COMPLEX NUMBERS  Beertino  John  Yeong Hui  Yu Jie
CONTENTS    Beertino                     Yeong Hui   Approaches/pedagogy         A level syllabus   Diophantus’s proble...
APPROACHES/PEDAGOGY Axiomatic Approach    Common in textbooks.    Start by defining complex numbers as numbers of the f...
APPROACHES/PEDAGOGY Utilitarian Approach    Briefly describe Complex Numbers lead to the theory of fractals    It allow...
APPROACHES/PEDAGOGY Totalitarian Approach! ( Just kidding )Back to Table of contents                   Diophantus’s problem
APPROACHES/PEDAGOGY Historical Approach Why this approach?    Real questions faced by mathematicians.    Build on pre-...
APPROACHES/PEDAGOGY So, how does the approach goes?    First, bring about the quadratic problem.        Tapping on prio...
DIOPHANTUS’S PROBLEM   Diophantus Arithhmetica (C.E 275) A right-angled triangle has area 7    square units and perimeter...
SOLUTION AND PROBLEMApproaches/Pedagogy   Back to Table of contents   Root of Function
ROOT OF FUNCTIONDiophantus’s problem   Back to Table of contents   Cubic Example
ROOT OF FUNCTIONDiophantus’s problem   Back to Table of contents   Cubic Example
CUBIC EXAMPLE  Root of Function   Back to Table of contents
CUBIC EXAMPLE  Root of Function   Back to Table of contents
LASTLY   Root of Function   Back to Table of contents
A-LEVEL SYLLABUSBack to Table of contents   Pedagogical considerations
PEDAGOGICAL CONSIDERATIONS   Operations in complex plane is   similar but not exactly the same   as vector geometry (see ...
PEDAGOGICAL CONSIDERATIONS   Multimodal Representation and usage of similarity in    vector geometry for teaching of comp...
BASIC DEFINITONS        First defined by Leonard Euler, a swiss        mathematician, a complex number, denoted        by ...
ARGAND DIAGRAM                                                       z=x+yi   Im(z)                                       ...
EXTENSION FROM REAL NUMBERS(ENGAGING PRIOR KNOWLEDGE)                                                     The Real Axis   ...
ADDITION OF COMPLEX NUMBERS Complex Addition    Addition of 2 complex numbers        z1 = x1 + y1i, z2 = x2 + y2i      ...
MMR IN ADDITION Multimodal Representation used: Pictorial    Geometric Interpretation        Vector Addition       Im(z...
SUBTRACTION OF COMPLEX NUMBERS Complex Subtraction    Difference of 2 complex numbers        z1 = x1 + y1i, z2 = x2 + y...
MMR IN SUBTRACTION Multimodal Representation used: Pictorial    Geometric Interpretation        Vector Subtraction     ...
UNIQUENESS OF COMPLEX NUMBERS If two complex numbers are the same, i.e. z1 = z2, then their real parts  must be equal, an...
A-LEVEL SYLLABUSBack to Table of contents   Pedagogical Considerations
PEDAGOGICAL CONSIDERATIONS   Operations in complex plane is similar but not exactly the same as    vector geometry (see c...
PEDAGOGICAL CONSIDERATIONS   Properties…      of complex multiplication assumed       (commutative, associative, distrib...
MULTIPLICATION OF COMPLEX NUMBERS    Complex multiplication        Multiplication of 2 complex numbers           z1 = x...
MULTIPLICATION OF COMPLEX NUMBERS    Complex multiplication        Scalar Multiplication           z = x + yi, k real n...
MULTIPLICATION OF COMPLEX NUMBERS    i4n = I, i4n+1 = i, i4n+2 = -1, i4n+3 = -I for any integer n        Explore using G...
DIVISION OF COMPLEX NUMBERS Complex division    Division of 2 complex numbers (Realising the denominator)        z1 = x...
DIVISION OF COMPLEX NUMBERS Solve simultaneous equations  (using the four complex number operations) Finding square root...
COMPLEX CONJUGATES Let z = x + iy. The complex conjugate of z is given by z* = x – iy.     Conjugate pair: z and z*    ...
COMPLEX CONJUGATES Properties:  Exercise for students (direct verification)    1. Re(z*) = Re(z); Im(z*) = -Im(z)       ...
LEARNING DIFFICULTIES/COMMON MISTAKES In z = x + yi, x and y are always REAL numbers Solve equations using z directly or...
A-LEVEL SYLLABUSBack to Table of contents   Pedagogical Considerations
PEDAGOGICAL CONSIDERATION Start with a simple quadratic equation    Example: x2 + 2x + 2 = 0. Get students to observe a...
PEDAGOGICAL CONSIDERATION Direct attention to discriminant of quadratic equation    What can we say about the discrimina...
PEDAGOGICAL CONSIDERATION Examples on Solving for Complex Roots of Quadratic  Equations    Expose students to different ...
PEDAGOGICAL CONSIDERATION Different version: What if we are given one complex root?Example:If one of the roots α of the e...
PEDAGOGICAL CONSIDERATION                                                        Good to Fundamental Theorem of Algebra  ...
PEDAGOGICAL CONSIDERATION                                                   Good to Visualizing Complex Roots            ...
PEDAGOGICAL CONSIDERATION Extending from quadratic equations to cubic equations    Can we generalize to any polynomial? ...
LEARNING DIFFICULTIES/COMMON MISTAKES X: complex roots will always appear in conjugate pairs ‘No roots’ versus ‘No real ...
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Complex Number I - Presentation

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Complex Number I - Presentation

  1. 1. COMPLEX NUMBERS Beertino John Yeong Hui Yu Jie
  2. 2. CONTENTS Beertino Yeong Hui Approaches/pedagogy  A level syllabus Diophantus’s problem  Pedagogical Consideration roots of function  Multiplication Cubic Example and Division of complex numbers  Complex conjugates Yu Jie A level syllabus Pedagogical considerations Basic definition & John Argand Diagram  A Level syllabus Addition and  Pedagogical considerations Subtraction of complex numbers  Learning difficulties Uniqueness of Complex Numbers
  3. 3. APPROACHES/PEDAGOGY Axiomatic Approach  Common in textbooks.  Start by defining complex numbers as numbers of the form a+ib where a, b are real numbers.Back to Table of contents Diophantus’s problem
  4. 4. APPROACHES/PEDAGOGY Utilitarian Approach  Briefly describe Complex Numbers lead to the theory of fractals  It allows computer programmers to create realistic clouds and mountains in video games.Back to Table of contents Diophantus’s problem
  5. 5. APPROACHES/PEDAGOGY Totalitarian Approach! ( Just kidding )Back to Table of contents Diophantus’s problem
  6. 6. APPROACHES/PEDAGOGY Historical Approach Why this approach?  Real questions faced by mathematicians.  Build on pre-existing mathematical knowledge,  Quadratic formula  Roots.Back to Table of contents Diophantus’s problem
  7. 7. APPROACHES/PEDAGOGY So, how does the approach goes?  First, bring about the quadratic problem.  Tapping on prior knowledge  Quadratic formula.  Roots of an equation.  Followed with definition of root.  Then sub value into root to get a cognitive conflict.  Give another example, this time it’s cubic  Tap on prior knowledge again  Completing Square to get to Completing Cube (Cardano’s Method)  Solve to get a weird answer.  Show that weird answer is 4, and get another cognitive conflict.Back to Table of contents Diophantus’s problem
  8. 8. DIOPHANTUS’S PROBLEM  Diophantus Arithhmetica (C.E 275) A right-angled triangle has area 7 square units and perimeter 12 units. Find the lengths of its sides.Approaches/Pedagogy Back to Table of contents Root of Function
  9. 9. SOLUTION AND PROBLEMApproaches/Pedagogy Back to Table of contents Root of Function
  10. 10. ROOT OF FUNCTIONDiophantus’s problem Back to Table of contents Cubic Example
  11. 11. ROOT OF FUNCTIONDiophantus’s problem Back to Table of contents Cubic Example
  12. 12. CUBIC EXAMPLE Root of Function Back to Table of contents
  13. 13. CUBIC EXAMPLE Root of Function Back to Table of contents
  14. 14. LASTLY Root of Function Back to Table of contents
  15. 15. A-LEVEL SYLLABUSBack to Table of contents Pedagogical considerations
  16. 16. PEDAGOGICAL CONSIDERATIONS  Operations in complex plane is similar but not exactly the same as vector geometry (see complex multiplication and division)  Building on Prior Knowledge  Rules-Based Approach vs Theoretical UnderstandingSyllabusBack to table of contents (Teaching) Basic Definition
  17. 17. PEDAGOGICAL CONSIDERATIONS  Multimodal Representation and usage of similarity in vector geometry for teaching of complex addition and subtraction  Algebraic proof for uniqueness of complex numbers and should it be taught specifically  No ordering in complex plane, not appropriate to talk about z1 > z2  ordering is appropriate for modulus, since modulus of complex numbers are real valuesSyllabusBack to table of contents (Teaching) Basic Definition
  18. 18. BASIC DEFINITONS First defined by Leonard Euler, a swiss mathematician, a complex number, denoted by i, to be i2 = -1 In general, a complex number z can be written as where x denotes the real part and y denotes the imaginary partPedagogical considerationsBack to Table of contents Argand Diagram
  19. 19. ARGAND DIAGRAM z=x+yi Im(z) x : Real Part P(x,y) y : Imaginary y Part Important aspect, common student error is forgetting Re(z) 0 x that x,y are both real valuedBasic DefinitionsBack to Table of contents Addition
  20. 20. EXTENSION FROM REAL NUMBERS(ENGAGING PRIOR KNOWLEDGE) The Real Axis Im(z) (x-axis) represents the real number z line. y |z| In other words the real numbers just have the imaginary θ part to be zero. Re(z) 0 x e.g. 1 = 1 + 0 iBasic DefinitionsBack to Table of contents Addition
  21. 21. ADDITION OF COMPLEX NUMBERS Complex Addition  Addition of 2 complex numbers  z1 = x1 + y1i, z2 = x2 + y2i  z1 + z2 = (x1 + y1i) + (x2 + y2i) = (x1 + x2) + (y1 + y2) i  Addition of real and imaginary portions and summing the 2 parts up  Geometric Interpretation (vector addition)RationaleMultimodal Representation: Argand DiagramEngaging prior knowledge: Addition for Real NumbersArgand DiagramBack to Table of contents Subtraction
  22. 22. MMR IN ADDITION Multimodal Representation used: Pictorial  Geometric Interpretation  Vector Addition Im(z) z1 z1+z2 z2 Re(z) 0Argand DiagramBack to Table of contents Subtraction
  23. 23. SUBTRACTION OF COMPLEX NUMBERS Complex Subtraction  Difference of 2 complex numbers  z1 = x1 + y1i, z2 = x2 + y2i  z1 - z2 = (x1 + y1i) - (x2 + y2i) = (x1 - x2) + (y1 - y2) i  Subtraction of real and imaginary portions and summing the 2 parts up  Geometric Interpretation (vector subtraction)RationaleMultimodal Representation: Argand DiagramEngaging prior knowledge: Subtraction for Real NumbersAdditionBack to Table of contents Uniqueness
  24. 24. MMR IN SUBTRACTION Multimodal Representation used: Pictorial  Geometric Interpretation  Vector Subtraction Im(z) z1-z2 z1 Re(z) -z2 0AdditionBack to Table of contents Uniqueness
  25. 25. UNIQUENESS OF COMPLEX NUMBERS If two complex numbers are the same, i.e. z1 = z2, then their real parts must be equal, and their imaginary parts are equal. Algebraically, let z1 = x1 + y1i, z2 = x2 + y2 i, if z1 = z2 then we have  x1 = x2 and y1 = y2 Geometrically, from the argand diagram we can see that if two complex numbers are the same, then they are represented by the same point on the argand diagram, and immediately we can see that the x and y co- ordinates of the point must be the same.Subtraction Back to table of contents
  26. 26. A-LEVEL SYLLABUSBack to Table of contents Pedagogical Considerations
  27. 27. PEDAGOGICAL CONSIDERATIONS  Operations in complex plane is similar but not exactly the same as vector geometry (see complex multiplication and division)  Limitations in relating to Argand diagram (pictorial) for teaching of complex multiplication and division in Cartesian form  Building on Prior Knowledge  Rules-Based Approach vs Theoretical UnderstandingSyllabus Back to table of contents Multiplication
  28. 28. PEDAGOGICAL CONSIDERATIONS  Properties…  of complex multiplication assumed (commutative, associative, distributive over complex addition)  of complex division assumed (not associative, not commutative)  of complex conjugates (self-verification exercise)  Notion of identity element, multiplicative inverse  Use of GC  Accuracy of answersSyllabus Back to table of contents Multiplication
  29. 29. MULTIPLICATION OF COMPLEX NUMBERS  Complex multiplication  Multiplication of 2 complex numbers  z1 = x1 + y1i, z2 = x2 + y2i  z1 z2 = (x1 + y1i) (x2 + y2i) = x1x2 + x1y2i + x2y1i - y1y2i2 = (x1x2 - y1y2 ) + (x1y2+ x2y1)i  Geometric Interpretation (Modulus Argument form) Rationale Engaging prior knowledge: Multiplication for Real NumbersPedagogical considerations Back to Table of contents Division
  30. 30. MULTIPLICATION OF COMPLEX NUMBERS  Complex multiplication  Scalar Multiplication  z = x + yi, k real number kz = k(x + yi) = kx + kyi  Geometric Interpretation (vector scaling)  k ≥ 0 and k < 0 Rationale Multimodal Representation: Argand Diagram Engaging prior knowledge: Multiplication for Real NumbersPedagogical considerations Back to Table of contents Division
  31. 31. MULTIPLICATION OF COMPLEX NUMBERS  i4n = I, i4n+1 = i, i4n+2 = -1, i4n+3 = -I for any integer n  Explore using GC (Limitations)  Extension of algebraic identities from real number system  (z1 + z2 )(z1 – z2 ) = z12 – z22  (x + iy)(x – iy) = x2 – xyi + xyi + y2 = x2 + y2  ALWAYS real Rationale Engaging prior knowledge: Multiplication for Real Numbers Cognitive process: AssimilationPedagogical considerations Back to Table of contents Division
  32. 32. DIVISION OF COMPLEX NUMBERS Complex division  Division of 2 complex numbers (Realising the denominator)  z1 = x1 + y1i, z2 = x2 + y2i . z1 x1 + y1i x1 + y1i ( x2 − y2i ) x1 x2 + y1 y2 x2 y1 − x1 y2 = = = + 2 i z2 x2 + y2i x2 + y2i ( x2 − y2i ) x2 + y2 2 2 x2 + y2 2RationaleEngaging prior knowledge: Rationalising the denominatorMultiplication Back to Table of contents Conjugates
  33. 33. DIVISION OF COMPLEX NUMBERS Solve simultaneous equations (using the four complex number operations) Finding square root of complex numberMultiplication Back to Table of contents Conjugates
  34. 34. COMPLEX CONJUGATES Let z = x + iy. The complex conjugate of z is given by z* = x – iy.  Conjugate pair: z and z*  Geometrical representation: Reflection about the real axis Multiplication: (x + iy)(x – iy) = x2 + y2 Division: Realising the denominatorRationale Bruner’s CPA Recalling prior knowledge, Law of recencyDivisionBack to Table of contents Learning Difficulties
  35. 35. COMPLEX CONJUGATES Properties: Exercise for students (direct verification)  1. Re(z*) = Re(z); Im(z*) = -Im(z) 7. (z1 + z2)* = z1* + z2*  2. |z*| = |z| 8. (z1z2)* = z1*z2*  3. (z*)* = z 9. (z1/z2)* = z1*/z2*,  4. z + z* = 2Re(z); z – z* = 2Im(z) if z2 ≠ 0  5. zz* = |z|2  6. z = z* if and only if z is realRationale Self-directed learningDivisionBack to Table of contents Learning Difficulties
  36. 36. LEARNING DIFFICULTIES/COMMON MISTAKES In z = x + yi, x and y are always REAL numbers Solve equations using z directly or sub z = x + yi Common mistake: (1 + zi)* = (1 – zi)  Confused with (x + yi)* = (x - yi)DivisionBack to Table of contents
  37. 37. A-LEVEL SYLLABUSBack to Table of contents Pedagogical Considerations
  38. 38. PEDAGOGICAL CONSIDERATION Start with a simple quadratic equation  Example: x2 + 2x + 2 = 0. Get students to observe and comment on the roots.Rationale: Bruner’s CPA Approach: Concrete Engaging Prior KnowledgeSyllabusBack to Table of contents Learning Difficulties
  39. 39. PEDAGOGICAL CONSIDERATION Direct attention to discriminant of quadratic equation  What can we say about the discriminant?Rationale: Engaging Prior Knowledge:  Linking to O-Level Additional Maths knowledge Involving students in active learning (Vygotsky’s ZPD)SyllabusBack to Table of contents Learning Difficulties
  40. 40. PEDAGOGICAL CONSIDERATION Examples on Solving for Complex Roots of Quadratic Equations  Expose students to different methods:  Quadratic Formula  Completing the Square MethodRationale:Making Connections between real case and complex caseGetting students to think activelySyllabusBack to Table of contents Learning Difficulties
  41. 41. PEDAGOGICAL CONSIDERATION Different version: What if we are given one complex root?Example:If one of the roots α of the equation z2 + pz + q = 0 is 3 − 2i, and p, q ∈ ℝ , find p and q.Rationale:Understanding and applying concepts learntSyllabusBack to Table of contents Learning Difficulties
  42. 42. PEDAGOGICAL CONSIDERATION Good to Fundamental Theorem of Algebra knowOver the set of complex numbers, every polynomial with real coefficients can be factored into a product of linear factors.Consequently, every polynomial of degree n with real coefficients has n roots, subjected to repeated roots.Rationale:Making Connections to Prior Knowledge in Real CaseSyllabusBack to Table of contents Learning Difficulties
  43. 43. PEDAGOGICAL CONSIDERATION Good to Visualizing Complex Roots know  Exploration with GeoGebraRationale:Stretch higher ability students to think furtherMotivates interest in topic of complex numbersSyllabusBack to Table of contents Learning Difficulties
  44. 44. PEDAGOGICAL CONSIDERATION Extending from quadratic equations to cubic equations  Can we generalize to any polynomial?  Recall: finding conjugate roots of polynomials with real coefficientsRationale:Making sense through comparing and contrastingSyllabusBack to Table of contents Learning Difficulties
  45. 45. LEARNING DIFFICULTIES/COMMON MISTAKES X: complex roots will always appear in conjugate pairs ‘No roots’ versus ‘No real roots’ Difficulty in applying factor theorem Careless when performing long division Application of ‘uniqueness of complex numbers’ does not occur naturallyPedagogical ConsiderationsBack to Table of contents
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