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Place Systems

Numbers are invisible values, that can be represented by number characters. The smallest unit, the number element, of a number is the cypher. Integer numbers can be represented to the upper limit of the number system with one place in all number systems. Beyond it, the place system has to be known, in which the number is represented, if two, three or more cyphers are necessary for representation of the number. The number image "1001" (one, zero, zero, one) e.g. means one thousand and one in the decimal system. The same number image, "1001", represents the value nine in the dual system and finally the same sequence of cyphers respectively the same sequence of number elements represents the value four thousand seventy nine in the hexadecimal system.

The best known place system is the decimal system. It consists of ten number characters, 0 (zero) to 9 (nine), the decimal point respectively the decimal point place and the neighbouring ten respectively tenth places. Accordingly every left standing place has ten times the value of the place to its right; vice versa every right standing place has a tenth of the value of its left standing place. This is likewise valid before and after the decimal point.

Three further place systems are important beneath the decimal system: the dual or binary system, the octal system and the hexadecimal system. Without these, especially the dual system today's computer technology barely would be imaginable.

How the place systems are realized ? Can they be methotically deducted by mathematical algorithms?

One possibility follows:

The integer rest is calculated for the quotient (n+x)/n for n = of integer numbers, n>=2 and x=0,1,2,3,..., infinite. On the occasion the cypher sequence of the integer rest occurs periodically with increassing x. Very special is, that the periodical rest for n=2 is 0 and 1. For n=3 the rest is periodically 0, 1 and 2, for n=4 is the periodically occuring rest 0, 1, 2 and 3, for n=z the cypher sequence is 0, 1, 2, ..., (z-1). Thus a period of the periodically rest sequence represents a place system in each case, beginning with the dual system.

Cyphers bigger than 9 are represented by capital characters of the latin alphabeth for the hexadecimal system.

For the:
value "10" the character A
value "11" the character B
value "12" the character C
value "13" the character D
value "14" the character E and
value "15" the character F are agreed upon.

This relatively clear system could be carried on for the complete latin alphabet with more or less success. Beyond it it would strike on limits and suitably combinations of cyphers or special characters not used otherwise have to be agreed upon.

Table for some place systems, of the quotient Z/N=(n+x)/n

 Z \ N 2 3 4 5 6 7 8 9 10 ...... 16 2 0 _ _ _ _ _ _ _ _ ...... _ 3 1 0 _ _ _ _ _ _ _ ...... _ 4 0 1 0 _ _ _ _ _ _ ...... _ 5 1 2 1 0 _ _ _ _ _ ...... _ 6 0 0 2 1 0 _ _ _ _ ...... _ 7 1 1 3 2 1 0 _ _ _ ...... _ 8 0 2 0 3 2 1 0 _ _ ...... _ 9 1 0 1 4 3 2 1 0 _ ...... _ 10 0 1 2 0 4 3 2 1 0 ...... _ 11 1 2 3 1 5 4 3 2 1 ...... _ 12 0 0 0 2 0 5 4 3 2 ...... _ 13 1 1 1 3 1 6 5 4 3 ...... _ 14 0 2 2 4 2 0 6 5 4 ...... _ 15 1 0 3 0 3 1 7 6 5 ...... _ 16 0 1 0 1 4 2 0 7 6 ...... 0 17 1 2 1 2 5 3 1 8 7 ...... 1 18 0 0 2 3 0 4 2 0 8 ...... 2 19 1 1 3 4 1 5 3 1 9 ...... 3 20 0 2 0 0 2 6 4 2 0 ...... 4 21 1 0 1 1 3 0 5 3 1 ...... 5 22 0 1 2 2 4 1 6 4 2 ...... 6 23 1 2 3 3 5 2 7 5 3 ...... 7 24 0 0 0 4 0 3 0 6 4 ...... 8 25 1 1 1 0 1 4 1 7 5 ...... 9 26 0 2 2 1 2 5 2 8 6 ...... A 27 1 0 3 2 3 6 3 0 7 ...... B 28 0 1 0 3 4 0 4 1 8 ...... C 29 1 2 1 4 5 1 5 2 9 ...... D 30 0 0 2 0 0 2 6 3 0 ...... E 31 1 1 3 1 1 3 7 4 1 ...... F I N T E G E R   R E S T

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