ABSTRACT
By K. Damnjanovic
The conception of the number systems to the variable base is different. The known number system are the systems to the constant base.
Any reel number may be presented with the following scheme:
|
an ... a(n-1) ...
a0
... b0 ... b1 ... bm |
(A) |
For the irrational number, as for a certain rational numbers m->∞. The investigation of the numbers systems to the variable base, may be interested for mathematics. For these systems, the scheme (A) is also valid. One of these systems is presented on the scheme (B). The base change for every position of the figure.
|
n! |
... |
4! |
3! |
2! |
1! |
. |
1/(2!) |
1/(3!) |
1/(4!) |
1/(5!) |
... |
1/[(n+1)!] |
|
|
n |
... |
4 |
3 |
2 |
1 |
. |
1 |
2 |
3 |
4 |
... |
n |
(B) |
|
ni |
... |
4i |
3i |
2i |
1i |
. |
1r |
2r |
3r |
4r |
... |
nr |
|
The lowest line marks a
position of figure in some number. So 11 mark the first position on the left
hand of the point. General position is marked by ni. The positions of figures on the
right hand of the point are marked by nr.
The middle line represents the greatest possible figure can be written on the observing
position. Generally, for position ni and nr a system “to the base (n+1)” is used.
In fact, different bases are used, according to precisely defined rule about
increasing base order: from “base two” to “base (n+1)”.
The
highest line represents a figure value for every position. For positions ni we have value n!. For positions nr we have 1/[(n+1)!].
For
the moment, we can consider three points concerning system presented on the
scheme (B):
1.
All of
the rational number may be written with finite number of “decimals” – right
hand of the scheme (A) that is entirely exactly.
Why? Because the rests during division in turn are multiplied by two, three,
four, five and so on, instead by a constant number in the systems to the
constant base.
2.
The
rational number “four divided by thirty-three” in system “to the base ten” is being
written as infinite, but a periodical series after the point: 0.12121212...
Infinit notation 0.12121212... can be written as
well in system “to the variable base”. This, evidently, will not be the number
“four divided by thirty-three”. Therefore, this notation does not represent a
some rational number. If it would be a rational number, series (of figures)
after the point would be finite, as it is for all rational numbers. It is an
irrational number. This is a fonction of the transcedental number e.
3.
System
“to the variable base” enables to us observing two families of irrational
numbers. The first family consists the number in system “to the variable base”
being written as infinite periodical series after the point. In second family
there are the numbers which “decimal” notations are random character, but
infinite too.
Thank
you, for your attention!
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texts are also available
Ó2001 All rights reserved
If
any questions or comments, please contact the author Konstantin
Damnjanovic
visitor.