Website owner:  James Miller

The foundations of arithmetic




   Soon after I graduated from college with a degree in 
   mathematics I started asking myself some questions about the 
   many rules one learns in arithmetic in grade school (rules for 
   operating on fractions, for example) and realized I couldn't 
   answer them.  The more I thought about the questions I had 
   posed for myself, the more I tried to answer them, the more 
   confused I became.  When I asked myself to prove this well 
   known rule or that one I couldn't.  I was confused by the 
   relationship between fractions and division and by the use of 
   the slash symbol for both.  I realized I had a degree in 
   mathematics and didn't even really understand arithmetic.  What 
   were the basic definitions and axioms of arithmetic?  What were 
   the derived theorems and results?  I didn't know.  The reason 
   for this is the way arithmetic is taught.  The textbooks and 
   teachers present the rules and show you how to apply them.  
   They give you a working knowledge of arithmetic.  But they 
   don't go into the reasons for the rules, the whys.  I got good 
   marks in arithmetic, was good in it, liked it, never had 
   trouble with it.  But I only knew how to use it, how to apply 
   the rules.  I didn't really understand it at a basic level.  So 
   realizing my lack of understanding of the basic fundamentals of 
   arithmitic I set myself a project of trying to set down in a 
   systematic and logical way the basic definitions and axioms of 
   arithmetic and then trying to derive the theorems and results 
   from them, in the manner of Euclidean Geometry.   



The first problem -- naming.

   The first problem that ancient man met in his encounter with 
   the number concept was a problem of naming.  For example 
   suppose an ancient farmer had 16 bushels of wheat that he 
   wanted to sell to a merchant but had no names for the integers.  
   How could he communicate the information to the merchant?  So 
   the first thing needed was names for the integers.  Now suppose 
   the integers 1, 2, 3, ...  have been named.  And suppose a 
   farmer has 16 2/3 bushels of wheat that he wants to sell to a 
   merchant.  How does he communicate this information?  He is 
   able to communicate the fact that he has between 16 an 17 
   bushels to sell but he does not have any word or symbol or 
   means to convey the fact that he has 2/3 of a bushel.  What 
   type of naming system could be devised that would name all the 
   possible fractional portions between zero and one?  Ancient man 
   had met a sizable problem.  Eventually someone came up 
   with the idea of using the already named integers to name the 
   fractional portions of unity.  The idea was to divide unity up 
   into a number of equal parts, say n parts, and then specify a 
   fractional portion through multiples of the n-th part of unity.  
   One can then specify any number as the sum of a whole number 
   plus a fraction.  Thus he can now specify that he has, for 
   example, 16 2/3 bushels of wheat. 


The first two steps:

Step 1.  Create a logical and easily used naming system for the 
   integers.

Step 2.  Create a system for designating the fractional parts of a 
   whole (i.e. of unity).



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NOTE. In our discussion when we speak of an integer we mean a 
   positive integer 1, 2, 3, 4, ...  .  Negative integers are 
   introduced in the study of algebra and are important there but 
   are not a part of arithmetic and have no meaning there. 

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   Different systems have been used for naming the integers and 
   the one we currently use traces back to the Hindus, employing a 
   zero and the decimal system.  




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  FRACTIONS AND MIXED NUMBERS
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   To define the concept of a fraction one must first define the 
   concept of a "unit fraction". 

Definition - Unit Fraction.  Conceptually a unit fraction is a 
   number such as 1/5, 1/6, 1/7, etc. (i.e. it is a fraction whose 
   numerator is 1).  We define 1/5 to be the fifth part of 1, 1/6 
   to be the sixth part of 1, and, in general, 1/n to be the n-th 
   part of 1.  Logically this concept comes first and the usual 
   concept of a fraction is built up from it.  For clarity of 
   thought and development we introduce the special notation n* 
   for 1/n.  Thus 1/5 is represented by 5*, 1/6 by 6*, etc. 

Definition - Fraction.  A fraction, designated by the symbol "m/n" 
   is by definition the quantity m(n*) -- which means it is m n*'s 
   (we can speak of m n*'s in the same way we can speak of m cows, 
   m pencils, or m dogs).  Thus the fraction 4/5 is, by 
   definition, 4(5*) and represents four "fifth parts of 1", the 
   fraction 5/6 is 5(6*) and represents five "sixth parts of 1", 
   etc.. 

Definition - Mixed Number.  A mixed number is, by definition, a 
   whole number (i.e. a integer) plus a fraction.  Thus the mixed 
   number 2 5/8 is defined as 2 + 5/8, the mixed number 1 2/3 is 
   defined as 1 + 2/3, etc.. 

   We now have a way of designating any positive rational number 
   p/q where p and q are integers.  Any rational number can be 
   expressed as a mixed number and vice versa.  Let us note that 
   any mixed number can be expressed as a fraction.  For example 
   the mixed number 3 5/6 can be expressed as 3x6(6*) + 5(6*) = 
   3x6/6 + 5/6 = 18/6 + 5/6 = 23/6. 



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ARITHMETIC DEVELOPED AS A SET OF PROCEDURES FOR SOLVING PROBLEMS
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   Ancient man faced a need to solve certain mathematical type 
   problems that he encountered in life.  He needed to measure 
   distances, areas, quantities of fluid or grain, weights; to 
   compute sums, differences, etc..  Let us list some of the 
   mathematical type problems he needed to cope with: 


1. To measure continuous quantities such as bushels of grain, 
   distances, areas, weights, etc.. 
2. To compute sums of whole numbers or sums of continuous 
   quantities. 
3. To compute differences of whole numbers or quantities.
4. To multiply an integer by another integer.
5. To multiply some quantity by an integer  e.g. How much is 5 
   times 8 7/8?
6. To find a fractional part of some quantity  e.g. How much is 
   3/8 of 8 7/8? 
7. To scale a quantity e.g. scale the quantity 8 7/8 by a factor 
   of 5 3/8 (this problem arises in computing the area of a 
   rectangle). 
8. To divide some quantity by an integer e.g. What is  the result 
   of dividing 8 7/8 by 7?   
9. To determine the number of times one quantity is contained in 
   another i.e. solve  the equation ax = b where the unknown x is 
   the number of times a is contained in b. 



   In the beginning early man probably didn't use symbols to 
   indicate the arithmetic operations of addition, subtraction, 
   multiplication or division as we do today.  And his procedures 
   for multiplication and division were possibly different from 
   those of our own.  He didn't yet have algebra with its use of 
   symbols and procedures and didn't think in terms of algebraic 
   ideas or techniques.  But he did have mathematical problems and 
   he needed methods and procedures for dealing with them.   He 
   developed procedures for solving his different mathematical 
   problems and from these procedures evolved arithmetic. 

   We now consider the arithmetic operations of addition, 
   subtraction, multiplication and division in a logical order 
   (presumably the order in which they naturally evolved). 



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  ADDITION
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   ADDITION.  SUM OF TWO OR MORE INTEGERS.  The ancients didn't 
     use the decimal system that we use for naming the integers 
     and because of that their procedure for adding numbers would 
     have been different than ours (the procedure needed is 
     dependent on the naming system).  They devised procedures 
     based on their naming system. 

   ADDITION.  SUM OF TWO OR MORE MIXED NUMBERS.  The sum of two or 
     more mixed numbers can be determined by adding the integral 
     parts and fractions separately, or by reducing each mixed 
     number to a fraction and then adding.   
     
     One can only add two fractions if they have the same 
     denominator.  If he wishes to add two fractions with 
     different denominators he must change the fractions to 
     equivalent fractions, both of which have the same 
     denominator.  To do this he needs the following theorem:    
     
   Theorem.

            p/q  =  np/nq   [ or p(q*) = np(nq)* ]

     where p, q and n are integers.

     In words:  We can multipy both the numerator and denominator 
     of a fraction by the same integer without changing the value 
     of the fraction (i.e. multiplying both numerator and 
     denominator of a fraction by the same integer gives an 
     equivalent fraction). 

         Example:     3/5  =  2x3/2x5  =  6/10

     Using the above theorem, if one wishes to add the fractions 
     a/b and c/d the sum will be 

             a/b + c/d  =  ad/bd + cb/db  =  (ad + cb)/bd




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   SUBTRACTION
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     In arithmetic you can only subtract a smaller number (integer 
     or mixed number) from a larger one.  Subtracting a larger 
     number from a smaller one doesn't make any sense and is not 
     allowed.  
   

   SUBTRACTION.  SUBTRACTING A SMALLER INTEGER FROM A LARGER 
     INTEGER.     As with addition the procedure used in 
     subtracting integers is dependent on the naming system used 
     in naming the integers. 



   SUBTRACTION.  SUBTRACTING A SMALLER MIXED NUMBER FROM A LARGER 
     MIXED NUMBER.  In general, to subtract a smaller mixed number 
     from a larger mixed number you reduce both mixed numbers to 
     fractions and then subtract.  To subtract them the 
     denominators of the fractions must be the same.  If they are 
     not you must change the fractions to equivalent fractions 
     whose denominators are the same. 

        Example.  5 3/7  -  2 2/3  =  5x7/7 + 3/7  -  2x3/3 + 2/3
                                   =  (5x7 + 3)/7  -  (2x3 + 2)/3
                                   =  38/7 - 8/3
                                   =  3x38/3x7 - 7x8/7x3
                                   =  (3x38 - 7x8)/7x3
                                   =  (114 - 56)/21
                                   =  58/21
                                   





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  MULTIPLICATION
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   MULTIPLICATION.  PRODUCT OF TWO INTEGERS.  The product m times 
     n, of two integers m and n, expressed as mn, is given by 
     repeated addition.  It is defined as  "n + n + n + ... "  
     where n is repeated m times. 
  
      Examples.  3x5  =  5 + 5 + 5
                 4x6  =  6 + 6 + 6 + 6 

 

   MULTIPLICATION.  The product of two numbers (i.e. two mixed 
     numbers) can be defined in terms of the products of two 
     fractions since mixed numbers can be converted to fractions 
     (any mixed number is equivalent to some fraction) and that is 
     the course we will take.  First we need to define what we 
     mean by the product of two unit fractions m* and n*.  We know 
     intuitively what the sum of two unit fractions is but do we 
     know what the product should be?  No.  It is something that 
     has to be defined.

   DEF. PRODUCT OF TWO UNIT FRACTIONS.  The product (m*)(n*) of 
     two unit fractions m* and n* is defined as 
     
                       (m*)(n*) = (mn)*   

     In our usual notation this means that we define the product 
     of 1/m times 1/n as the quantity given by 1/mn ( e.g. 1/5 X 
     1/3 = 1/15). 


   PRODUCT OF TWO FRACTIONS.  We are now in a position to define 
     the product of any two fractions.  By definition, the product 
     of two fractions m(n*) and p(q*), expressed as m(n*)p(q*), is 
     given as: 

                    m(n*)p(q*) = mp(nq)*
                    
     In our usual notation this says (m/n)(p/q) = mp/nq.

     
   Products of mixed numbers.   Mixed numbers can be multiplied by 
     converting them to fractions and multiplying the fractions.  
     They can also be multiplied by multiplying each term of one 
     number by each term of the other number and adding i.e. mixed 
     numbers e + f/g and h + i/j can be multiplied as: 

       (e + f/g)(h + i/j) = eh + ei/j + fh/g + fi/gj

     Either method of evaluating a product of two mixed numbers 
     gives the same result. 

     Examples.  (3/5)(2/7) = 3x2/5x7 = 6/35
                (2 3/8)(3 2/3) = 2x3 + 2(2/3) + (3/8)3 + (3/8)(2/3)
                               = 6 + 4/3 + 9/8 + 6/24
                               = 6 + (4x8)/(3x8) + (9x3)/8x3 + 6/24
                               = 6 + 32/24 + 27/24 + 6/24
                               = 6 + 65/24  
                               = 8 17/24



  NOTES ON THE PRODUCT OF TWO FRACTIONS.  Intuitively the product 
     (m/n)(p/q) of a fraction m/n times a fraction p/q represents 
     a scaling of the second fraction by the first.  For example 
     when the multiplier is 1/3, the product is 1/3 the size of 
     the multiplicand; when the multiplier is 5/6, the product is 
     5/6 the size of the multiplicand; when the multiplier is 5 
     1/4, the product is 5 1/4 times the size of the multiplicand.  
     To see why this is true note that 

                     (m/n)(p/q) = mp/nq

     is equivalent to 

                     (m/n)(p/q) = m(1/n)(p/q)

     where (1/n)(p/q) represents the n-th part of p/q and "m" 
     plays the role of a scaling factor that scales this n-th part 
     of p/q. 






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   DIVISION
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  DIVISION.  QUOTIENT OF TWO MIXED NUMBERS.  To divide two mixed 
     numbers it is necessary to first reduce them to fractions.  
     The quotient a/b of two numbers a and b is defined as the 
     solution to the equation bx = a.  Thus the quotient is x = 
     (1/b)a.  When a and b are fractions the rule for computing 
     the quotient is "invert the divisor and multiply". 
     
      Example 1.  The quotient of 5/8 divided by 2/3 is the 
     solution to the equation 

                   2/3 x = 5/8

        so x = (5/8)(3/2)  which we obtained by multiplying the 
        equation by 3/2.

      Example 2.  Find the quotient of 7/8 divided by 2/5 (we 
        invert the divisor and multiply):  

             (7/8)/(2/5) = (7/8)(5/2) = 5x7/8x2 = 35/16



   NOTES ON DIVISION.

     When the divisor is an integer, n, it should be noted that 
     this definition coincides with the concept of division as an 
     operation that divides the dividend into n equal parts i.e. 
     an operation that  yields a quantity that is the n-th part of 
     the dividend.  For example, 23/5 represents the 5-th part of 
     23 (the part obtained by dividing 23 into 5 equal parts). 
   
     Division also gives the number of times the denominator is 
     contained in the numerator.  Or equivalently, the quotient 
     represents the number of divisors in the dividend.  For 
     example, 9/2 represents the number of 2's contained in 9.  Or 
     said another way, the quotient is the scaling factor that 
     must be applied to the denominator to yield the numerator.    

     The well known rule for dividing a fraction a/b by the 
     fraction c/d, invert the divisor and multiply, follows 
     directly from the equation (c/d)x = a/b.  We solve for x by 
     multiplying both sides of the equation by d/c to get x = 
     (d/c)(a/b). 






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   INTUITIVE INTERPRETATIONS   
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      Operation                             Interpretation
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  (integer) x (integer)                 repeated addition

  (integer) x (any number)              repeated addition

  (any number) x (any number)           scaling of second number by 
                                          first

  (any number) / (integer)              divide numerator into equal 
                                          parts

  (any number) / (any number)           gives number of times the
                                          denominator is contained
                                          in the numerator