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Positional Numeral Systems

A positional (numeral) system is a system for representation of numbers by an ordered set of numerals symbols (called digits)  in which the value of a numeral symbol depends on its position. For each position a unique symbol or a limited set of symbols is used. The value of  a symbol is given by the weight of its position expressed in the bases (or radices) of the system.  The resultant value of each symbol is given by the value assigned to its position (e.g. by a product of  bases) and modified (e.g. multiplied) by the value of the symbol. The total value of the represented number in a positional number is then sum of the values assigned to the symbols of all positions.

Probably the first scientific treatment of other numeral systems than the decimal one can be found in Blaise Pascal’s De numeris multiplicibus  [1]   ↑ which was presented to the Academie Parisienne in 1654  and first published in 1665 as supplement to his Traité du Triangle Arithmétique.  For a detailed treatise on the history of non-decimal numeral systems consult [2] .

Example: The Maya number system is principally a vigesimal number system (uses a base of 20).  The symbols of a digit was composed from three basic symbols for  zero (a shell-shaped glyph), one (a dot), and five (a bar). For example, the numbers from one to four are represented as one, two, three or four dots, seven is a bar and two dots, ten is represented by two bars and nineteen is three bars and four dots as 3 x 5 + 4 x 1 = 19 .

The bases of each position, written from right to left, are typeset structure, where all typeset structure’s equal to 20 in standard mathematical calculations, but in calendric calculations typeset structure. Thus  typeset structure with typeset structure represents in mathematical calculations the value typeset structure, but in connection with calendric calculations its value is typeset structure.

If the bases typeset structure,  in a positional system are different, the system is called mixed-base positional system, otherwise a uniform-base.

The standard decimal system uses ten symbols typeset structure.  Contemporary  computers use binary, octal, and hexadecimal systems. In the hexadecimal system we need 16 symbols for representation in each position. In the standard notation the decimal numerals typeset structure plus first six capital letters typeset structure are used.

References

[1]  Pascal, B. (1908). De numeris multiplicibus ex sola characterum numericorum additione agnoscendis (De caracteres de divisibilité des nombres deduit de la somme de leurs chffres). In Oeuvres 3 (pp. 311-339). Paris: Brunschvicg et Boutroux.

[2]  Glaser, A. (1981). History of binary and other nondecimal numeration (Rev. ed.). Los Angeles: Tomash Publishers.

Cite this web-page as:

Štefan Porubský: Positional Numeral Systems.

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