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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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Caution: Site advice is approximately
correct, for some circumstances, not all. That leaves room for thought |
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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.
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Chapter 18
Arithmetic Rules and Patterns
Previous: Sums as Factors -Product of Sums -
Distributive Law
The Non-Zero Product Rule from Decimal Arithmetic.
If you use decimal arithmetic to add two positive numbers
together, the result will be positive. If you use decimal arithmetic to
multiply two positive numbers together, the result will again be positive.
This implies the Nonzero Product Rule indicated below in the setting of real
numbers. .
We add another rule to those you have seen, namely if a and b
are both nonzero real (signed decimal) numbers or real quantities then their
product ab is also nonzero.6
This rule can be rewritten in contrapostive form: If a product ab of
two real numbers a and b is zero then at least one of the factors a
and b must be zero. The further explanation of this observation is an
intellectual IOU. To collect, read about the contrapositive for one-way
implication rules in the chapters on logic and reason below.
When a and b are real numbers. The contrapositive way of saying
If a and b are both nonzero then the product ab must be
nonzero
is
If the product ab of two factors a and b is zero,
then at least one of the factors a and b must be zero.
This zero product rule is used to obtain and justify the formula for the
solution of the quadratic formula. If you have not met the quadratic formula,
don't worry about it now. Later is sufficient.
The presence of (supposedly) never disobeyed rules and properties in
mathematics requires a knowledge of logic, that is rule- and pattern-based
reason. In mathematics, the never-disobeyed rules are stated in terms of
shorthand notation. So in mathematics after arithmetic, a knowledge of both
algebraic shorthand notation and logic is needed.
More Chapter Sections: [ Up ] [ 18 Changing Formulas ] [ 18. Proper Use of Equal Sign ] [ 18. Replacement & Substitution ] [ 18 Real Numbers & Quantities ] [ 18 Rules for Algebra ] [ 18 Sums as Factors I ] [ 18 Sums as Factors II ] [ 18 Addition Properties ] [ 18 Sum Associative Property ] [ 18 Sums and Number 0 ] [ 18 Replacing Subtraction by Addition ] [ 18 Times Properties ] [ 18 Sum Grouping and Ordering ] [ 18 Product Associative Property ] [ 18 Products with the Number 1 ] [ 18 Product Grouping and Ordering ] [ 18 Power Rules ] [ 18 To Divide, Multiply ] [ PS: Rules for Fractions and Division ] [ 18 Inconsistent Nttn ]
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www.whyslopes.com
Volume 2, Three Skills for Algebra -
Preview, starter & further lessons for logic and algebra
to (i) improve work & study skills; (ii) to to ease or avoid
algebra (math) fears & difficulties; and (iii) to fill gaps in the
exposition of mathematics.
Foreword, Chapters and Appendices follow.
Foreword
1. Introduction
2. Implication Rules
3. Chains of Reason
4. Romeo and Juliet
4. Induction Mathematical
5 Knowledge Islands
6 Old Language
7 Arith Skill Check
7. The Next Chapters
8 The Three Skills
8 VNR-Concise-Encyclopedia
PS. What is a Variable
9. Algebra Talk
10 Two More Skills
11 Why Shorthand
12 Shorthand Usage
13 What's Next
14 Compound Interest
15 Linear Equations
PS I. Distributive Law
PS II. Polynomials
16 Painless Proofs
17 Pythagoras
18 Rules of Algebra
19 Functions & Sets
20 Degrees & Radians
21 What's Next
22. Arith & Geometric Sums
23 Summation Notation
24 Your Money
25 Induction & Recursion
26 What's Next
27 Pronouns in Logic
28 Occurrence Tables
29 Contrapositive
30 Truth Tables
31 Indirect Reason
A. Advice For Learning
Words Before Symbols:
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
Complex number: starter lesson
Solving Linear Equations:
A. Letters and Lengths
B. & C. Solving Linear Eq'ns
with stick diagrams.
(i) x + 20 = 29
(ii) 2x + 5 = 20
(iii) 3x + 10 = 32
(iv) 5a + 16 = 3a+ 24
(v) (½)x + 8 = 24½
(vI) (¾)a + 16 = (¼)a+ 24
(vii) (¾)q + 17 = 32
(viii) 13 =[2/3]x +7 twice
(x) Animated Examples
(i) Integral Coefficients (A)
(ii) Integral Coefficients (B)
(iii) Fractional Coefficients
(iv) With
Parameters
Problem Solving with Linear
Equations in one or many
unknowns, and in essentially
one unknown - Symbols before
words.
C. Solving Linear Eq'ns
without
Stick Diagrams
D.
Problems in
essentially one unknown
E: 2D Systems - Sub Methods.
F. Larger Systems
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