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Appetizers and Lessons for Mathematics and Reason
by A. Selby, Ph. D.
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 20+ pages in French: Algèbre  
 Définition d'une variable  Logique
La raison basée sur les  règles et modelés

www.whyslopes.com > Volume 3, Why Slopes & More.Math, 1995  >   23 Proofs of.     [ Back ] [ Next ]

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.



The product a·Q is collinear with the vector Q as a is real while the product ib·Q is perpendicular to Q since ib has angle 90 degrees with the positive (real) axis.

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 previous sections show how multiplying points (a,b) in the plane with polar coordinate-based add the angle, multiple the lengths rule, led to the expression for the product in terms of the rectangular components, that is, the real and imaginary parts of the factors. This section provides an alternate approach. The expressions are consequences of the distributive law for complex multiplication.

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

For every triple of complex numbers (A,B,C), the right distributive law (the right form)
A·{B+C} = A·B+A·C
and the left distributive law (the left form)
{B+C}·A = B·A+B·C
applied twice in succession would imply the following in terms of rectangular coordinates if they held.
(a,b)·(c,d)
=
(a,b)·{(c,0)+(0,d) }
=
(a,b)·(c,0)+(a,b)·(0,d)
       if the right distributive law holds,
=
{(a,0)+(0,b)}·(c,0)+{(a,0)+(0,b)}·(0,d)
=
{(a,0)·(c,0)+(0,b)·(c,0)}
       +{(a,0)·(0,d)+(0,b)·(0,d)}
       if the left distributive law holds,
=
{(ac,0)+(0,bc)}
       +{(0,ad)+ b[1,90°]·d[1,90°]}
=
{(ac,0)+(0,bc)}+{(0,ad)+ bd[1,180°]}
=
{(ac,0)+(0,bc)}+{(0,ad)+ bd(-1,0)]}
=
{(ac,0)+(0,bc)}+{(0,ad)+ (-bd,0)}
=
{(ac,bc)}+{(-bd,ad)}
=
(ac-bd,bc+ad)
In terms of complex number notation, the foregoing says that
(a+ib)·(c+id) = (ac-bd)+(bc+ad)i
where i = Ö(-1). Therefore
(a+ib)·(c+id) = (ac-bd)+(bc+ad)i
holds for all real numbers a, b, c and d IF the left and right distributive laws hold.

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.

Why Slopes
and 
More Math
 Volume 3
Printed in Canada
ISBN 0-9697564-3-7
Read slowly,  Volumes 2 & 3 may ease or avoid  calculus difficulties.  Take the risk.

Chapters and Appendices

Content Guide
Foreword
2nd Content Guide
1. Introduction
Geometric Calculus Preview (1983)
2. Algebraic 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

Appendices:

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.

What is a Variable?
Introduction
Variation between Examples
Variation of Letters
A letter denotes a variable
Cases of Double Variation
Three Notions of a Variable
Constants, Parameters
& Variables
Talking about numbers
Dependent or Independent
Variable, a Matter of Choice

 

For Parents & Teachers: Speaking Skills, Reading & Writing,  Preparing for Science,  ends, values and methods for work and study,  parent- friendly mathematics booklets for ages 4-14.

 - Math Education Essays   (opinions, possibilities, references)  

-  POMME, a two level program for future skill development in schools and colleges worldwide. Address content & motivation gaps with ends, values & methods for skill development to say which way to go, how and why. - Present Day Curriculum:
 (A) Secondary I Mathematics
consolidate  fractions and measurement, skills and sense consolidation,
 (B) Secondary II Mathematics
year of algebra and proportionality
(C) See too the following:
- Arithmetic & Number Theory Practices (horribly put, but useful) 
-  Algebra and Logic SubProgram
(well put, extremely useful)  

For Avid Readers in School & Out - Online Books 
   1.  Elements of Reason. 1996 
1A. Pattern Based Reason  1995 
1B. Math Curriculum Notes 1996 
2. Three Skills for Algebra  1995 
3.Why Slopes & More.Math 1995
Tour their  forewords.   

Calculus Prep or Help: See Volumes 2 & 3, and this bigger Calculus Guide.  

 
 

Senior High School  & 
Calculus Students

 
 

Free Live Lesson
- Operations with Decimals -  Comparison, Subtraction and Long Division - Click here to attend.

 
 

For Senior High School Mathematics & Calculus 

Intro to Solving Linear Equations 
- a different paths for junior and even senior high school students.   

5 wordy Logic Chapters
4 curious Algebra Chapters
Words before & besides symbols. A Key Algebra forward & backwards Chapter    

First Calculus Preview (1st intro)
Four Calculus Chapters  (2nd intro)
Intro to Complex Numbers (long)
Intro to Mathematical Induction (romantic & wordy at first)

  
 

Many More Topics 

 1. Decimal Arithmetic  Reference!
2. Integers - Intro to Signed No.s
3.  Fractions - fully explained.
4.  Fractions  with Units  
5.   Number Theory, 
6.    Solving Linear Equations  
7  Formulas Use Forward & backwards -  
8.  Proportionality, Back- & For-wards.   
9. Logic Chapters:   
10.  Euclidean-Geometry  
11.  Slopes & Equations of Straight Lines.  (Take I. See take II below)
12.  Why Study Slopes. 
13. Maps, Plans,  Similarity & Trig,  
  (Take II included here)
14.  Quadratics: Starter lessons
15.  Polynomials: Starter lessons 
16 Why Factor Polynomials:  
17   Functions - Forwards & Backwards.  
18.  Exponents, Radicals & logs.  
19. Complex Numbers before trig (new advance/ starter lesson)
20.  DC Electric Circuits Etc 
21. Real  Analysis 
22. The Olde Complex No, Trig
& Vector Section.
23. More Calculus Stuff
- written after Volumes 2 and 3.

More For Instructors
- Education Essays   (opinions, possibilities, references)  
-  POMME, a two level program for instruction K1-14
- Math & Logic  How-TOs 
1. Arithmetic
2. Algebra
3. More Algebra
4.  Beginner Geometry
5.  More Geometry
6. Calculus 
7. Show Work or Logic 
 These may be too dense for students.

Level I Material: New Stuff
Time and Date Matters
 Level I Arithmetic. 
Money Matters
Measurement Matters
Matters of Chance (Risk Control)
 Logic Chapters (leave what's not clear in Level I to Level II)
Using/Making Maps and Plans.
(A variant of Maps, Plans,  Similarity & Trig,  to appear here).

 
 

Skill Development Tips
For All

Standards: (A) Take care to avoid the domino effect of errors & approximations; (B) Do and record steps in an  manner  that allows skill mastery to be seen or corrected. Anything represent substandard work.  

Key Numerical Methods
- To multiply signed numbers, prefix the product of their signs to the product of their lengths or unsigned parts. The product is negative if the no of negative sign in it is odd.  
- To add signed numbers with like signs, prefix the common sign to the sum of the lengths.
- To add signed numbers with opposite signs, prefix the sign of the longest to the difference: length of longest minus length of shortest.
- Should we study roots and powers of real numbers with formulas involving exponential and log.
- How does adding and multiplying points in the plane and rotating the midpoint of a line segment lead to mastery of complex numbers and the thought-based development of their properties, all before trig?

- New Axioms for High School Mathematics: In accounting, totals of assets and debts may be calculated by dividing the assets and debts into non-overlapping (disjoint) groups and then adding subtotals. In general, sums (and products) of counts and  numbers,  positive and negative numbers included,  can  be obtained by adding subtotals (and multiplying subproducts, respectively). These practices may be cast as axioms in secondary mathematics. Then operations on polynomials are easily implied  justified by these "axioms" and the geometric introduction of column methods for expanding a products of two sums.  While set theory in pure mathematics may imply the above axioms in university mathematics programs instruction, an earlier and more accessible explanation based on easily accepted and understood geometric and counting practices  derivation of the above axioms is possible at the high school for students heading for college programs in science. 

In Volume 2: Prep for Calculus
 - What is the difference between saying A if B and saying A if and only if B. Being aware of the difference will sharpen ye wits. 
- What is a chain of reason?
-Are your arithmetic skills OK? 
-Have words been missing in the introduction of algebra?
- Can ye talk about numbers & quantities varying apart from or before the use of letters & functions?
- Do ye know about the forward & backward use of formulas?
-Contrapositive: is that backward use of  A if B?
-What is a variable x? Answer before speaking of function f(x) = x.
-What a twist! There are no rules of algebra for subtraction and division. But if you replace them by addition of -x and multiplication by 1/x, rules of algebra (properties of arithmetic) can be used. 

In Volume 3: Calculus Slowly? 
-Why are slopes studied and polynomials factored in high school?
-   Volume 3 suggests how to ease or delay algebra shock in calculus *& beyond.   In Calculus, derivatives and integrals introduced and defined by limits, but calculated without when possible by using differentiation rules forwards and backwards. The second site calculus section may help in differential calculus.

 

 

 

www.whyslopes.com >  Volume 3, Why Slopes & More.Math., 1995  >   23 Proofs of.     [ Back ] [ Next ]

Road Safety Message   Walk on a side walk. If that is not possible, try  not to  walk on a road with your back to the traffic.
Try to see what  trucks, cars, buses or bicycles are coming, so that you may step out of their way.  Put safety first. .

Support for Technical Mathematics from Number Theory to Calculus Prep
A. More Arithmetic a must for algebra etc D. Logic In Mathematics G. Algebra with Take Home Value I. Vectors & Functions
Decimal Lesson - Reference  
Counting & Addition   (8 lessons)
Comparison to Subtraction  (9 lessons)
Multiplication ( 11 lessons)
Long Division  (12 lessons)
Decimals and Primes (8 lessons)
-Primes & Composites 
-Primes Factorization
-Greatest Common Divisors & Multiples. 
-Prime Factorization Aids  (Learn how to find factors quickly)
-Prime Factorization Examples 
-Counting & Generating. Factors. 
-Divisibility Rules and Remainders for Division by 2, 3, 5, 9 and 11.
Integers (12 lessons) Intro to Signed Numbers
 Fractions (< 20 lessons)  Essential Skills & Concepts 
Ratios & Fractions (3 lessons):  Similarities & Differences   
Units in calculations Fractions  with Units
B.  Basic Algebra
Solving Linear Equations  
- in one unknown. Intro  with stick diagrams?
the normal way  & with good nttn. (the nttn that reappears in Gaussian Elimination. |
-in more unknowns: simultaneous equations essentially one unknown. the let algebra do the work view of  word problems.
  - still in more unknowns:  Gaussian Elimination via substitution, by equality or comparison, by operations on equations. 
C. More Algebra
Words before symbols: See if U like the lengthy chapters 8 to 12 in Volume 2, Three Skills for Algebra  
What is a Variable.  The answer here  is a simple prequel to the modern mathematics viewpoint.
First, every rule & pattern U meet in math, logic & science will be used forwards and backwards.  Get a head start with this theme by reading  Chapter 14 in Three Skills for Algebra.  Second, in the study of Proportionality Relations (3 dense lessons here) finding the proportionality constant gives an initial  backward  use of the proportionality formula.
 Talking about words before symbols and the forward and backward use of formulas gives words to make algebra simpler & clearer.  
If you can not read or write precisely, you will have difficulty in following instructions.  One wordy remedy  is given by chapters 2 to 5  in Three Skills for Algebra. Where does Logic or a geometric model for reason Appear in Mathematics? The answer lies in  Euclidean-Geometry    In North America, Euclidean Geometry disappeared from high school mathematics as it was too hard. The light treatment here is a possible remedy.
E.  More Geometry
The Pythagorean Theorem. Chapter 17 from  in Three Skills for Algebra uses algebra and geometry   to show why the  Pythagorean equation  for right triangles holds. Its forward and backward use  is common exercise..  At a more theoretical level, the Pythagorean theorem leads the discovery that not all lengths can be  fractional multiples of a unit length. That geometrically implies a  need for and even existence of irrational numbers.
Analytic Geometry: Common Practices with  Maps and Plans drawn to scale  give coordinate-dependent base  for senior high school development of similarity, trig, vectors and straight lines.   
Complex Numbers: This lesson on Complex Numbers  draws on Euclidean and Analytic geometry. Sbortcuts simplifiy  trig identities, the cosine law; and   trig formulas for 2D dot- and cross-products. 

F. Logarithms, Exponentials,
Roots & Powers

Logarithms, exponentials, rational and real powers for secondary students. This  complete Operational Viewpoint. (Sufficient for the precalculus forward and backward use of compound growth and decay formulas in biology, physics, chemistry,  personal finance, and calculus. To learn more, if you study calculus,  see chapter 19 of Volume 3, Why Slopes and More.Math

In Volume 2, Three Skills for Algebra, chapters
  1. Geometric Sums Etc,
  2. Notation For Sums,
  3. Personal Money Maths and
  4. Some Finite Mathematics
identify methods useful in money computations, methods needed for calculus. Your teachers or other writer may present the same ideas with greater clarity and detail - A site to do.

H. Polynomial & Quadratics

Analytic Geometry:   -  Slopes and Lines - Take 1.   Take 2 appears in site section Maps and Plans.   Two views are better than one.  I may combine them later.  -In my school days, slopes appeared year after year.   This Why  Slopes calculus preview on graphs of functions y = f(x) explains why.  Enjoy.
Quadratics and Polynomials: -  Operations on Polynomials: Meet a light and ultraquick geometric introduction to  multiplication, addition and subtraction of polynomials. Then see how the foregoing combine to permit long division of polynomials.    Compare Fractions  with Units. Enrichment: A Plus:  The Geometric introduction here gives or is almost identical to a justification for column methods in decimal arithmetic. 
Geometric Derivation of the Quadratic Formula:   The account here gives a starter lesson for the more algebraically harder geometric-free derivation. If you study physics, chemistry or trigonometry, you will need to know about quadratics, their factorization and the quadratic formula.
Technical Value: The study of polynomials  high school mathematics has technical value as part of the senior high school mathematics preparation for calculus.  This simple account of Why Factor Polynomials   (Chapters 2 to 6 in Volume 3 .Why.Slopes.&.More.Math.) will give a context for the study of polynomials,  their factorization, and sign analysis of functions, all in a way that should improve your algebraic thinking and reasoning skills.  		Vectors in the Plane (2 simple lessons)
- Navigation with vectors or arrows
- Sum of Motions
- more lessons to be added later.
Operations on movement or vectors along the line and in the plane have value in mathematics in defining and implying the properties of real and complex numbers before the assumption of those properties as axioms.  Vectors and their properties appear in physics, its mathematical description and formulation. 
 - Functions - Forwards & Backwards.  Here is a full technical reference (24 lessons) for use in a calculus or precalculus course as needed. In it, the set viewpoint of functions expression of modern pure mathematics.  comes from the set-based codification and
In the mathematics education reforms of the 1960s in North America, primary and secondary school mathematics were expressed in terms of sets. That expression has now retreated from primary and secondary school texts. But it still lingers on, and can be very useful, a source of clarity and precision, in the situations where it should be retained: Counting with the aid of sets and functions; the description of functions; the high school account of probability theory; and in the discussion or illustration of ideas in logic. 

J. Pre-Calculus Skill Check
-  Arithmetic Skill Check.  In the calculus courses I taught 1983-89, too many students had weak skills in arithmetic. I would give and carefully correct these exercises to tell students what they needed to review and master.  
-  All the skills and concepts in  Chapters 1 to 24 or Volume 2, Three Skills for Algebra: Look for those you do not understand and fill the gaps. Do so quickly while balancing this advice with  your other duties.  Good luck.

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