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YOU are better than YOU think. Show
yourself how:
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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
Do not leave here without it - Logic
mastery will develops critical thinking, improve reading and
writing, and give a firmer base for work and studies at many levels.
Good luck.
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Caution: Site advice
is approximately correct, for some circumstances, not all.
Site How-TOs are
logically developed, but not tried and tested. That leaves
room for thought and refinement.. |
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After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside
site area on solving
linear2007 Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
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.
|
Explore collaborative whiteboards
from groupboard,
twiddla or
scriblink.
|
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Chapter 4. Longer Chains of Reason
Previous Section: Chapter Entrance
The principle of mathematical induction stated below describes the above
(previous) ladder idea in the algebraic shorthand notation favored in
mathematics. The last part of this chapter will not make sense to you if you are
not familiar with this shorthand notation. If this is the case, you may skip
this description of mathematical induction. If you read it, and you find that
you do not understand it, you could return to it later after you have seen the
following chapters on algebra. They explain the use of shorthand in algebra.
Mathematical Induction
We assume that when or if we have counted to any number n,
we can count to the next one as well. Just add one to the count n. This
gives the next number in our count which is written n+1. This offers a
way to begin counting all the whole numbers 1, 2, 3, 4 and so on.
Suppose or imagine for each whole number n, there is a situation An.
This gives a step on the ladder. Now the next whole number after a whole number n
is given by adding 1, that is n+1. So the next step after An
is written as An+1. The principle of mathematical
induction says the following:
If
- for each whole number n, there is a situation An;
- every time the situation An occurs, the next
situation Am = An+1 with
m = n+1 must also occur; and
- the first A1 situation occurs,
then all the situations An (where n is
a whole number) occur.
The word occurs can be replaced by the expression can be reached. The
principle of mathematical induction is quite simple. It requires the following:
(1) there is a ladder; (2) on the ladder, from each step we can reach the next;
and (3) the first step is reachable. When these three requirements are met, the
principle of mathematical induction says: all the steps can be climbed or
reached. That is all there is to this inductive principle.
Question. What can be said about the reachability of An
where n ³ 4 if we find a ladder for which
requirements (1) and (2) are met, and we somehow know (3¢)
that A4 is reachable? Hint: Imagine a ladder where the first
three steps are broken, but the fourth is somehow climbable. Is the ladder
climbable?.
Next Chapter: 5. Islands and Division of Knowledge
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[ Back ] [ Next ]
Three Skills for Algebra
www.whyslopes.com
Foreword, Chapters
& Appendices
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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