Counting Numbers
(Counting the "number" of elements of a non-empty set)
1 2 3 4 5 6 7 8 9 ... 
infinity stasrts here!

What is a number? Too early to answer!
For a rigorous defintion, we must wait for the invention of Set Theory.

For now we just call them "counting numbers".
They are applicable only to "whole" objects,
and they are "positive".


Measauring Numbers
(Measuring the magnitude of a quantity)

Using something named a "unit",
we can reduce the measuring of a quantity,
to counting the "whole" units of that quantity contained in it.

Measuring is more than just counting the units.
We must deal with the fraction, remained after counting out all the whole units.

For that, we need numbers with a whole part, and a fraction part.
We call them "measuring numbers".


Addition
addition of any two numbers, is always possible,
no matter adding counting numbers or measuring numbers.

Even addition of a counting number and a measuring number is possible.
Because any counting number is also a measuring number.



Subtraction
reverse of addition, not always possible.
we can only subtract a "smaller" number from a "greater" one.



Multiplication
repeated addition
multiplication of anything, means adding it to itself multiple times

for counting numbers it is a "product"
you can multiply any two counting numbers

but for measuring numbers it is different.
we can only multiply a measuring number by a counting number.

product of two measuring numbers have not a meaning (yet).
so we continue with the arithmetic of counting numbers.


Division
inverse of multiplication
it is not always possible.

...

    
Archimedean property
No matter how small a quantity is,
if you add it to itself enough times,
it will eventually exceed any other given quantity,
no matter how large.













arithmetic of whole numbers
  addition of any two numbers
  subtraction: reverse of addition, not always possible
    we can only subtract a smaller from a larger
  multiplication of any two numbers: repeated addition
  division: inverse of multiplication, not always possible
    quotient: divisibility, birth of number theory
    remainder: for the cases where division is not possible

representation of whole numbers, using digits and place values


Archimedean property
No matter how small a quantity is,
if you add it to itself enough times,
it will eventually exceed any other given quantity,
no matter how large.

no quantity is infinitely small,
no quantity is infinitely large.

for any two quantities A and B,
where A can be arbitrarily large,
and B can be arbitrarily small,
there always exist a "natural" number n,
such that n.B>=A

the set of Natural numbers is not bounded above.

In Eucld's proof using repeated subtraction (anthyphairesis),
he continuousely subtracted the smaller line segment B from the larger segment A,
A-B A-2B A-3B ... A-nB
The Archimedean property is the logical guaranmtee that this sequence cannot decrease indefinitely while remaining positive.

There must exist a specific step q where (q+1).B>A
This forces the subtraction process to stop.
The step just before this threshold yields the unique quotient q and the remainder remainder r 
ensuring that 
A=B.q+r where 0<=r
(Megethos)
measuring: can be converted to counting using a "unit"

fractions - an extension to whole numbers, suitable for measuring

commensurable magnitudes


incommensurable magnitudes