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YOU are better than YOU think. Show yourself  how:  

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 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;

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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
Continued

Before reading the second proof below, explore the geometry-vectors-trig section of this website created after this book was written.  The latter section presents a simpler path for the exposition geometry, vectors and trig. The second proof below was part of the exploration of ideas that led to the simpler path.

Second Proof

Step 1. Suppose P = a+i b and Q = c+id are at unit distance from the origin of the plane. Then a²+b² = 1 and c²+d² = 1. (Units of length are omitted. The product P·Q has length 1.) The diagram below shows that the product P·Q located by adding angles (and multiplying lengths) coincides with the vector given by the sum of a·Q and i b·Q. This implies the distributive law (a+ib)·z = a·z+ib·z for the situation depicted. The argument holds regardless of the quadrants in which P is located.

Step 2. The foregoing implies the distributive law (a+ib)·z = a·z+ ib·z for case a²+b² = 1 = |z|. Since the polar coordinate defined product of points in the plane is commutative, the foregoing law is two-sided. Hence z = c+id with c and d both real and c²+d² = 1 implies

(a+ib)(c+id) = (a+ib)·z = a·z + ib·z       by step 1 = a·(c+id) + ib·(c+id) = a·c+a·id + b·ic+ib·id       by step 1 twice = ac+iad+ibc+i2bd = ac-bd+i(ad+bc)        as i2 = -1

due to the definition of the product of complex numbers and the associativity (proof?) of the addition of points in the plane. The real part of the product is ac-bd and the imaginary part is ad+bc.

Step 3. From the polar coordinate definition of the product of complex numbers, we observe

[cos(q)+isin(q)][cos(b)+isin(b)] = cos(q+b)+isin(q+b).

Now the angle sum formulas for sine and cosine:

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

and

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

follow from the expressions derived above for the rectangular coordinates of the product of two unit magnitude points in the plane. How to compute the real and imaginary parts in a product of any pair of complex numbers in terms of the real and imaginary parts of the factors is given next.

Distributive Law and Consequences

The next pages offer another way to obtain the distributive law for multiplication without assuming the factors have unit length. The next pages further imply the formulas for the real and imaginary parts of a product, and from them the angle sum formulas. This shows links between complex numbers and trigonometry can be obtained in different ways.

The Commutative Law

Since the order of multiplication of positive numbers, and the order of addition of real numbers is immaterial, the add the angles, multiply the lengths rule implies

[r1,q1]·[r2,q2] = [r1r2,q1+q2] = [r2r1,q2+q1]

Therefore

[r1,q1]·[r2,q2] = [r2,q2]·[r1,q1]

This says that the order of multiplication is not important. Therefore, the right and left distributive laws imply each other (why?). So if one holds, then so does the others.

Use of The Distributive Laws

Now

cis(a) = cos(a)+isin(a)

The property cis(a)·cis(b) = cis(a+b) follows from the add the angles, multiply the lengths definition of complex multiplication and not both factors have unit lengths. But this property cis(a)·cis(b) = cis(a+b) can be rewritten in terms of rectangular coordinates or complex number notation as

(cos(a),sin(a))·(cos(b),sin(b)) = (cos(a+b),sin(a+b))

Computation of the real and imaginary parts of the left hand side implies the angle-sum formulas for the cosine function

cos(a)cos(b)-sin(a)sin(b) = cos(a+b)

and for the sine function

cos(a)sin(b)-sin(a)cos(b) = sin(a+b)

respectively.

Proof of Distributive Laws

Plan. The proof of the distributive law A(P+Q) = AP+AQ will be based on the observation (the physical assumption) that multiplication by

A = [r,q] = [r,0]·[1,q] = [1,q]·[r,0]

can be done into two steps. One step is a rotation through the angle q while the other is a multiplication by the stretch factor or shrinkage factor r = [r,0]. Multiplication by a stretch factor and rotation through an angle will be shown to be distributive operations over addition.

Distributive Law For Stretch Factors. Now let P = (a,b) and Q = (c,d). Now

(r,0)*(P+Q) = (r,0)*[(a,b)+(c,d)] = (r,0)*(a+c,b+d) = (r{a+c},r{b+d}) = (ra+rc,rb+rd) = (ra,rb)+(rc,rd) = (r,0)*P+(r,0)*Q

Therefore A(P+Q) = AP+AQ when A = (r,0) for some r > 0. This argument assumes the distributive law for multiplication of the sum of two real numbers by another.

It can be illustrated by tiling the plane with parallelograms - copies of the parallelogram determined by the arrows [1/(n)] P and [1/(n)]Q (where n ³ 1 is a whole number). Such an illustration might be sufficient corroboration for some pre-algebraic students.

Distributive Law for Rotations. A parallelogram corresponding to the map addition of the arrows associated with P = (a,b) and Q = (c,d) is indicated below.

We assume that the parallelogram and the two triangle forming it are rigid bodies. This implies that after a rotation, that the map addition of the vectors forming the sides before and after rotation will yield the diagonal arrow before and after rotation, respectively. See the next diagram.

The following diagram shows that the triangle vertices P, Q and P+Q rotated respectively into the triangle vertices P¢, Q¢ and P¢+Q¢. 

This suggests that

[1,q]·(P+Q) = P¢+Q¢ = [1,q]·P+[1,q]·Q

and hence that multiplication by the factor [1,q] is distributes over the addition of arrows.

End of the Proof. Observe that

A(P+Q) = ([r,0]·[1,q])·(P+Q) = [r,0]·([1,q]·(P+Q)) = [r,0]·([1,q]·P+[1,q]·Q) = [r,0]·([1,q]·P)+[r,0]·([1,q]·Q = ([r,0]·[1,q])·P +([r,0]·[1,q])·Q = A·P +A·Q

Remark. The formal or proper presentation of mathematics relies on no diagrams and on no physical interpretation or reasoning. The preceding presentation of complex numbers was informal. It relied on geometric ideas (assumptions) to make a link between polar and rectangular coordinates. But the conclusions drawn above can be obtained in a geometric-free manner (no diagrams) and drawn solely from assumptions about arithmetic. See the diagram-free description of the complex numbers and trig functions in the university-level book Principles of Mathematical Analysis by W. Rudin, McGraw-Hill 1964, for more details.

www.whyslopes.com
Volume 3,  Why Slopes and More Math -  

Foreword, One Calculus  preview and Online Chapters: (V) signals video (RealPlayer Format)  to watch 

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.

 

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