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Cardinality, countable, Power set, rational numbers, irrational numbers, Countability of rational numbers, Cantor diagonal, Problems with the concept of infinity, Applications of the number system in the real world, Infinitely large, Infinitely small, 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 translated yet.
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 8.

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 'extension' 1/q.

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 unit [0,1].
The number 7/12 can be considered as consisting of 7 parts of the 'extension' 1/12.

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 8073176679 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
numerator: 141421356237309504880
denominator: 100000000000000000000

This is a finite approximation with a limited precision. If a better approximation is needed, the used numbers for the numerator and denominator grow.

14142135623730950488016887242 /10000000000000000000000000000 = 1.4142135623730950488016887242
is a better approximation
numerator: 14142135623730950488016887242
denominator: 10000000000000000000000000000
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 approximated by 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 enlarged.

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.

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