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