X2 T01 01 definitions (2010)

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  • 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

    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

    Slide Number 1Slide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Slide Number 29Slide Number 30Slide Number 31Slide Number 32Slide Number 33Slide Number 34Slide Number 35Slide Number 36Slide Number 37Slide Number 38Slide Number 39Slide Number 40Slide Number 41Slide Number 42Slide Number 43Slide Number 44Slide Number 45Slide Number 46Slide Number 47Slide Number 48Slide Number 49Slide Number 50Slide Number 51Slide Number 52Slide Number 53Slide Number 54Slide Number 55Slide Number 56Slide Number 57Slide Number 58Slide Number 59Slide Number 60