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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

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The  cis or exponential functions

Recall the unit circle definitions of the sine and cosine functions. Let
cis(q) = cos(q)+isin(q) = exp(iq)
of a purely imaginary argument. It is now easy to say how and why the exponential property
cis(A)·cis(B) = cis(A+B) or equivalenly
exp(iA)·exp(iB) = exp(i(A+B))
follows immediately from the above add the angles, multiply the lengths definition of complex multiplication. Hint: both factors have unit lengths.

 The notation exp(iA) is employed because it is possible to define the exponential exp(z) of a complex number z = a + ib where  a and b are both real. The definition for those of you who know about exponentials exp(a) of real numbers a is as follows: 

exp(a + ib) = exp(x) {cos(b) + i sin(b)).

For more details visit Calculus and Beyond.

Many trig identities follow from the above property (exponential property) and the equality of two different ways to compute products of complex numbers. 

Cosine and Sine Addition Formulas

Preliminary Exercise: In the identity exp(iA)·exp(iB) = exp(i(A+B)), express the left hand side in terms of the real and imaginary parts of the factors.

Solution:

 exp(iA)·exp(iB) = {cos(A) + i sin (A)} {cos(B) + i sin (B))

= cos(A)cos(B)-sin(A)sin(B) + i{cos(A)sin(B) + sin(A)cos(B)}

exp(i(A+B)) =  cos(A+B) + i sin(A+B).

The solution to the above exercise, implies the cosine and sine addition formulas:

cos(A+B) =  cos(A)cos(B)-sin(A)sin(B)

sin (A+B) = cos(A)sin(B) + sin(A)cos(B)

Trig Identities from Properties of Complex Calculations

Formulas for cos(2A) and sin (2A)

From cis(2A) = cis(A)*cis(A) we found

cos(2A)+ i sin(2A)  = (cos (A) + i sin(A)) (cos (A) + i sin(A))

 = cos2(A)-sin2(A) + i 2cos(A)sin(A)

Therefore comparison of rectangular coordinates/components (or real and imaginary parts) yields the double angle formulas

cos(2A) =  cos2(A)-sin2(A)

sin(2A)  =  2 i cos(A)sin(A)

Formulas for cos(nA) and sin (nA), the case n =3.

Observe

 exp(i3A) = cos(3A) + i sin (3A) = exp(iA) exp(iA) exp(iA)

Therefore

exp(i3A) = {cos(A) + i sin (A)}3 =

=   .cos3(A) + 3cos2(A) i sin(A) + 3 cos(A) {i sin(A)}2 + (i sin(A))3

= .cos3(A) - 3 cos(A) sin2(A) + i{3cos2(A) sin(A) - sin3(A))

Equality of corresponding real and imaginary parts gives

sin (3A) = 3cos2(A) sin(A) - sin3(A) = 3( 1- sin2(A))sin(A) - sin3(A)

= 3 sin(A) - 4 sin3(A)

cos(3A) = .cos3(A) - 3 cos(A) sin2(A) = cos3(A) - 3 cos(A) (1-cos2(A))

=  4  cos3(A) - 3 cos(A) 

Exponential or cis Expressions for Trig Functions

From the two equations

exp(iA) = [cos(A),sin(A)] = cos(A) + i sin(A)

exp(-iA) = [cos(-A),sin(-A)] = cos(A) - i sin(A)

we see

exp(iA) + exp(-iA) = 2 cos(A)

and

exp(iA) - exp(-iA) = 2i sin(A)

Therefore

(1/2)[exp(iA) + exp(-iA)] =  cos(A)

and

(1/2i) [exp(iA) - exp(-iA)] =  sin(A)

Therefore

                    tan(A) = sin(A)/cos(A)  = (1/i) { [exp(iA) - exp(-iA)]/[exp(iA) + exp(-iA)] }

or              

      tan(A) = sin(A)/cos(A)   =  -i{[exp(iA) - exp(-iA)]/[exp(iA) + exp(-iA)]}

So all trig functions may be expressed in terms of exp(iA) and exp(-iA).  The substitution of exp(iA) and exp(-iA) expressions for trig functions turns the proof or derivation of trig identities into simple algebraic exercises involving complex numbers and the add the angles rule for multiplication of exp(iA) with exp(iB).     While highschool students may be taken through the exercises of proving trig identities before meeting complex numbers, the above quick explanation of complex numbers  and its links to trig imply a quick route through highschool mathematics courses on algebra and trig.  In retrospect, the presentation of trig in highschool has been harder than need-be.                               

More motivation and more applications of trig, tied to right triangles and complex numbers,  come from engineering and physics.  The description of electric currents and devices depends on sine and cosines, or more simply, if you know complex numbers and exp(iA) = exp(iA). The latter functions also appear in the theoretical treatment of statistics and of higher mathematics (analysis). .  

Extra: Exponentials of Complex Numbers

If you have met the exponential ea = exp (a) of a real number a then you may be please to know that the exponential of a complex number a + ib (where a and b are real) is given by

exp(a + ib) = ea exp(ib) = ea cos(b) + i ea sin(b)

Take this as a computational definition. Higher mathematics in science, engineering, statistics and economics is often best seen with  a knowledge of  the exponentials  ea  or eib or ea+ib .    One may write ex instead of exp(x), or vice versa.

 

 

www.whyslopes.com

Analytic Geometry 
& Functions 

[#] shows no. of lessons.
  
Up
Equal Sign Use/Abuse
Real Numbers
Simplify Square Roots
Absolute Value |x|
Say More Than
Theory of Inequalities
|x| Eq'ns & Inequalities
Rectangular Coords 1, 2&3D
Distance Formulas - 1, 2 & 3D
Shortest Path
Triangle Inequality
Point Addition & Real Multiples
Polar Coordinates
Radians
(A) Vectors
(A) Coordinate Arithmetic
(A) Navigation on Maps
(A) Addition Geometrically
(PT) Translations
(PT) Dilatations
(A) Rotation
(C) Complex No. Intro
(C) Distributive Law - Applied
(C) Properties
(C) Complex Conjugates
(C) Pythagoras Thm, New Proof
(T)Trig on Unit Circle
(T) Complex No.s &Trig
(T) cis or exponential FNS
(T) Dot & Cross Products
(T) Cosine Law
(T) Pythagoras Converse

Section Entrance

Links
More Links
learn more
Two Treatments of Geometry
BIG Table of Contents
conic sections briefly

 

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.

 

 

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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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