x2 t01 03 argand diagram (2013)

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

y4

(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

y4

(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

y4

(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

y4

(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

y4

(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

y4

(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

y4

(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

F = -(4 - i)= - 4 + i

F

NOTE: Opposites are rotated 180°

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

y4

(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

F = -(4 - i)= - 4 + i

F

NOTE: Opposites are rotated 180°

G = i(4 - i)= 1+4i

GNOTE: Multiply by i results in 90° rotation

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

y4

(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.

F = -(4 - i)= - 4 + i

F

NOTE: Opposites are rotated 180°

NOTE: Multiply by i results in 90° rotation

G = i(4 - i)= 1+4i

G

Locus in Terms of Complex Numbers

Horizontal and Vertical Lines

Locus in Terms of Complex Numbers

Horizontal and Vertical Lines

y

x

c

cz Im

Locus in Terms of Complex Numbers

Horizontal and Vertical Lines

y

x

y

x

c

cz Im

k

kz Re

Circlesy

x

R

R

R

R2Rzz

Circlesy

x

R

R

R

R2Rzz

e.g.Find and describe the locus of points in the Argand diagram; (i) 4 4 49z i z i

Circlesy

x

R

R

R

R2Rzz

e.g.Find and describe the locus of points in the Argand diagram; (i) 4 4 49z i z i (4 ) (4 ) (4 )(4 ) 49zz i z i z i i

Circlesy

x

R

R

R

R2Rzz

e.g.Find and describe the locus of points in the Argand diagram; (i) 4 4 49z i z i (4 ) (4 ) (4 )(4 ) 49zz i z i z i i

2 2

4( ) (4 )(4 ) 49

4( ) (4 )(4 ) 49

4( ) (4 )(4 ) 49

8 2 16 1 49

zz z z iz iz i i

z z z z iz iz i i

z z z z iz iz i i

x y x y

Circlesy

x

R

R

R

R2Rzz

e.g.Find and describe the locus of points in the Argand diagram; (i) 4 4 49z i z i (4 ) (4 ) (4 )(4 ) 49zz i z i z i i

2 28 16 2 1 49x x y y

2 2

4( ) (4 )(4 ) 49

4( ) (4 )(4 ) 49

4( ) (4 )(4 ) 49

8 2 16 1 49

zz z z iz iz i i

z z z z iz iz i i

z z z z iz iz i i

x y x y

Circlesy

x

R

R

R

R2Rzz

e.g.Find and describe the locus of points in the Argand diagram; (i) 4 4 49z i z i (4 ) (4 ) (4 )(4 ) 49zz i z i z i i

2 24 1 49x y

2 2

4( ) (4 )(4 ) 49

4( ) (4 )(4 ) 49

4( ) (4 )(4 ) 49

8 2 16 1 49

zz z z iz iz i i

z z z z iz iz i i

z z z z iz iz i i

x y x y

2 28 16 2 1 49x x y y

Circlesy

x

R

R

R

R2Rzz

e.g.Find and describe the locus of points in the Argand diagram; (i) 4 4 49z i z i (4 ) (4 ) (4 )(4 ) 49zz i z i z i i

14:centre ,units7:radius

Locus is a circle

2 2

4( ) (4 )(4 ) 49

4( ) (4 )(4 ) 49

4( ) (4 )(4 ) 49

8 2 16 1 49

zz z z iz iz i i

z z z z iz iz i i

z z z z iz iz i i

x y x y

2 24 1 49x y

2 28 16 2 1 49x x y y

Circlesy

x

R

R

R

R2Rzz

e.g.Find and describe the locus of points in the Argand diagram; (i) 4 4 49z i z i (4 ) (4 ) (4 )(4 ) 49zz i z i z i i

14:centre ,units7:radius

Locus is a circle

2Rzz Locus is a circle

centre ωradius R

2 2

4( ) (4 )(4 ) 49

4( ) (4 )(4 ) 49

4( ) (4 )(4 ) 49

8 2 16 1 49

zz z z iz iz i i

z z z z iz iz i i

z z z z iz iz i i

x y x y

2 24 1 49x y

2 28 16 2 1 49x x y y

1 1( ) 1iiz z

1 1( ) 1iiz z

z z zz

1 1( ) 1iiz z

z z zz 2 22x x y

1 1( ) 1iiz z

z z zz 2 22x x y

2 2

2 2

2 0( 1) 1x x yx y

1 1( ) 1iiz z

z z zz 2 22x x y

2 2

2 2

2 0( 1) 1x x yx y

centre: 1 0,radius: 1 unit

Locus is a circle

1 1( ) 1iiz z

z z zz 2 22x x y

2 2

2 2

2 0( 1) 1x x yx y

centre: 1 0,radius: 1 unit

Locus is a circle

y

x1 2

1 1( ) 1iiz z

z z zz 2 22x x y

2 2

2 2

2 0( 1) 1x x yx y

centre: 1 0,radius: 1 unit

Locus is a circle

excluding the point (0,0)

y

x1 2

Cambridge: Exercise 1C; 1 ace, 2 bdf, 3, 4 ace, 5 bdfh, 6, 10 to 15

1 1( ) 1iiz z

z z zz 2 22x x y

2 2

2 2

2 0( 1) 1x x yx y

centre: 1 0,radius: 1 unit

Locus is a circle

excluding the point (0,0)

y

x1 2