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.
Equi-Sized Groups - Repeated Addition,
Multiplication and its Properties
How many elements are there in a finite set of equi-sized groups?
Product Rule - Counting or Denoting How Many in Equal-Sized Groups
The number of elements in M groups, each of size N, gives a count denoted by M*N and called M times N or the product of M and N.
For example 1 times N is N and 2*N is N +N, 3*N is N+N+N. We will see later how to denote counts by decimals and how to mutliply decimals.
Commutative Law for Multiplication
Geometric Reason for 5 x 3 = 3 x 5
5 columns (groups) of three pluses
+ + + + + + + + + + + + + + + can also be viewed of 3 rows (groups) of 5 pluses without losing or gaining pluses - the tally marks. Therefore 5*3 = 3*5 = two ways of counting them. Here we use the assumption that any two ways of counting the same set of elements must yield the same number.
Geometric Reason for MxN = N x M
Now if M groups, each of size of N, are stacked in N equi-height columns, where each group element occupies a single square in that column, then all squares being the same size, we obtain a rectangle. Now we divide the total number present into N rows (groups) of M elements each. So that implies the total count or tally initially given by M*N is also given by N*M.
The foregoing as is or recast implies the commutative law of multiplication namely
Commutative Law: N*M =M*N whenever M and N are whole numbers.
Students of pure mathematics can recast the foregoing into set theory form where the number of elements in the Cartesian product A x B of two sets is M*N if A and B have M and N elements, respectively. The number of elements in the Cartesian product defines M*N.
Associative Law for Multiplication
In general, for any three whole numbers M, N and P,
N(MP) = (NM)P)
Proof: Suppose we divide a set of cubes T = N(MP) cubes into N one-cube high layers of M by P cubes, and stack the layers. The result is a rectangular box of cubes N cubes high, M cubes wide and P cubes long. Now we may recount the by viewing the box as NM columns, each P cubes long.. That yield T = (NM)P as well.
Corollary: N(MP) = (NM)P = (MN)P = M(NP) = P(MN) = (PM)N
We can take the value of NMP to be the common value of the six ways to compute the number of small boxes (cubes with sides of unit length) in a larger box with dimensions N units, M units and P units.
Remark. In more advanced mathematics, the foregoing may be rewritten algebraically to give a formal proof. By mathematically induction, we may argue that a product of r whole number factors yields a single result independent of the order and grouping. Thus grouping and regrouping view as different ways to count a r-dimensional hyper-box implies order of multiplication does not affect results.
Consequences of Associative Law:
In our discussion of decimal methods below, we will need the property that
(a 10m) x (b 10n) = (ab) 10m+n
That follows as (a 10m) x (b 10n)
= (a x 10m) x (b 10n) due to a change in notation
= a x (10m x (b 10n)) due to a 1st use of associative law
= a x (b x (10m x 10n)) due to a 2nd use of associative law or six ways to compute the product 10m x (b 10n)
= a x (b x (10m+n)) as 10m+n =10m x 10n
= (a b) x 10m+n due to a 3rd use of associative law
= (a b) x 10m+n due to a change in notationNote: Some authors might mention the commutative law in going from one way to compute the product 10m x (b 10n)to another.
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 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:
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