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YOU are better than YOU think. Show
yourself how:
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Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work
and study
Learn to read notes and textbooks like
a lawyer, so that no nuance, no subtlety and no clause escapes your
attention. |
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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.
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11. Distributive Law for Real Numbers
Summary: The distributive law for real numbers
implies that changes of scale or direction in coordinate systems determined
the by selection of unit vectors in the addition of vectors in the line or
plane do not affect the result. This essay shows the converse, namely if
the addition of vectors in the line or plane is not affected by a change of
scale and/or direction in coordinates systems then the distributive law holds
of the addition and multiplication of real numbers.
Option: Readers may restrict directed line
segments to being a rational multiple of a unit length and assume or require
or changes of scale involve rational multiples rather than real numbers.
Theorem: If a, b and c are real numbers then (a+b)c = ac+ bc
and
c(a+b) = ca+ cb.
First Proof: If c = 0 then both sides are equal and there is nothing
to prove. So assume without loss of generality that c is non-zero.
Let m = c k where k is a non-zero vector. Then
both m and k can be employed as unit vectors for a real
number line.
Now (ac+bc) k
Therefore (a+b)c = ac+ bc. by the unique measurement assumption for
unit vectors. The equality c(a+b) = ca+ cb now follows as multiplication is
commutative.
Alternate Proof - Changing the Coordinate Scale
Let unit vector k be a unit vector for a straight line -
a real number line.
Each point P on a straight line may be identified with its
position vector, with a unique multiple pk with tail at the origin and
head at P. Let Q be another point likewise identified with it position
vector qk. Then P+Q = (p+q)k can be identified with the position vector
of another point T.
That being said, the addition of the position vectors is
independent of the selection of unit vector k. Let k = c m
where m is another nonzero vector. Then
P = pk = p (c m) = (pc) m
Q = qk = q (c m) = (qc) m
P+Q = (p+q) k = (p+q) (c m) = ((p+q)c)
m
But P+Q = (pc) m + (pc) m = (pc+qc)
m as well.
Therefore unique measurement assumption implies the two
expression the coefficients of m in the representations of P+Q, that is, in
((p+q)c) m = (pc+qc) m
must be equal. Therefore
(p+q)c = pc +qc.
The latter provides a second proof of the distributive law.
Remark: In the above proof, p and q are the
coordinates of P and Q relative to the choice of k = cm as a unit
vector for a coordinate system. Likewise pc and qc the coordinates of P
and Q relative to the choice of m as a unit vector for a coordinate
system. The distributive law implies the coordinates of sum P +Q can be
calculated relative to k and then transformed (multiplied by c) or the addends
can be transformed first and then added. So the sum of vectors can be
calculated directly or in any unit -vector based coordinate system. The
distributive law is equivalent to the latter invariance. | |
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 Theorem
Primes & Composites
Prime Factorization Aids
Prime Factorization Examples
Counting Whole No. Factors
Arithmetic Videos
Square Roots & Primes
Long Division Continued
Fractions & Decimals
Fractions as Decimals
1 = 0.999 Recurring
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
Related Site Pages:
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
Food for thought: 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..
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