Year 13 Calculus‎ > ‎

Complex Numbers

> COLLEGE ALGEBRA                                                                

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 COMPLEX NUMBERS

Complex Joke


Why do we need complex numbers?ARTICLE
Indices/surds   
Expand (x+1)(x-6)(x+2)

Expanding three brackets


Expand (a+b+c+d)^2 
Factorising x^1/2 - x^3/2

Factorising Difference of Squares

Factorising pq+pr-qs-rs 
Factorising x^2+7x+12 
Factorising 4x^2-44x+112 
Factorising 2x^2+11x+12

Factorising Quadratics with a not 1

Factorising x^16 - x^8

Factorising Difference

Factorising mixed 
Factorising x^3+8

Solving quadratics 4x^2=9 
Solving quadratics x^2+7x-3=0

Cpmpleting Square




Solving ineq's 2(1-4x)/5 +6 less than (1-x)/2 
Change the subject y=4/3x 
Algebraic fractions (7x-14)/(5x-10) 
Algebraic fractions add/ subtract

Add and Subtract Algebraic Fractions



 EXTENSION 

Changing Subject Rational Functions



Partial Fraction Decomposition

 REVISION (2 LESSONS):MANIPULATE FUNCTIONS 
Domain/range 
Function or not? 
Transformation of functions 
Even/odd or periodic? 
Composite functions

Composition Functions

Inverse functions 
Rational functions (ax+b)/(cx+d) 
Domain/range of rational functions 
Sketch rational functions 
EXTENSION  
 REVISION (2 LESSONS):EXP AND LOG FUNCTIONS 
Sketch e^(-x) or e^(2x)+1 
Solve log base 3 of(x)=4 
Simplify 3log(4)+2log(2) 
Solve exp eq's 6^(x)+18=40 
Sketch ln(x)-2 
Applications of log's MODELLING LOGS ACTIVITY

EXTENSION
 
  TOPIC 4 FALLS INTO THE CATEGORY OF CONICS
TEACH ONLY POLAR COORDINATES 
 
 REMAINDER AND FACTOR TH (3 LESSONS) 
Long division 

Long Division Algorithm


Remainder theorem

Substitution technique



Dividing polynomials and the remainder theorem

 Remainder theorem
Factor theorem

Factor Theorem


 EXTENSION

 BINOMIAL TH (5 LESSONS) (NOT IN THE SYLLABUS)
-Evaluate factorials

Pascal Triangle

-Combinations 
-Binomial coefficients

Expanding Binomial


-Expand (y+1/y)^6
 
-Approximation (10.1)^5 
-General term 

 EXTENSION 
  

Binomial Theorem

SURDS (4 lessons)
Rational or irrational?

Rational or Irrational

Simplify surds sqrt(72) 

Simplifying Surds

Add/subtract surds 3sqrt(64) -2sqrt(32) 
Multiply 2sqrt(32) x sqrt(27) 

Multiplying Surds

Rationalise 3/2sqrt(5)

Rationalise Denominators


Rationalise 4/(sqrt6+sqrt3)
 
EXTENSION
 QUADRATICS (3 LESSONS) 
Discriminant 
Complete square 
Applications of quadratic 
Inequations 
Prove inequalities 
 
EXTENSION
 
 COMPLEX INTRODUCTION (6 LESSONS)

Why do we need them?


Equating imaginary numbers 
Add/subtract 
Multiply 

Powers of Complex


Powers of i

Imaginary Units and Its Powers


IMAGING SONG
Conjugates 
Division

Dividing Complex

Modulus 
Mixed APPLICATIONS OF COMPLEX NUMBERS

EXTENSION
CLASSROOM ACTIVITIES!

z^5+243=0 and z^5-243=0 


COMPLEX & POLYNOMIALS (5 LESSONS)
 INTERPRETATION OF COMPLEX
Fundamental theorem of Algebra FACTORISING USING CALCULATOR
Solving quadratics by completing square 
Solving quadratics by using formula 
Polynomials given roots 
Using factor theorem

Factorising Cubics


Using conjugate root theorem 

 EXTENSION
 PHANTON GRAPHS (BRILLIANT!)

QUARTIC EQUATIONS
 DE MOIVRE'S & COMPLEX ROOTS (4 LESSONS)De moivre
Rectangular to polar and vice versa 

Polar Coordinates (r,arg)


Degrees to Radians


Multiply polar numbers 

Euler's Proof


Euler's Identity using Taylor Series


Multiplying Polar Form


Polar numbers and trig applications
Why polar form?
Polar form
Multiplying in Polar Form 
De Moivre's (3+i)^6 DE MOIVRE'S

De Moivre's Backwards



SQUARE ROOT OF -1
Symmetry property

Locus

 ROOTS OF A POLYNOMIAL APP!
EXTENSION

Solving Quartics with Real Solutions