Complex Numbers012 x

Solving Quadratics

Complex Numbers012 x

Solving Quadratics

solutionsrealno12 x

Complex Numbers

In order to solve this equation we define a new number

012 xSolving Quadratics

solutionsrealno12 x

Complex Numbers

In order to solve this equation we define a new number

012 xSolving Quadratics

solutionsrealno12 x

numberimaginary an is1or 1 2

i ii

Complex Numbers

In order to solve this equation we define a new number

012 xSolving Quadratics

solutionsrealno12 x

numberimaginary an is1or 1 2

i ii

ixx

x

112

All complex numbers (z) can be written as; z = x + iy

All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part

All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part

yz

xz

ImRe

All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part

yz

xz

ImRe

izge 53 ..

All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part

yz

xz

ImRe

izge 53 .. 3Re z 5Im z

All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part

yz

xz

ImRe

izge 53 .. 3Re z 5Im z

(2) If Re(z) = 0, then z is a pure imaginary numberiige 6,3 ..

All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part

yz

xz

ImRe

izge 53 .. 3Re z 5Im z

(2) If Re(z) = 0, then z is a pure imaginary numberiige 6,3 ..

(3) If Im(z) = 0, then z is a real number4,,,

43 .. ege

All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part

yz

xz

ImRe

izge 53 .. 3Re z 5Im z

(2) If Re(z) = 0, then z is a pure imaginary numberiige 6,3 ..

(3) If Im(z) = 0, then z is a real number4,,,

43 .. ege

(4) Every complex number z = x + iy, has a complex conjugateiyxz

All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part

yz

xz

ImRe

izge 53 .. 3Re z 5Im z

(2) If Re(z) = 0, then z is a pure imaginary numberiige 6,3 ..

(3) If Im(z) = 0, then z is a real number4,,,

43 .. ege

(4) Every complex number z = x + iy, has a complex conjugateiyxz

izge 72 .. iz 72

Complex Numbersx + iy

Complex Numbersx + iy

Real Numbersy=0

Complex Numbersx + iy

Real Numbersy=00

NumbersImaginary y

Complex Numbersx + iy

Real Numbersy=00

NumbersImaginary y

Rational Numbers

Irrational Numbers

Complex Numbersx + iy

Real Numbersy=00

NumbersImaginary y

Rational Numbers

Irrational Numbers

Fractions Integers

Complex Numbersx + iy

Real Numbersy=00

NumbersImaginary y

Rational Numbers

Irrational Numbers

Fractions IntegersNaturals

Zero

Complex Numbersx + iy

Real Numbersy=00

NumbersImaginary y

Rational Numbers

Irrational Numbers

Fractions IntegersNaturals

Zero

Negatives

Complex Numbersx + iy

Real Numbersy=00

NumbersImaginary y

Rational Numbers

Irrational Numbers

Fractions IntegersNaturals

Zero

Negatives

0,0NumbersImaginary

Pure

yx

Complex Numbersx + iy

Real Numbersy=00

NumbersImaginary y

Rational Numbers

Irrational Numbers

Fractions IntegersNaturals

Zero

Negatives

0,0NumbersImaginary

Pure

yx

NOTE: Imaginary numbers cannot be ordered

Basic OperationsAs i a surd, the operations with complex numbers are the same as surds

Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition

ii 2834 i 4

Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition

ii 2834 i 4

Subtraction ii 2834

i512

Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition

ii 2834 i 4

Subtraction ii 2834

i512

Multiplication ii 2834

Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition

ii 2834 i 4

Subtraction ii 2834

i512

Multiplication ii 2834

2624832 iii

Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition

ii 2834 i 4

Subtraction ii 2834

i512

Multiplication ii 2834

2624832 iii 63232 i

i3226

Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition

ii 2834 i 4

Subtraction ii 2834

i512

Multiplication ii 2834

2624832 iii 63232 i

i3226

Division (Realising The Denominator)

i

i28

34

Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition

ii 2834 i 4

Subtraction ii 2834

i512

Multiplication ii 2834

2624832 iii 63232 i

i3226

Division (Realising The Denominator)

i

i28

34

ii

2828

464624832

ii

Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition

ii 2834 i 4

Subtraction ii 2834

i512

Multiplication ii 2834

2624832 iii 63232 i

i3226

Division (Realising The Denominator)

i

i28

34

ii

2828

464624832

ii

i

i

348

3419

681638

Conjugate Basics 1 2 1 21 z z z z

Conjugate Basics 1 2 1 21 z z z z

1 2 1 22 z z z z

1 1

2 2

3 z zz z

Conjugate Basics 2 24 zz x y 1 2 1 21 z z z z

1 2 1 22 z z z z

1 1

2 2

3 z zz z

Conjugate Basics 2 24 zz x y 1 2 1 21 z z z z

1 2 1 22 z z z z

1 1

2 2

3 z zz z

15 zz zz

Conjugate Basics 2 24 zz x y 1 2 1 21 z z z z

1 2 1 22 z z z z

1 1

2 2

3 z zz z

15 zz zz

1e.g.

4 3i

Conjugate Basics 2 24 zz x y 1 2 1 21 z z z z

1 2 1 22 z z z z

1 1

2 2

3 z zz z

15 zz zz

1e.g.

4 3i4 316 9

i

4 325 25

i

Complex EquationsIf two complex numbers are equal, then their real parts are equal and their imaginary parts are equal

ibaibaei 2211If ..

21

21

then

bbaa

Complex EquationsIf two complex numbers are equal, then their real parts are equal and their imaginary parts are equal

ibaibaei 2211If ..

21

21

then

bbaa

Complex EquationsIf two complex numbers are equal, then their real parts are equal and their imaginary parts are equal

ibaibaei 2211If ..

. . (i) 2 3 4 2e g i x iy i

21

21

then

bbaa

Complex EquationsIf two complex numbers are equal, then their real parts are equal and their imaginary parts are equal

ibaibaei 2211If ..

. . (i) 2 3 4 2e g i x iy i

21

21

then

bbaa

2 2 3 3 4 2x iy ix y i

Complex EquationsIf two complex numbers are equal, then their real parts are equal and their imaginary parts are equal

ibaibaei 2211If ..

. . (i) 2 3 4 2e g i x iy i

21

21

then

bbaa

2 2 3 3 4 2x iy ix y i 2 3 4 and 3 2 2x y x y

Complex EquationsIf two complex numbers are equal, then their real parts are equal and their imaginary parts are equal

ibaibaei 2211If ..

. . (i) 2 3 4 2e g i x iy i

21

21

then

bbaa

2 2 3 3 4 2x iy ix y i 2 3 4 and 3 2 2x y x y

4 6 8 9 6 6

x yx y

Complex EquationsIf two complex numbers are equal, then their real parts are equal and their imaginary parts are equal

ibaibaei 2211If ..

. . (i) 2 3 4 2e g i x iy i

21

21

then

bbaa

2 2 3 3 4 2x iy ix y i 2 3 4 and 3 2 2x y x y

4 6 8 9 6 6

x yx y

5 1414 5

x

x

Complex EquationsIf two complex numbers are equal, then their real parts are equal and their imaginary parts are equal

ibaibaei 2211If ..

. . (i) 2 3 4 2e g i x iy i

21

21

then

bbaa

2 2 3 3 4 2x iy ix y i 2 3 4 and 3 2 2x y x y

4 6 8 9 6 6

x yx y

5 1414 5

x

x

14 16 , 5 5

x y

iiwziiwz ii

432 342

iiwziiwz ii

432 342

iiwziiwz

8624342

iiwziiwz ii

432 342

iiwziiwz

8624342

iz

iz

35

32

52 3

iiwziiwz ii

432 342

iiwziiwz

8624342

iz

iz

35

32

52 3

iiwi 34235

32

iiw34

3102

i

iw

35

32

64

610

iiwziiwz ii

432 342

iiwziiwz

8624342

iz

iz

35

32

52 3

iiwi 34235

32

iiw34

3102

i

iw

35

32

64

610

iziw35

32 ,

35

32

Cambridge: Exercise 1A; 1 to 10 ace etc, 11bd, 13 to 20

Patel: Exercise 4A; 28 ac