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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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9. Changing the unit vector
Changing the Coordinate Scale
A unit vector k may be replaced by positive real
multiple m
= q k. When q is a a natural number or fraction, we may argue that
A = s m implies A = sq k . So each multiple s of m
correspond to a multiple sq of k
Changing the Coordinate Scale and Direction.
Defining Products of Real Numbers
from a change of units
Saying how to compute a number defines it.
Suppose k is employed as a unit vector. Let m
= q k for some real number q and let p be a real number as
well. Then p m = b n for some unique real number
b by measurement assumption. We take pq (q times q) to be the number
c. That defines our product
Product Definition: Let pq = b if bk =
p(qk) for the unit vector k
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This computational method for pq raises a few questions.
- Is it consistent with the previously encountered method for multiplying pairs of fractions and unsigned real numbers with infinite decimal
expansions?. The answer needs to be Yes
- Does the product depend on the choice of unit length k? The
answers needs to be No.
Showing the answers are as required is also left for the reader for the
time being. In what follows, we assume pq is defined for pairs of real numbers pq.
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Exercise: Show the foregoing product definition leads to the law of
signs:
(positive)(positive) gives a positive
(positive)(negative) gives a negative
(negative)(positive) gives a negative
(negative)(negative) gives a positive
Exercise: Each real number can written as a sign (direction)
prefixed to an unsigned real number - the magnitude of the real number
Show the product can be given by multiplying signs and magnitudes
separately. From this show, earlier obtain properties of fractions and real
numbers imply multiplication is commutative and associative. One step in this
process is to observe or to show that the multiplication of signs is
commutative and associative.
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www.whyslopes.com
Number Theory
Start of Number Theory
Origins of Counting or Tallying
Adding Wholes
Multipling Wholes
Distributive Law Preamble
Distributive Law for Wholes
Consequences
More Consequences
What is a Fraction
Compound Fractions
Number Theory
Continued
Decimal Place Value
Comparison Method
Addition Method
Subtraction Methods
Multiplication Methods
Division Methods
Remainder Arithmetic I
Primes & Composites
Primes Factorization
Primes & Composites
Prime Factorization Aids
Prime Factorization Examples
Counting Whole No. Factors
Arithmetic Videos
Square Roots
Fractions & Decimals
Fractions as Decimals
1 = 0.999 Recurring
Long Division Continued
Ratio of Simple Fractions
Ratio of Decimal Fractions
Unsigned Reals Numbers
Signed Coordinates
Plane Vectors
Horizontal Vectors
How to Add Reals
How to Multiply Reals
Distributive Law for Reals
Remainder Arithmetic II
Related Site Pages:
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