# X2 T01 01 definitions (2010)

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• Complex Numbers012 x

• Complex Numbers012 x

solutionsrealno12 x

• Complex Numbers

In order to solve this equation we define a new number

solutionsrealno12 x

• Complex Numbers

In order to solve this equation we define a new number

solutionsrealno12 x

numberimaginary an is1or 1 2

i ii

• Complex Numbers

In order to solve this equation we define a new number

solutionsrealno12 x

numberimaginary an is1or 1 2

i ii

ixx

x

1

12

• 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 OR0732 xx

2193 i

2193or

2193 ixix

• 073 .. 2 xxge

2193

22893

x OR0732 xx

2193 i

2193or

2193 ixix

04

1923 2

x

• 073 .. 2 xxge

2193

22893

x OR0732 xx

2193 i

2193or

2193 ixix

04

1923 2

x

04

1923 2

2

ix

• 073 .. 2 xxge

2193

22893

x OR0732 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 OR0732 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 4Subtraction

• Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition ii 2834

i 4Subtraction ii 2834

• Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition ii 2834

i 4Subtraction ii 2834

i512

• Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition ii 2834

i 4Subtraction ii 2834

i512Multiplication

• Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition ii 2834

i 4Subtraction ii 2834

i512Multiplication

ii 2834

• Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition ii 2834

i 4Subtraction ii 2834

i512Multiplication

ii 2834 2624832 iii

• Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition ii 2834

i 4Subtraction ii 2834

i512Multiplication

ii 2834 2624832 iii

63232 i

• Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition ii 2834

i 4Subtraction ii 2834

i512Multiplication

ii 2834 2624832 iii

63232 ii3226

• Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition ii 2834

i 4Subtraction ii 2834

i512Multiplication

ii 2834 2624832 iii

63232 ii3226

Division (Realising The Denominator)

• Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition ii 2834

i 4Subtraction ii 2834

i512Multiplication

ii 2834 2624832 iii

63232 ii3226

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 4Subtraction ii 2834

i512Multiplication

ii 2834 2624832 iii

63232 ii3226

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 4Subtraction ii 2834

i512Multiplication

ii 2834 2624832 iii

63232 ii3226

Division (Realising The Denominator)

i

i28

34

ii

2828

464624832

ii

i

i

348

3419

681638

• Exercise 4A; 1 to 16 evens

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