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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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Whole Number Comparison with Decimals
Ordering of whole numbers is learnt by comparison of 1, 2, 3 and
then multi-digit decimal forms or representations.
The digits 0 to 9 in ascending or increasing order are: 0, 1, 2,
3 ,4, 5 ,6 7, 8, 9
Counting with decimals gives polynomials
a0 + a1101 + a2102
+ a3103 + a4104 + a5105
+ .... = a5a4a3a2a1a0
in which the coefficients , a5, a4, a3,
a2, a0 are given by digits 0 to 9.
If two whole numbers are being compared via their decimal representations two
cases are possible.
Example 1: Decimal Representation have different lengths:
4912 is greater that 956
as the leading digit in the first occupies the thousand column while the
leading digit in the second occupies the hundreds column.
Example 2: Decimal Representation have same lengths:
34589 is less than 56721 as the leading digit 3 is less than 5.
Learning by Examples: In a mathematics classes over a few years, many
examples would illustrate the comparison of whole numbers by their one, two,
three and longer decimal representations. How to describe more clearly in words
fail me at the present time. In an advanced mathematics courses, one might
discuss Lexicographic Ordering:
Lexicographic Ordering: Given two decimal
representations anan-1 .... a2a1a0
and bmbm-1 .... b2b1b0 of
with n and m digits, we may pad one with leading zeroes to get m = n.
Then anan-1 .... a2a1a0
is greater than bmbm-1 .... b2b1b0
when and only when there is a whole number k such that ak >
bk and aj = bj whenever n = m >
j > k (Something more could and possibly should be said here.)
Comparison by Conversion Method:
- 956 is greater than 436 has the first number has the same number of units,
but more tens and more hundreds - no conversion required.
- 10000 is greater than 8269 as 1000 = 9999 + 1 = 9990 + 10 has more
thousands, more hundreds, more tens and more ones.
- 3234635 is greater than 18722 as 32465 = 3233635 + 1000 = 323365 + 999990
+10 has more units, more tens, more hundreds, more thousands and so on than
18742 (maximal conversion method for comparison)
- 32 is greater than 18722 as 34635 = 24635 + 1000 = 24635 + 9900 + 100 has
more units, more tens, more hundreds, more thousands and so on than 18722
(minimal conversion method for comparison)
Associated Subtraction by Conversion Method:
The minimal conversion method for comparison implies the common (minimal)
conversion method for subtraction.
3 2 3 4 6 3 5
- 1 8 7 2 2
by conversion, that is by writing 30000 as 29000 + 1000 gives
2 9 10
3 2 3 6 6 3 5
- 1 8 7 2 2
Here the 3 in the ten thousand position is crossed-out and replaced by
2, 9 and 10 in the top row to indicate the replacement or conversion
of 30000 by/to 29000 + 1000. Here position indicates place value.
Following the conversion, in each column, the rows to be added combined have
more ones, tens, hundreds and so
the minuend than the subtrahend. So the difference is easily computed
2 9 10
3 2 3 6 6 3 5
- 1 8 7 2 2
3 2 1 7 9 1 3
Remark: If a conversion method other than minimal is
employed, the subtraction will result in carries.
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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
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..
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