complex number i - presentation
TRANSCRIPT
COMPLEX NUMBERS
Beertino
John
Yeong Hui
Yu Jie
CONTENTS
John A Level syllabus Pedagogical cons
iderations Learning difficu
lties
John A Level syllabus Pedagogical cons
iderations Learning difficu
lties
Yu Jie A level syllabus Pedagogical cons
iderations Basic definition
& Argand Diagram Addition and
Subtraction of complex numbers
Uniqueness of Complex Numbers
Yu Jie A level syllabus Pedagogical cons
iderations Basic definition
& Argand Diagram Addition and
Subtraction of complex numbers
Uniqueness of Complex Numbers
Yeong Hui A level syllabus Pedagogical Cons
iderations Multiplication
and Division of complex numbers
Complex conjugates
Yeong Hui A level syllabus Pedagogical Cons
iderations Multiplication
and Division of complex numbers
Complex conjugates
Beertino Approaches/pedagogy Diophantus’s problem roots of function Cubic Example
Beertino Approaches/pedagogy Diophantus’s problem roots of function Cubic Example
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 problemBack to Table of contents Diophantus’s problem
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 problemBack to Table of contents Diophantus’s problem
APPROACHES/PEDAGOGY
Totalitarian Approach! ( Just kidding )
Back to Table of contents Diophantus’s problemBack to Table of contents Diophantus’s problem
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 problemBack to Table of contents Diophantus’s problem
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 problemBack to Table of contents Diophantus’s problem
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 FunctionApproaches/Pedagogy Back to Table of contents Root of Function
SOLUTION AND PROBLEM
Approaches/Pedagogy Back to Table of contents Root of FunctionApproaches/Pedagogy Back to Table of contents Root of Function
ROOT OF FUNCTION
Diophantus’s problem Back to Table of contents Cubic ExampleDiophantus’s problem Back to Table of contents Cubic Example
ROOT OF FUNCTION
Diophantus’s problem Back to Table of contents Cubic ExampleDiophantus’s problem Back to Table of contents Cubic Example
CUBIC EXAMPLE
Root of Function Back to Table of contentsRoot of Function Back to Table of contents
CUBIC EXAMPLE
Root of Function Back to Table of contentsRoot of Function Back to Table of contents
LASTLY
Root of Function Back to Table of contentsRoot of Function Back to Table of contents
A-LEVEL SYLLABUS
Back to Table of contents Pedagogical considerationsBack to Table of contents Pedagogical considerations
PEDAGOGICAL CONSIDERATIONS
SyllabusBack to table of contents (Teaching) Basic DefinitionSyllabusBack to table of contents (Teaching) Basic Definition
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 Understanding
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 values
SyllabusBack to table of contents (Teaching) Basic DefinitionSyllabusBack to table of contents (Teaching) Basic Definition
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 part
BASIC DEFINITONS
Pedagogical considerationsBack to Table of contents Argand DiagramPedagogical considerationsBack to Table of contents Argand Diagram
ARGAND DIAGRAM
Basic DefinitionsBack to Table of contents AdditionBasic DefinitionsBack to Table of contents Addition
0
P(x,y)
x
y
Re(z)
Im(z)z = x + y i
x : Real Part y : Imaginary
Part
Important aspect,common studenterror is forgettingthat x,y are bothreal valued
EXTENSION FROM REAL NUMBERS (ENGAGING PRIOR KNOWLEDGE)
Basic DefinitionsBack to Table of contents AdditionBasic DefinitionsBack to Table of contents Addition
0
z
|z|
x
y
Re(z)
Im(z) The Real Axis (x-axis) represents the real number line.
In other words thereal numbers justhave the imaginarypart to be zero.
e.g. 1 = 1 + 0 i
Argand DiagramBack to Table of contents SubtractionArgand DiagramBack to Table of contents Subtraction
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 Numbers
Multimodal Representation used: Pictorial Geometric Interpretation
Vector Addition
Argand DiagramBack to Table of contents SubtractionArgand DiagramBack to Table of contents Subtraction
MMR IN ADDITION
z1z1
z2z2
z1+z2z1+z2
Re(z)Re(z)
Im(z)Im(z)
00
SUBTRACTION OF COMPLEX NUMBERS
AdditionBack to Table of contents UniquenessAdditionBack to Table of contents Uniqueness
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)
Rationale
Multimodal Representation: Argand Diagram
Engaging prior knowledge: Subtraction for Real Numbers
Multimodal Representation used: Pictorial Geometric Interpretation
Vector Subtraction
MMR IN SUBTRACTION
AdditionBack to Table of contents UniquenessAdditionBack to Table of contents Uniqueness
z1z1
-z2-z2
z1-z2z1-z2
Re(z)Re(z)
Im(z)Im(z)
00
UNIQUENESS OF COMPLEX NUMBERS
Subtraction Back to table of contentsSubtraction Back to table of contents
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.
A-LEVEL SYLLABUS
Back to Table of contents Pedagogical ConsiderationsBack to Table of contents Pedagogical Considerations
PEDAGOGICAL CONSIDERATIONS
Syllabus Back to table of contentsMultiplication
Syllabus Back to table of contentsMultiplication
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 Understanding
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 answers
Syllabus Back to table of contentsMultiplication
Syllabus Back to table of contentsMultiplication
MULTIPLICATION OF COMPLEX NUMBERS
Pedagogical considerations Back to Table of contentsDivisionPedagogical considerations Back to Table of contentsDivision
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)
RationaleEngaging prior knowledge: Multiplication for Real Numbers
MULTIPLICATION OF COMPLEX NUMBERS Complex multiplication
Scalar Multiplication z = x + yi, k real number k z = k(x + yi)
= kx + kyi Geometric Interpretation (vector scaling)
k ≥ 0 and k < 0
RationaleMultimodal Representation: Argand DiagramEngaging prior knowledge: Multiplication for Real Numbers
Pedagogical considerations Back to Table of contentsDivisionPedagogical considerations Back to Table of contentsDivision
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 ) = z1
2 – z22
(x + iy)(x – iy) = x2 – xyi + xyi + y2 = x2 + y2 ALWAYS real
RationaleEngaging prior knowledge: Multiplication for Real Numbers
Cognitive process: Assimilation
Pedagogical considerations Back to Table of contentsDivisionPedagogical considerations Back to Table of contentsDivision
DIVISION OF COMPLEX NUMBERS
Complex division Division of 2 complex numbers (Realising the
denominator) z1 = x1 + y1i, z2 = x2 + y2i .
RationaleEngaging prior knowledge: Rationalising the denominator
Multiplication Back to Table of contents ConjugatesMultiplication Back to Table of contents Conjugates
1 1 1 1 1 2 2 1 2 1 2 2 1 1 22 2 2 2
2 2 2 2 2 2 2 2 2 2 2
( )
( )
z x y i x y i x y i x x y y x y x yi
z x y i x y i x y i x y x y
+ + − + −= = = +
+ + − + +
DIVISION OF COMPLEX NUMBERS
Solve simultaneous equations (using the four complex number operations)
Finding square root of complex number
Multiplication Back to Table of contents ConjugatesMultiplication Back to Table of contents Conjugates
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 denominator
Rationale Bruner’s CPA Recalling prior knowledge, Law of recency
DivisionBack to Table of contents Learning DifficultiesDivisionBack to Table of contents Learning Difficulties
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 real
Rationale Self-directed learning
Division Back to Table of contents Learning DifficultiesDivision Back to Table of contents Learning Difficulties
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 contentsDivisionBack to Table of contents
A-LEVEL SYLLABUS
Back to Table of contents Pedagogical ConsiderationsBack to Table of contents Pedagogical Considerations
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 Knowledge
SyllabusBack to Table of contents Learning DifficultiesSyllabusBack to Table of contents Learning Difficulties
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 DifficultiesSyllabusBack to Table of contents Learning Difficulties
PEDAGOGICAL CONSIDERATION
Examples on Solving for Complex Roots of Quadratic Equations Expose students to different methods:
Quadratic Formula Completing the Square Method
Rationale:Making Connections between real case and complex caseGetting students to think actively
SyllabusBack to Table of contents Learning DifficultiesSyllabusBack to Table of contents Learning Difficulties
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 learnt
SyllabusBack to Table of contents Learning DifficultiesSyllabusBack to Table of contents Learning Difficulties
PEDAGOGICAL CONSIDERATION
Fundamental Theorem of Algebra
Over 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.
Good to know
Rationale:Making Connections to Prior Knowledge in Real Case
SyllabusBack to Table of contents Learning DifficultiesSyllabusBack to Table of contents Learning Difficulties
PEDAGOGICAL CONSIDERATION
Visualizing Complex Roots Exploration with GeoGebra
Good to know
Rationale:Stretch higher ability students to think furtherMotivates interest in topic of complex numbers
SyllabusBack to Table of contents Learning DifficultiesSyllabusBack to Table of contents Learning Difficulties
PEDAGOGICAL CONSIDERATION
Extending from quadratic equations to cubic equations Can we generalize to any polynomial?
Recall: finding conjugate roots of polynomials with real coefficients
Rationale:Making sense through comparing and contrasting
SyllabusBack to Table of contents Learning DifficultiesSyllabusBack to Table of contents Learning Difficulties
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 naturally
Pedagogical ConsiderationsBack to Table of contentsPedagogical ConsiderationsBack to Table of contents