Complex NumbersSolving Quadratics

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

073 .. 2 xxge

073 .. 2 xxge

2193

22893

x

073 .. 2 xxge

2193

22893

x

2193 i

073 .. 2 xxge

2193

22893

x

2193 i

2193or

2193 ixix

073 .. 2 xxge

2193

22893

x OR

0732 xx

2193 i

2193or

2193 ixix

073 .. 2 xxge

2193

22893

x OR

0732 xx

2193 i

2193or

2193 ixix

04

1923 2

x

073 .. 2 xxge

2193

22893

x OR

0732 xx

2193 i

2193or

2193 ixix

04

1923 2

x

04

1923 2

2

ix

073 .. 2 xxge

2193

22893

x OR

0732 xx

2193 i

2193or

2193 ixix

04

1923 2

x

04

1923 2

2

ix

0219

23

219

23

ixix

073 .. 2 xxge

2193

22893

x OR

0732 xx

2193 i

2193or

2193 ixix

04

1923 2

x

04

1923 2

2

ix

0219

23

219

23

ixix

ixix219

23or

219

23

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

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

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

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 number

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 number

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 conjugate

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 ..

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

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

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

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 Operations

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

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

ii 2834

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

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

ii 2834 i 4

Subtraction 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

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

ii 2834 i 4

Subtraction ii 2834

i512

Multiplication

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

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)

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

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

Exercise 4A; 1 to 16 evens