg

Mod-Arg Form

y

xO

ModulusThe modulus of a complex number is the length of the vector OZz = x + iy

x

y 22

222

yxr

yxr

zr

22 yxz

ArgumentThe argument of a complex number is the angle the vector OZ makes with the positive real (x) axis

zarg

xyz 1tanarg zarg

ige 44ofargument andmodulus theFind ..

22 4444 i

ige 44ofargument andmodulus theFind ..

22 4444 i

ige 44ofargument andmodulus theFind ..

2432

22 4444 i

ige 44ofargument andmodulus theFind ..

44tan44arg 1i

2432

22 4444 i

ige 44ofargument andmodulus theFind ..

44tan44arg 1i

2432

1tan 1

22 4444 i

ige 44ofargument andmodulus theFind ..

44tan44arg 1i

2432

1tan 1

22 4444 i

ige 44ofargument andmodulus theFind ..

44tan44arg 1i

2432

1tan 1

4

22 4444 i

ige 44ofargument andmodulus theFind ..

44tan44arg 1i

2432

1tan 1

4

Every complex number can be written in terms of its modulus and argument

22 4444 i

iyxz

ige 44ofargument andmodulus theFind ..

44tan44arg 1i

2432

1tan 1

4

Every complex number can be written in terms of its modulus and argument

22 4444 i

iyxz

ige 44ofargument andmodulus theFind ..

44tan44arg 1i

2432

1tan 1

4

Every complex number can be written in terms of its modulus and argument

sincos irr

22 4444 i

iyxz

ige 44ofargument andmodulus theFind ..

44tan44arg 1i

2432

1tan 1

4

Every complex number can be written in terms of its modulus and argument

sincos irr sincos ir

22 4444 i

iyxz

ige 44ofargument andmodulus theFind ..

44tan44arg 1i

2432

1tan 1

4

Every complex number can be written in terms of its modulus and argument

sincos irr sincos ir

The mod-arg form of z is;

22 4444 i

iyxz

ige 44ofargument andmodulus theFind ..

44tan44arg 1i

2432

1tan 1

4

Every complex number can be written in terms of its modulus and argument

sincos irr sincos ir

The mod-arg form of z is; sincos irz

22 4444 i

iyxz

ige 44ofargument andmodulus theFind ..

44tan44arg 1i

2432

1tan 1

4

Every complex number can be written in terms of its modulus and argument

sincos irr sincos ir

The mod-arg form of z is; sincos irz

rcisz

22 4444 i

iyxz

zzrarg

where;

ige 44ofargument andmodulus theFind ..

44tan44arg 1i

2432

1tan 1

4

Every complex number can be written in terms of its modulus and argument

sincos irr sincos ir

The mod-arg form of z is; sincos irz

rcisz

iie.g. 44

iie.g. 44

424 cis

iie.g. 44

424 cis

iii 3

iie.g. 44

424 cis

iii 3

24

13 22

z

iie.g. 44

424 cis

iii 3

24

13 22

z3

1tanarg 1z

iie.g. 44

424 cis

iii 3

24

13 22

z3

1tanarg 1z

6

iie.g. 44

424 cis

iii 3

24

13 22

z3

1tanarg 1z

6

623 cisi

iie.g. 44

424 cis

iii 3

24

13 22

z3

1tanarg 1z

6

623 cisi

Exercise 4B; evens