www.whyslopes.com:  Parents:  Help Your Child/ Teen Learn
Appetizers and Lessons for Mathematics and Reason  (Français)
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Online Volumes (Book Orders)
1,  Elements of Reason. (1996)
1A. Pattern Based Reason  (1995)
1B. Math Curriculum Notes (1996)
2. Three Skills for Algebra  (1995)
3.-Why Slopes & More Math  - (1995)

Read chapters 1-5, 8-12, 14, 16 & 17 
in  Three Skills for Algebra 

More Site Areas 
1.  Solving Linear Equations  (04-2005)
2.  Fractns, Rates Proportns, Units  (2006)
3.  Algebra, Odds & Ends, HS level-(2001)
4.  Euclidean Geometry  & Cmplx NOs-new 
5.  Analytic Geometry/Functions (2006)
6.  Number Theory. (2006-7)
7.  Complex Numbers (2001)
8.  Calculus Introduction (2005)
More Site Areas 
99   Real  Analysis (1995)
10. Secondary IV(?) maths (2006-7)
11. Math Education Essays  (2006-7)
12. LaTeX2HotEqn: (2004)
13. Electric Circuits Etc  (2007)
14. Quebec Math Ed (2004 -8))
15  Prequel to the How-TOs (06-2008)
16-Virtual-Classroom (Take a peek)
Tutor -Teacher-How TOs-(08- 2008)
1. Arithmetic Reference
2. Algebra 
3. More Algebra 
4. Geometry  
5. More Geometry
6. Calculus
7. Logics in Maths
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YOU are better than YOU think. Show yourself  how:

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 For better work & study skills, read logic chapters 1 to 5  in  Three Skills for Algebra. Sooner is better. Good luck.

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 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

Do not leave here without it -  Logic mastery  will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.

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Caution: Site advice is approximately correct, for some circumstances, not all. Site How-TOs are logically developed, but not tried and tested. That leaves room for thought and refinement..

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After logic  (a) continue reading Three Skills for Algebra, chapters 8 to 14  and do so alongside site area on solving linear2007 Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes  & More Math, chapters 2 to 6;


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.


Explore collaborative whiteboards from groupboardtwiddla  or scriblink.


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 .... a2a1a is greater than bmbm-1 .... b2b1b when and only when there is a whole number k such that ak >  bk and aj = bwhenever 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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Number Theory

Back
Up
Next

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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