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Area pages
Words Before Symbols
(what is a variable?)
All of Volume 2
except
for chapter 18
Foreword
1. Introduction
2. Implication Rules
3. Chains of Reason
4. Induction (Longer Chains)
5 Knowledge Islands
Assuming Consistency
6 Old Language
Arithmetic WebVideos
7 Arith Skill Check
A Arith Problems
Solutions
B Caclulator Problems
Solutions
C More Arith Problems
Solutions
D Algebra Problems
Solutions
8 The Three Skills
9 Numbers & Quantities
9 Everyday Words
9 Words Math Usage
9 Precision or Not
9 Numbers & Quantities
9 Further Readings
Words Before Symbols
10 Two More Skills
10 Shorthand
10 Changing Calculations
10 Find a Number
11 Why Shorthand
12 Shorthand Usage
12 Symbols & Pronouns
12 Symbol Overuse
12 Symbols & Numbers
13 What's Next
14 Compound Interest
14 The Formula
14 Direct Use
14 Indirect Use I
14 Indirect Use II
14 Further Notes
15 Linear Equations
15 Algebra Solutions
15 Triangular Systems
15 Making Triangular
15 With 3 Unknowns
15 Rules and Advice
16 Painless Proofs
17 Pythagoras
19 Functions & Sets
20 Degrees & Radians
21 What's Next
22. Sums
22 Arithmetic Sums
22 Geometric Sums
23 Sum Shorthand
24 Your Money
24 Periodic Deposits
24 Account Tracking
24 Pension Plans
25 Inductive Proof.Eg
25 Factorial Definition
25 Product Notation
25 Notation for Sums
26 What's Next
27 Pronouns in Logic
28 Occurrence Tables
29 Contrapositive
30 Truth Tables
31 Indirect Reason
A. Appendices
A How to Learn
A How to Read
A. What to do in School
A. How to Study Math
Appendices above offer advice approximately correct and
repetitive for some circumstances, not all.
See more Volume 2 Postscripts in Right Margin
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Consistency and Reality
Here is a postscript to volume 1A, Pattern Based Reason,
and to the logic chapters in Volume 2, Three Skills
for Algebra. See too the last chapters of 2
on direct and indirect methods of reason, statements in logic and their
truth tables, (this postscript is not in the printed version).
This postscript may be best read later, after or besides the last chapter in
Volume 2. The online versions of 1A, 2 and 3 include postscripts
not in the printed version.
For Consistency Sake
Law of the Excluded Middle: A or Not A.
Let A be the statement that some situation occurs. Then a story or
theory that suggests a statement A is both true and false is inconsistent. So
for the sake of consistency in our present and further reason, we may require
and assume the statement
A AND Not A
to be false - NEVER TO OCCUR. So in our story or theory in its present and
further development, we require
A OR not A
to be true but not both at any instance (except during a brief transition
period).
So A requires not (not A) for consistency with A AND not A, and not (not A)
requires A at any instance (except during a brief transition period).
Remark: The discussion of transition time suggests
the law of excluded middle might be broken momentarily when situations are
time-dependent or place dependent. For example, in counting people in a
room that has a door, we cannot say a person is all in or all out because of
the middle possibility of a person being part in and part out. So a person has
three static states namely, in, out and partly both, and two transition
state namely, going from in to out, and going from out to in. During these
transitions, the middle state of partly in and partly out occurs for a short
or long period of time.
The CONTRAPOSITIVE.
The first situation
A AND not B
is inconsistent with the implication rule
IF A THEN B.
So in circumstance where the latter implication rule IF A THEN B. holds (is
not disobeyed), we conclude or require the first situation
A AND not B
not to occur. The non-occurrence of A AND not B in turn implies the
original implication
IF A THEN B
and the contra positive implication
IF not B THEN Not A
Since both imply not( A AND not B), the two implications are equivalent
to each other and to the non-occurrence of A AND not B.
Proof by Absurdity
alias proof by contradiction
In telling a story or developing a theory, we may look at the consequences of
our assumptions - the situations we tend to assume as holding or being true.
If a chain of reason implies that a situation C occurs and does not occur, then
the story or theory is inconsistent - becomes absurd. For the sake of
consistency, the story or theory needs to be revised or abandoned.
- Example One: A detective in solving a crime may have a
suspect. Then he may found the suspect has an alibi which directly or
indirectly implies she did not committed the crime. So the alibi and
suspicion are inconsistent - that is incompatible. The detective may drop
the suspicion or challenge the alibi. Lawyers for the prosecution and
defense may erect competing chains of reason, and leave it to a jury or
judge to decide which one, if any, appears to be true.
- Example Two: Assume any infinite decimal expansion locates a point
or distance on a real number line. Assume further that each ratio of two
whole numbers can be expressed a ratio of two whole numbers with no
common divisors? The Pythagorean theorem then suggests in an isosceles
right triangle, the ratio of the hypotenuse to each of the others sides, the
legs, by length given by the square root of 2. Is that square root
equal to a rational number? The suspicion or assumption that YES, the square
root of 2 equals a rational number implies an inconsistency.
Namely, that in any ratio or fraction that represents the square root of 2,
the denominator and numerator will both be multiples of 2. So the square
root of two cannot be rational.
The Pythagoreans in finding the inconsistency in example 2 had a problem.
They assume lined segments in the plane represented numbers and they
assumed all such lengths were rational multiples of each other. When these
assumptions or their consequences clashed, reconciliation was not obvious.
Their view of numbers collapsed and a replacement was not available. That was a
serious problem for the Pythagorean school in their theory of knowledge was
based on and assumed rational numbers and only rational numbers.
Today, however, we have an advantage or two. One advantage is a our
assumption that infinite decimals expansions represent rational and irrational
numbers. Physically, if you imagine ruler with a unit length and its
division into tenths, hundredths, thousandths and so on, then you can count the
maximum number of units, tenths, hundredths, thousandths and that may fit
in a line segment. The result is a sequence of numbers which provide a
better and better approximation to the line segment length. You can
further imagine that sequence of approximations given by two, three, four, five
and more decimal places in a decimal expansion locate the end of a line segment.
The decimal representation of numbers does not depend on nor require the
"numbers" to be rational.
Aside: The number 1.0000 represent a single unit of
length. It also represents the limit of the sequence 0.9, 0.99, 0.999, 0.9999
and so on. The sequence of decimal approximations 0.99999 (9 recurring) is
shorthand for a sequence of lengths L(j) =1 - 10-j < 1
where the lengths are increasing, and where the differences d(j) = 1 -
L(j) = 10-j is getting smaller and smaller as j increases. So
the sequence approaches 1. Because the difference tends to zero, the
number 1 has several decimal expansions
- 1, 1.0, 1.00 (0 recurring finitely many times)
- 1.000 (0 repeating indefinitely)
- 0.99999 (9 recurring)
where the last one represents 1 as the limit of a
sequence of approximations 0.9, 0.99, 0.999, 0.9999, 0.99999, ... .
Reality Versus Fiction
The author of a story in a book or a play creates an imaginary world for us
to visit in our minds. The story may or not be consistent with our
knowledge of real life. More and less can be suggested in a story than occurs in
real-life. Stories can be fictional, half-fictional, approximately true to
life or true.
Stories may explain or describe how things came to be. Stories
may provide lessons a for reader directly or through the words and
interpretations of another. Stories may give us ideas of what to do or
not. Stories have plots and chains of events or reasons to follow, real or
not. Stories presented on stage as plays may include not only words but also
actors and props to make the plot or reenactment easier to follow. Actors have
scripts to follow. Actors are defined by their names, costumes and actions.
Most of us, many of us, have the ability to follow a story, its sequence of
scenes with words and events, and to recognize what is real or pretend. We
can learn stories, invent them and tell them to others via spoken and written
words. Stories can be told and retold in ways that are almost repeatable and
reproducible. Our knowledge of a culture may come from its stories and myths.
In cooking and construction, plans and recipes give or suggest sequences of
steps or actions to take to arrive at results. The steps and the results should
be repeatable and reproducible. Technical know-how is based on rules and
patterns to follow plus some judgment as to when they can be applied. Trying to
apply rules and patterns when items they require are missing usually leads to
bad results.
In mathematics, science and technology, as in daily life, there are stories
to follow. These stories, normally called theories, describe a situation (say
what is what is assumed) and describe as a well assume methods for arriving at
results or conclusions in a step by step way. The authors of these
stories or theories would like their consistency with reality. A theory is
inconsistent with reality if it says two exclusive events occur at the same time
or if predictions based on it fail. Unfortunately, the author of theory to say
what should happen may capture a pattern in theory that works in some
circumstances, but not all. So a theory may be applicable and sufficiently
consistent with reality to be useful in some circumstances - those it reflects -
while failing in others.
Knowledge in mathematics and science and technology is based on theory and
practice. A method or procedure describe in a lab or controlled circumstances
how following certain steps will give a result. Those steps and the
results, done carefully enough, appear to give repeatable and reproducible
independent of the doer. Methods that work in practice may be described and
accumulated, and used one at a time and one after another to follow steps and
arrive at results, one at a time and one after another. A skilled
practitioner may recognize when one method can replace another because it gives
the same result or a more convenient result.
Geometry was codified in the works of Euclid, about 300 B. C. The
codification consisted of assumptions or definitions about points, straight
lines, circles, triangles and the geometric figures composed from the latter.
The resulting theory or theories was presented not on stage, but on paper (a
prop) with the aid of rulers and compass (more props) to provide construction
methods and to suggest and describe results and conclusions one at a time and
one after another. Students and teachers and philosophers could follow
explanations one at a time and one after another in way that follow some or all
of the strands of thought in Euclid's work. The codification provides a
mechanical knowledge of geometry because each of us in following the steps
should verify that the steps are valid, that the implication rules used in each
step are justly applied.
The foregoing gives rule- and pattern-based chain of reasons independent of
the followers and authors. All that provides a model for making and
arriving at conclusions with rules and patterns not only in geometry, but also
in other disciplines where rules and patterns are valued as guides. But this
model for reason has its limitations.
Rules and patterns describing what we have observed, drawn from experience,
are not absolute. We do not know if they are fully reliable, or we may not
precisely when they apply, if at all. When rules and patterns are not reliable,
a risk appears. What they suggest, one at a time and one after another, may not
be consistent with reality. None the less, recognizing rules and patterns in a
subject provides a means for accumulating know-how for arriving at results, and
a limited know-why. The latter is given by the chain of reason or
suggestion with rules and patterns, reliable or not, that led to a result.
(Implication rules and suggestions in a theory may themselves rely on the need
for a theory to be consistent. See above). Volume 1A, Pattern
Based Reason, gives a further description of the benefits, origins and
limitations of rule and pattern based thought. Not all is certain.
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Online Volumes
1Elements of Reason
1A. Pattern Based
Reason
1B. Mathematics
Curriculum Notes
2. Three Skills
for Algebra
3. Why
Slopes and
More Math
More Site Areas:
Helping
Your Child or
Teen Learn.
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Linear Equations with Stick Diagrams
Fractions,
Ratios, Rates, Proportions
& Units
Euclidean
Geometry
Analytic
Geometry
Number
Theory.
Calculus
Intro
Complex
Numbers
Quebec Maths Education
Real Analysis
LaTeX2HotEqn
Good news:
Site pages identify what you need to study.
Bad news: Site pages do not explain
everything
Worse news: Learning takes time, yours.
Volume 2, More Extras
(not in printed version)
1b Problem Solving Methods
2. Exact Arithmetic
3. What is a Variable?
4.. Square Roots
5. Straight Lines
6.. Quadratics
7. Trig and Complex No.
8. Complex Applet
9. History of No.s
10. ln(x) and exp(x)
13. Rename the > Sign
14. Problems: Quadratics
15. Problems: Algebra Test
16. Problems: Linear Eqns I
17. Problems: Linear Eqns II
18. Problem Solving Hints
19. Functions & Sets
20. Independent Variables
21. Why Logic
22. Why Math
23. 15 Times Table
24. 20 Times Table
25. Algebra Formulas
26. On Learning Maths
27. Maths in Biology
28. Navigation +Time
29. A Quibble: What is Algebra
30. Logic in Maths
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