Complex Number I - Presentation

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.


APPROACHES/PEDAGOGY

Totalitarian Approach! ( Just kidding )


APPROACHES/PEDAGOGY

Historical Approach Why this approach?

Real questions faced by mathematicians. Build on pre-existing mathematical knowledge,

Quadratic formula

Roots.


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.


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


LASTLY


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


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


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


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



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


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


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


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)



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



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



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



Visualizing Complex Roots Exploration with GeoGebra

Good to know

Rationale:Stretch higher ability students to think furtherMotivates interest in topic of complex numbers



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


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

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Complex Number I - Presentation

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