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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
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
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
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
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