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# The foundations of arithmetic

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Soon after I graduated from college with a degree in
many rules one learns in arithmetic in grade school (rules for
operating on fractions, for example) and realized I couldn't
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.  It was probably one that went
unsolved for centuries.  However, 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.  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

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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 (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

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