Countability of rational numbers,
with the concept of infinity,
Applications of the number system in the real world,
The number Pi
Notes to the terms
countability, limit, intersection of infinitely many sets
Some general remarks
Links are assigned partially to German Parts, which have not been
Some of the English Parts have been translated automatically, I think
theys must be overworked.
As far as rational and irrational numbers are considered, I restrict
the consideration to the positive range.
Infinity in the great (large) and infinity in the small
I distinguish between Infinity in
the Great and Infinity in
the small .
The infinity in the Great is
expressed by the fact that there exist no greatest natural number.
Natural numbers are for example 1,2,3,4,5,6,7,8 ...
The three dots indicate that infinitely many natural numbers follow the
The Infinity in the small is
expressed by the fact that a unit can be divided into parts, for
example 1/2, 1/3, 7/12 ... are parts of a unit. There are no
restrictions to devide a unit
into pieces, arbitrary small parts can be defined. Each fraction of the
form p/q with integers p and q, p > 0, q > p, defines a Part of a unit.
The numerical value
of p/q is measured by
p parts of the 'extension' 1/q,
with q > 0.
A rational number p/q can be considered to exist of
p parts of the
As an Example consider the
interval [0,1]. The rational number 1/12 describes the 12th part of the
unit [0,1], the rational number 7/12 describes the 7/12th part of the
The number 7/12 can be considered as consisting of 7 parts of the
Also rational numbers p/q which are greater than 1, can be described by
finitely many parts of the 'extension' 1/q:
Example: The number 27/12
consists of 27 parts of the 'extension' 1/12.
In this form of representation, there exists no largest natural number
and no smallest part of a unit.
For a physical realization of the rational
numbers no smallest parts exist.
The irrational numbers
contribute nothing new to the
infinity in the large. The meaning of irrational numbers,
for example, states, that there are numbers between 0 to 1, which can
not be expressed as parts of the unit in the specified form p/q.
Maybe an irrational number can be considered as existing of infinitely
many parts of the 'extension' 0 (a further reasoning is given below).
The value of an irrational number x in the interval [0,1] can be
approached by rational numbers with an arbitrary precision. More
precisely, there is a sequence of rational numbers pn/ qn
,with natural numbers pn and qn, which
converges against the
irrational number x.
The numerators pn and denominators qn of the
approximating sequence of rational numbers pn/qn
will exceed all bounds (consider the example below).
Irrational numbers can be found in any real number interval.
Example square root of 2
Consider, for example the irrational number
= 1,4142135623 7309504880 1688724209 6980785696 7187537694
7379907324 7846210703 8850387534 3276415727 …
The three points suggest that inifinitely many digits follow the last 7.
The value of the square root of 2
can be approximated by the following term:
141421356237309504880 /100000000000000000000 = 1.41421356237309504880
This is a finite approximation with a limited precision. If a better
approximation is needed, the used
numbers for the numerator and
14142135623730950488016887242 /10000000000000000000000000000 =
is a better approximation
The sequence 1 / qn in
this representation converges to zero, without ever reaching it.
Using the above introduced concept of 'extension', an irrational number can be described by
infinitely many parts of the 'extension' 0. A finite sum of parts with
the 'extension' 0 results in 0.
Note: the term 'extension' is
not in mathematics,
as it was used here.
Motivation: The numerical value
of irrational numbers appears in the specified view as an infinite
summation of parts of zero size, while rational numbers can be
described by a
finite summation of parts which have a non-zero 'extension'.
The square root of 2 can be
141421356237309504880 parts of the expansion 1 / 100000000000000000000,
resulting in the fraction 141421356237309504880/100000000000000000000.
For "getting closer" the number of digits in the
numerator and denominator of the approximating fraction must be
In order to represent the square
root of 2 exactly, I imagine that numerator and denominator
are infinite great natural numbers.
In theory all digits in the numerator can be calculated, using a well
defined formula, Square Root of 2 , although
there exist infinite many. In this sense infinity has a structure.
It is difficult to know how many zeros shall appear in the denominator,
the statement "infinite many" is not helpful.