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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

More Site Areas 
1.  Solving Linear Equations  2005
2.-Fractions-Rates-Proportns-Units-2006
3.  Algebra, Odds & Ends, HS level-2001
4.-Euclidean-Geometry/Complex No.s 
5.  Analytic Geometry/Functions 2006
6.  Number Theory. 2006-7
7.  Complex Numbers More 2001
8.  Calculus Introduction 2005
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9   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 Education 2004
15-Prequel-to-the-How-TOs-06-2008
How TOs/ Ref.-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;


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 Division in General and With Decimals

Division - How times does one object (measure) fit into another.

When we have a whole number of objects, we may decide to form or try to form equi-sized or equipollent groups.  The division of a whole number N by another whole number d counts the maximum number of times q that groups of size d can be formed and yields a remained less the divisor d. What is left-over may be zero or a group of count r < M.   Division in the first instance may be appear as a physical operation.  We say d divides N when and only when the remainder after division is zero.  Here N = qd+r 

The foregoing discussion of division may be repeated or recalled later when needed.  The concept of multiplication is needed next.  

When fractions are known and allowed as multipliers, the division of a whole number N by another whole number d is given by the proper or improper fraction N/d as is or expressed in lowest terms or expressed as a mixed number.   In the latter case,  the number of times that d goes into N is  N/d exactly, with no remainder.  That being said if N/d is not a whole number, it equals  (qd+r) = q +r/d where r = remainder  for the number of wholes times q that d goes in N.

Decimal  Methods for Division

Motivating Questions.

  • How many times must 6 be added to itself to obtain 48?

  • How many times must 2 be added to itself to give 6?

  • How many times must 2 be added to itself to give 48 ?

 

Example 1. What number added to itself 3 times, gives 963 = 9 hundred + 6 tens + 3 ones. 

The answer is 3 hundred + 2 tens + 1 ones =321. 

(Related question: How many times must 3 be added to itself to give 963?)

Example 2. How many times does 7 go into 963 = 9 hundred + 6 tens + 3 ones? 

The Euclidean Division Method gives the answer.

     9 = 7 x 1 + 2.

     Thus 9 hundred = (7 x 1 + 2) hundred
     Thus 963 = (7x1+2) hundred + 63 =7 x 1 hundred + 263
                                                   Thus 963 = 7 x (1 hundred) + 263.
                                                      --------------------------

     2 is smaller than 7. 
     But 7 x 3 = 21 and 7 x 4 =28 > 26.
     and 26 = 7 x 3 + 5. Therefore
     263 = (7x3 +5) tens +3 = 7 x 3 tens +53
                                                      Thus 263 = 7 x (3 tens) + 53
                                                      -------------------------------

     5 is smaller than 7. But 7 x 7 = 49 and 7 x 8 = 56 > 53.
     Therefore     53 = 7 x 7 ones + 4 ones.         
                                                 Thus  53 = 7 x (7 ones) + 4 leftover.
                                                 ------------------------------------

     4 is "too small to be divided by 7" 
     without the use of fractions.     

                          

     Now

         963 = 7 x (1 hundred) + 263                           

             = 7 x (1 hundred) + 7 x (3 tens) + 53

             = 7 x (1 hundred) + 7 x (3 tens) + 7 x (7 ones) + 4 leftover

             = 7 x (1 hundred + 3 tens + 7 ones) + 4 leftover

             = 7 x 137 + 4.

             = 959 + 4   

     Our conclusion is that 7 goes into 963, 137 times completely, with

     4 leftover = remainder.

        
In shorthand notation, we write the above calculation
 more compactly and briefly as follows.

     

                  137
                 ----                      
              7 | 963
                 -700
                  ---
                  263
                 -210
                  ---
                   53
                   49
                   --
                    4 <--- the remainder.



      Conclusion: The Euclidean Division Method justifies the

      long division algorithm. (Algorithm is just another word

      for method.)

                                                            4
      If you know about fractions:       963 = 7 x ( 137 + --- )
                                                            7         

Example 3. How many times can 23 go into 478155?

We will apply the  Euclidean Division Method
     

      47 = 23 x 2 + 1 leftover

            Thus  470000= 23 x 20000 + 10000.

            Adding 8155 gives

                                     478155 = 23 x 20000 + 18155 

                                     ---------------------------

      18 is smaller than 23 

      but 7 x 23 = 161 and 181 - 171 = 20 is

      smaller than 23. Thus

      181 = 23 x 7 + 20 leftover                                   

        Thus 18100 = 23 x 700 + 2000 leftover

        Adding 55 gives 

                                    18155 = 23 x 700 + 2055 leftover

                                    ------------------------

                                                

      20 is smaller than 23. But 8 x 23 = 160 +24 = 184

       Thus 200 = 23 x 8 + 16 leftover

             2000 = 23 x 80 + 160 leftover.

              So adding 55 gives   

                                    2055 = 23 x 80 + 215 leftover

                                    ---------------------

      

     21 is smaller than 23. But

       9 x 23 = (10-1)x23 = 230 - 23 =207.

       Thus                              

          210 = 9 x 23 + 7      or

                                     215 = 23 x 9 + 8 leftover

                                     -----------------

    

      Here the remainder 12 is less than 23.                                

      

      Now we substitute:

        478155 = 23 x 20000 + 18155 

               = 23 x 20000 + 23 x 700 + 2055

               = 23 x 20000 + 23 x 700 + 23 x 80 + 215

               = 23 x 20000 + 23 x 700 + 23 x 80 + 23 x 9 + 12 leftover

               = 23 x (20000 + 700+ 80 + 9) + 12 leftover.

               

      We conclude 23 goes into 478115, 

            20789 times completely with 12 leftover             



      The foregoing calculations can be written more compactly

      in the left column. Read it first. Consider it as summary of the above.
        
                                          _________________________________ 
                                         |
       20789                             |       2078
    ---------                            |       --------
23 |  478155                             |  23 | 478155
    - 455000   as 23 x 2 = 46 --> 20000  |       46     (23 x 2 -> 2)
      ------                             |     - --   
      18155                              |*       18   
     -16100   23 x 7 = 161   --> 700     |        00    (23 x 0 -> 0)
      -----                              |        --- 
       2055                              |*       181 
      -1840   23 x 8 = 184   --> 80      |        161      (23 x 7 -> 7) 
       -----                             |        --- 
        215   23 x 9 = 207   --> 9       |         205     
       -207                              |         184     (23 x 8 -> 8)
       ----                              |         ---
          8   Last Leftover or           |          215    
              remainder is 8.            |          207    (23 x 9 -> 9)
                                         |          ---
                                         |            8    (less than 23).
                                         |                  Stop.  
                                         |_________________________________

   

The rightmost column shows the usual long division algorithm, one that was taught in elementary school in the 1960s. Variations of it may be found. You should be able to see how the steps on the right correspond to those on the left. In the rightmost column, you will see rows with * in them. The row in-between in them is usually omitted to lessen the amount of writing. The row is included here to help in the comparison of the Euclidean Division Method and that which I met in elementary school in the 1960s. The right hand column is a more cryptic implementation and variation of the Euclidean Division Method.

                                                     8  
Conclusion      478155 = 20789 x 23 + 8 = (20789 +  -- ) x 23.
                                                    23                  

                   
Continuing the Division Process
The remainder 8 is smaller than 23, but 3 x 23 = 69. So

 80 = 23 x 3 + 11 and therefore, dividing by 10, yields   
                                      8 = 23 x 0.3 + 1.1

                                     --------------------

This gives                   



   478155 = 20789 x 23 + 8

          = 20789 x 23 + 23 x 0.3 + 1.1

          = 20789.3 x 23 + 1.1



The remainder has become 1.1 instead of 8. It is much smaller.

We can do this again, and again. For example: 4 x 23 = 92. So


        110 = 23 x 4 + 18



and therefore division by 100 gives

        1.1 = 23 x .04 + .018


This yields again


   478155 = 20789 x 23 + 8

          = 20789.3 x 23 + 1.1        

          = 20789.34 x 23 + .018


The remainder has become smaller. This division process can be recorded
in the shorthand form described above.
 

 

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Number Theory

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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

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