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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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Prime Decomposition of Composite Numbers.
A prime decomposition (a.k.a factorization) of a composite number > 1 is a
product of primes which results in the whole number. The prime
decomposition of a prime number is the prime number.
According to our previous discussion, the order or regrouping of factors in a
product does not affect the results. So we may arrange the prime factors in a
product so that equal factors are adjacent and so that larger prime factors come
after the smaller prime factors in the product. We may call that an ascending
prime decomposition, and in it count the number of times each prime factor is
occurs or is repeated. That count gives the multiplicity of the prime factor
p.
Unique Prime Decomposition (Factorization)
Theorem: If whole number N = AB is a product of two
small whole numbers A and B, both greater than one, and prime number p
is divides N with no remainder then p divides at least one of the factors A
and B with no remainder.
Proof: Let a be the remainder after division of A by p.
Let b be the remainder after the division of B by p. Then a and b are
whole numbers > 0 and < p with ab= 0 modulo p. Now ab is
> 0 and < p**2. So ab=0 modulo p implies the product ab = p or ab=0.
Now is p is prime. So it is not the product of two smaller numbers.
Therefore, ab=0 is the only possibility. But the latter cannot be if
both remainders a and b are nonzero. So at least one of the remainders a and b
must be done. That completes the proof.
Corollary I: If a prime factor p has multiplicity k in one prime
decomposition of a whole number N then it has multiplicity at least k in
any other prime decomposition.
Corollary II: If a prime factor p has multiplicity r in one prime
decomposition of a whole number N and another multiplicity t in another prime
decomposition then the two multiplicities r and t are equal.
The foregoing arguments imply each whole number N has only one ascending
prime decomposition and that multiplicity (power) of each prime factor depends
on the whole number N. If a prime does not divide N, we may optionally as
an accounting convenience include it the prime factorization with the power
zero. We may also take the prime decomposition of unity (one) to be 1 as
another accounting convenience.
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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 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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