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 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.
Proper and Improper Factors
for Whole Numbers
Let N be a whole number. Then a whole number A < N is a whole number factor of N when and only when N = AB for some other whole number B.
Each whole number N has two improper factors, namely itself and the number 1. All other factors, should they exist are proper. Prime number have no proper factors.
Our aim is to count the whole number factors of a number N with the aid of its prime number decomposition. The example below is followed by the general case.
Example: Let N = 60. Then
N = 10*6= (5*2)*(3*2) = (5*2)(3*2).So the prime decomposition of 60 is 5*3*22.
Now every factor A of N = 60 cannot be a product of primes other than the primes 5, 3 and 2 appearing in the prime decompositions of N= 60 Moreover, in the prime decomposition of A, the primes 5 can appear at most once each and the prime 2 can appear at most two times. So A = 5p3q2r where p may have one of two values, 0 and1, q may have one of two values 0 and 1 and r may have one of three values 0, 1, 2 and 3.
The product rule for counting implies there are 2 *2 * 3 = 12 factors of the form 5p3q2r. The factors given by p = q =r =0 and p = q = 1 and r = 2, that is, the minimum and greatest allowed values of the indices p, q and r are the improper 1 and N = 120.
Generalization
The foregoing example shows how to generate all possible factors, proper and improper, when the number of different primes and the number of times each appears are all small numbers. The above approach has merit in that it shows how to ensure the list of factors is complete in a systematic fashion.
Counting Integral Whole Number Factors: If the prime decomposition of a whole number
N = p1m(1)p2m(2) ... ptm(t)
for t primes p1, p2 , .... pt where each prime pk occurs m(k) times then
- Each whole number factor has the form p1q(1)p2q(2) ... ptq(t) where 0 < q(k) < m(k) for k = 1 to t.
- There are exactly M = [m(1)+1][m(2)+1] ..... [m(t)+1] proper and improper factors of N.
Note claim B follows from A and a or the product rule for counting.
Example: The whole number
N = 21 32 73 112 135 has
M = (1+1)*(2+1)*(3+1)*(2+1)*(5+1)
=2*3*4*3*6 =432
factors, proper and improper. Their generation and listing is best left to a computer program.
www.whyslopes.com
Number TheoryStart 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 Place Value Reinforcement 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 Infinite Decimals Expansion Arithmetic 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..