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Appetizers and Lessons for Mathematics and Reason
by A. Selby, Ph. D.   Feedback & Questions

 20 pages in French: Algèbre  
 Définition d'une variable  
La raison basée sur les  règles et modelés

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Decimal Expansions of Fractions

Finite or Infinite & Periodic

The long division process applied to a simple fraction

M
N

will terminate before or after the decimal point and give a whole number or a decimal fraction in decimal form if the simple fraction is equivalent to a decimal fraction. 

The long division will not terminate if the simple fraction

M
N

is not equivalent to a decimal fraction.  In this case,  long division in the first gives

M = k N +r 

where the remainder r is a whole number between 1 and N-1.  The remainder r will be nonzero as we are not in the decimal fraction case. Continuing the long division process to q decimal places after the decimal point gives 

10q  *M = kq*N +  rq

where the q-th remainder rq is a whole number between 0 and N and is kq is a  whole number..  

Now the infinite sequence of remainders takes values in the interval [1, N-1] of whole numbers - N-1 possible values in all. Therefore the remainders must repeat. If not, the interval [1, N-1] would have infinitely values.

Pigeon Hole Principle: Now the leading N remainders rq  where 1 < q < N sequence of remainders  takes values in the interval [1, N-1] of whole numbers - N-1 possible values in all. Therefore the remainders must repeat. If not, the interval [1, N-1] would include N distinct values.  

 It follows that the decimal expansion of M/N must start repeating on or before the N first place. So the period of the decimal expansion is N or less.

Check Logic later: Am I misleading myself in the above argument?

Now for each whole number q of decimal places, the equation 

10q  *M = kq*N +  rq

implies 

M
---
N		  =  kq
-----
10q 		 + rq
-----
10q 

where rq is in the interval [1,N-1] repeats and eventually has period P in q, that is,   rq+P =  rq for q sufficiently large, and where km for m > q agrees with kq to q decimal places.  Therefore,

M
N

has a periodic, non-terminating decimal expansion

 aTaT-1 ....a1.b1b2b3b4b5b6  .... 

in which the first q decimal places are provided by the decimal fraction 

kq
-----
10q 

We assume that a non-terminating decimal expansion may be associated with a single point on a coordinate (half) line:

A Cauchy sequence f(n)  has the following decimal property: For each whole number k, there is a whole number N with the following property: all terms in the sequence after the first N-1 agree with each other to at least k decimal places. This property allows us to define and compute in principle an infinite decimal expansion. This expansion is assumed to define a unique real number: the limit L of the Cauchy sequence.

The limit of a repeating non-terminating, infinite decimal expansion is given by a simple fraction. For a proof, study the geometric series. 

Exercise: If two non-terminating periodic decimals differ at the k-th place in their expansion then the two decimals converge to different limits. Explain or see why.

 

Number Theory & Practices

Fractions as Decimals

A. Start of Number Theory
Section Entrance
Origins of Counting
Adding Wholes
Multipling Wholes
Distributive Law  Preamble
Distributive Law for Wholes
Consequences
More Consequences
What is a Fraction
Compound Fractions
Extrinsic Numbers Theory
Origins of Counting or Tallying

B. More Number Theory
& Practices

Arithmetic Videos
Decimal Place Value
Place Value Reinforcement
Addition Method
Comparison Method
Subtraction Methods
Multiplication Methods
Division Methods
Long Division Continued
Remainder Arithmetic I
Primes & Composites
Primes Factorization Theorem
GCMs and LCMs from Primes
Prime Factorization Aids
Prime Factorization Examples
Counting  Whole No.  Factors
N-th Roots and Primes
Fractions & Decimals
Fractions as Decimals
1 = 0.999 Recurring
Infinite Decimals Expansion Arith
Ratio of Simple Fractions
Ratio of Decimal Fractions
Unsigned Reals Numbers
Signed Coordinates
Plane Vectors
Horizontal Vectors
Adding Vector Multiplies
Adding Signed Numbers
Multiplying Signed Numbers
Distributive Law for Reals
Real Numbers Axioms
Remainder Arithmetic II

See too complex numbers.

For Senior High School  & Calculus Students

  <| (o)   (o)   |> 
 \     | |      / 
\___ _/
||
 -/[]\- 
||
   / \_ 

Words  to clearly introduce algebra and variables have been missing in course design. For people who cannot do algebra, 
the missing words may explain or ease their difficulties.  Volume 2 ,Three Skills for Algebra,  in Chapters 8 to 14 & 18 etc, puts words before symbols to providing the missing words in a way that enrich the comprehension of all.  Those words form the middle part of a algebra (and logic) lessons aimed at helping or improving all of  high school mathematics and also calculus course design & delivery. 

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.  If your  calculus   questions is not answered here, submit it. Over time, that may complete the site development of calculus. 

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

Mostly For High School

Intro to Solving Linear Equations 
- a different paths for junior and even senior high school students. Question for Tutors: When do you use and when you skip the stick diagram method here?

Fraction Skills,  thought-based  development, Ages 10 to 14 may need a tutor.  Students who have to understand in order to do may like the development in all or part. 

For Senior High School Mathematics & Calculus

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)

Tutors & Instructors: These lessons introduce skills differently Would you recommend them? 

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 for- & 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.

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).

For Instructors
- Education Essays   (opinions, possibilities, references) 
- Free Advice and Directions for teaching primary & high school maths will be given in online meeting place with voice & whiteboard.   
- 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.

Offering ideas to change education makes this site different.  Nothing ventured, nothing gained.  Site material is mathematically  correct, and where not, please report errors. The two level program POMME in the site entrance implies multiple paths for instruction. Supporting those paths in turn implies a clear destination  for site development and perhaps a new name.

 

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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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