YOU are better than YOU think. Show yourself  how:  

      |      
//  _   _ \\
/\             /\
  <|  (o)   (o)   |> 
 \     | |      / 

 -/[]\- 
||
   / \_ 
 ||||||||||||||||||||||||||||

 Logic chapters 1 to 5  re- appear not in sequence, as is or longer,  in  Volume 1A,  Pattern Based Reason, Bon Appetite.

Logic Mastery
 Amazing, Amusing, Amorous,  Delicious, Delightful, Edifying, Strengthening Elixir. 
It eases work & learning difficulties Makes the hard easier. Opens eyes. Leads to greater precision.
in reading and
writing

Logic mastery makes the hard, easier. Logic mastery  leads to better, stronger and richer comprehension.  Logic mastery  improves reading and writing.  Logic mastery ease learning difficulties.  Logic mastery gives a headstart.  In sum, logic mastery  will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.

After logic,   (a) continue reading Three Skills for Algebra, chapters 8 to 14  and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes  & More Math, chapters 2 to 6;

      |      
//  _   _ \\
/\             /\
<|   (o)   (o)  |> 
|      | |     |
   \             /   
\    =   /

 -/[]\- 
||
  _ / \     
 ||||||||||||||||||||||||||||

What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts.

Try the Twiddla Whiteboard. In principle, it  allows to people to draw and chat together online on a copy of this webpage or a clean sheet. The chat may be via text or audio.  Visit www.twiddla.com to set up whiteboards to work with the webpage of your choice.

For online automated help in senior high school maths & calculus, visit  quickmath.com  For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com  With  overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck.

Chapter 23
Complex Numbers - Links to Trig

This applet (online only) illustrates the ideas in this chapter.

Further sections of this chapter are given by the web pages . Use the next, previous and up buttons in these links to move between these sections or to return this page.

The addition of complex numbers or points in the plane was given by means of their rectangular coordinates while multiplication was given in terms of polar coordinates. It is still an exercise perhaps in trig, an application of the angle sum formulas, to obtain expressions for the rectangular coordinates of a product in terms of the rectangular coordinates of the factors. Another exercise or alternative, is to justify and then apply the distributive law for complex multiplication over addition. The application bypasses the exercise or link with trig just indicated in favor some algebraic manipulation. The alternatives are given. Which approach to favor may be a matter of taste. This chapter assumes you are familiar with the unit circle definition of sines and cosines and with the addition of vectors.

The  cis or exponential functions

Applying the Angle Sum Formula

Now the add the angles, multiples the length rule implies the product (a+bi)(c+id) has polar coordinates [Rr,q+b] = (x,y). The rectangular coordinates (x,y) of the product are therefore given by

The angular sum formula for sine says

sin(q+b) = sin(q)cos(b)+cos(q)sin(b)

This implies the imaginary part

y = Rr sin(q+b) = Rr (sin(q)cos(b)+cos(q)sin(b)) = (R sin(q))(r cos(b))+(R cos(q))(r sin(b)) = bc+ad

This expresses the imaginary part y of the product x+iy = (x,y) = (a+bi)(c+id) in terms of the real and imaginary parts of the factors, that is their rectangular coordinates.

Product Rule in terms of Rectangular Coordinates

The foregoing yields the expression

(a+ib)(c+id) = (ac-bd)+i(bc+ad)

for the product of two points in the plane in terms of their rectangular coordinates, alias real and imaginary parts.

Properties of Complex Numbers

The assumption that the addition and multiplication of positive real numbers a, b, and c are commutative and associative operations implies the following. The length and angle defined multiplication of complex numbers is a commutative and associative operation as well. Details are omitted. They are left as an exercise.

Another Exercise. Employ the expresson (a+ib)(c+id) = (ac-bd)+i(bc+ad) to show that the distributive law for real numbers and the above expressions for the real and imaginary part of the product of two complex numbers implies the distributive property a(b+c) = ab+ac holds for any triplet a = a₁+ia₂, b = b₁+ib₂ and c = c₁+ic₂ of complex numbers.

www.whyslopes.com
Volume 3,  Why Slopes and More Math - Preview, starter & further lessons for calculus to ease or avoid algebra shock in instruction & self-instruction

Foreword, One Calculus  preview and Online Chapters: (V) signals video (RealPlayer Format)  to watchChapters 2 to 6: offer a very simple preview of calculus and a context for earlier study of  slopes and factored polynomials 

Area Entrance & Hub
Foreword
Chapter Descriptions
1. Introduction
2. Calculus Starter Lesson
2. Second Preview Begins
2 Skier in Motion (V)
2 The Skier (V)
2. Position Dependent (V)
3 Slope & Extrema (V)
4 Single Factor Analysis (V)
4 Two Factor (V)
4 More Factors (V)
4 With Divisors (V)
5 Maxima & Minima Tests
6 Jumps & Discontinuities
8 Review  (optional)
9 On Calculus Studies
11 Slope of Slope
13  Acceleration
14 Limits & Error Control (V)
14 Limit of a Fn.
14. Limited Error Control
14 Signif. Digits
14 Cauchy Limits
14 Sequence Limits
14 Decimal Arith.
15 What is Slope (V)
15 Slope Calculation (V)
15 Slope, a Limit
15 Tangent Lines
15 Linear Approx.,
15 Limits via Algebra (V)
15 Recap.
PS.Chain Rule for Polys
PS Chain Rule- General  (V) -
PS More Chain Rule (V)
PS - Sign Analysis (V)
16 What is Velocity
17  What is Area
18 Integration
18 Area Calculation
18  Fn DefN, 6 Ways
19 Logs & Powers
19 Natural Log.
19 Exponential Fn.
20 What's Next
21 Add Vectors
22 Complex #'s
23 Complex #'s
23 Trig Identity
23 Proofs of.
24 Complex Logs etc

Units in Calculations:
7 Velocity
7 Varying Velocity Example
7. Velocity Calculation
7 Changing Units
7 Same Velocity  Motions
10 Slopes without Units.
10 Units & Slopes
10  Units in Cost vs. Quantity
10  How Units  Appear
10 Unit  Elimination
10 Partial Elimination
10 Interest & Units
12 More on Units
Content Guide

Enriched material: The Appendices of Volume 3 are located in the Real  Analysis  Area.

Pigeon Hole Principle
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
One Side Range Theorem
Range On One Side Theorem
Integration & Lipschitz
 Continuity

These appendices continue the
decimal viewpoint of limits, error
control and continuity begun
in Chapter 14. The One Sided
 Range Theorem is a postscript,
not in printed version.

---

Online Volume 2, Three Skills for Algebra, Chapters 1 to 25 - skip 18., verbalizes and explains key skills and concepts, those needed in calculus, again to make the hard easier. A visual understanding of complex numbers may help - serve as back ground info,  in partial fraction decomposition.

 

www.whyslopes.com
[Top of this Page] [Site Exit] [ Back ] [ Area Entrance & Hub ] [ Next ]
[Comments, Reactions, Feedback][ Road Safety Message ]
: Favourite Sites:  BBC News  and mathematics portion of  English National Curriculum  

All trademarks and copyrights on this page are owned by their respective owners.
Copyright to comments & contributions are owned by the Poster. 
The Rest © 1995 onward by site author,   Alan Selby, a 1983 McGill. Ph. D. in mathematics
All Rights Reserved.