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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
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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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Prime Decomposition of Whole Numbers
This lesson focuses on obtaining prime factors and prime
decomposition of whole numbers. Prime decompositions are called prime
factorizations as well.
Recognition of prime factors and the prime decomposition of
whole numbers aid calculations of LCM, GCD and LCD in arithmetic with whole
numbers and fractions. They also lead to cosmetic simplification of square
roots. The recognition of common factors in numerators and denominators of
fractions alone or in products helps with the reduction of fractions and via
cancellation leads to efficient methods for multiplying fractions.
The question of whether or not, a whole number is prime may be related to its
decimal representation.
A whole number is not prime if is a proper multiple of 2, 3, 5, 7 or 11, or
equivalently if it remainder, modulo these small primes is zero. The multiple
one of each of these primes is considered improper. The decimal-based rules for
recognizing multiples of these primes (or calculating remainders) can be used to
recognize prime factors.
The numbers 2, 3, 5, 7 or 11 are the smallest primes. A theorem of Euler
shows there the number of primes is unlimited. Given any finite sequence
of prime numbers, their product plus one is a prime or is a multiple of a prime
not in the sequence.
Squaring the five primes 2, 3, 5, 7 or 11 squared give the sequence 4, 9, 25,
49 and 121 of whole numbers.
Theorem: If N = AB is product of two whole numbers A and B
where A > B > 1 then A2 > N and
N2 > B.
Proof: N = AB < AA = A2 .Therefore A2
> N. Similarly N = AB > BB = B2 .Therefore
B2 < N.
The above theorem implies if N is a product of two whole numbers, both
greater than one, than the smallest squared will be less than or equal to
N and the largest squared will be greater than or equal to N. Now the smallest
factor is a prime or it a multiple of a prime. In either case there is prime
whose square is less than or equal to the smallest factor squared and hence less
than or equal to the original number N.
Theorem: If N = AB is product of two whole numbers A and B where
A > B > 1 then N has a prime factor p with square p2
< N.
Contrapositive Consequence: If N has is not a proper multiple of all
primes p with square p2 < N then N cannot be decomposed in
to a product of two smaller whole numbers, and hence N is prime.
Squaring the first six primes 2, 3, 5, 7 or 11 and 13
squared give the sequence 4, 9, 25, 49, 121 and 169 of whole
numbers.
The Contrapositive consequence implies the following.
If N < 169 = 132 is not divisible by any of the primes 2, 3,
5, 7 or 11 then N is a prime number.
Proof: If N < 169 is divisible by a prime less
than itself, then there is a prime p with square p2 < N
< 169 which divides into p. So p has to be one of the first five
primes 2, 3, 5, 7 or 11 as all further primes have square > 169 >
N
So to check if a whole number N < 169 is a prime, is enough to compute the
remainders for N when N is divided by 2, 3, 5, 7 or 11. The latter can be done
with the help of divisibility rules for recognizing multiples of these primes,
with the help of the 10 or 12 times table, or with the aid of a
calculator.
Calculator Rounding Hazard: If some prime
p, N/p = q for a whole number q according to your calculator, due to the
possibility of rounding, you need to compare pq and N. Most calculators can
compute a product of whole numbers exactly. So if the product pq is not
equal to N, you know that some rounding error led you to think p was a divisor
and N was the exact multiple q of p.
Rule of Thumb
In learning and applying algebra exactly, one only needs to
compute with fractions or ratios of whole numbers less than 100 or so. So the
efficient ability to find recognize prime factors of whole numbers less than 169
(or 100) appears sufficient for most purposes in high school and college
mathematics and science courses.
Curiosity
Theorem: If N = AB is product of two whole numbers A and B
where A > B > 1 then N has a prime factor p <
N with square p2 > N.
Contrapositive Consequence: If N has is not a proper multiple of
all primes p with square p2 > N then N cannot be
decomposed in to a product of two smaller whole numbers, and hence N is
prime.
The "density" of primes relative to the set of whole numbers
gets smaller as the primes increase. So if you are searching for
primes factors of a large number N with the aid of a calculator or
computer program, checking for divisibility of N starting with the largest
primes p satisfying p < N with square p2 >
N might involve less work (fewer divisions) than starting with the
smallest primes p with square p2 < N. |
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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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