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Non-standard positional numeral systems here designates numeral systems that may be denoted positional systems, but that deviate in one way or another from the following description of standard positional systems:

- In a standard positional numeral system, the base b is a positive integer, and b different numerals are used to represent all non-negative integers. Each numeral represents one of the values 0, 1, 2, etc., up to b-1, but the value also depends on the position of the digit in a number. The value of a digit string like $d\_3d\_2d\_1d\_0$ in base b is given by the polynomial form

- $d\_3times\; b^3+d\_2times\; b^2+d\_1times\; b+d\_0$.

- For instance, in hexadecimal (b=16), using A=10, B=11 etc., the digit string 1F3A means

- $1times16^3+15times16^2+3times16+10$.

- Introducing a radix point "." and a minus sign "–", all rational numbers can be represented.

This article summarizes facts on some non-standard positional numeral systems. In most cases, the polynomial form in the description of standard systems still applies.

Certain historical numeral systems like the Babylonian (standard) sexagesimal notation or the Chinese rod numerals could be classified as standard systems, if the 60 resp. 10 distinct numerals are considered as digits, unconventionally counting the space representing zero as a numeral. However, they could also be classified as non-standard systems (more specifically, mixed-base systems with unary components), if the primitive repeated glyphs making up the numerals are considered.

- The value of a digit does not depend on its position. Thus, one can easily argue that unary is not a positional system at all.
- Introducing a radix point in this system will not enable representation of non-integer values.
- The single numeral represents the value 1, not the value 0=b-1.
- The value 0 cannot be represented (or is implicitly represented by an empty digit string).

It can be generalized on other complex bases: Complex base systems.

For calendrical use, the Mayan numeral system was a mixed radix system, since one of its positions represents a multiplication by 18 rather than 20, in order to fit a 360-day calendar. Also, giving an angle in degrees, minutes and seconds (with decimals), or a time in days, hours, minutes and seconds, can be interpreted as mixed radix systems.

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Last updated on Tuesday September 30, 2008 at 04:09:00 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Tuesday September 30, 2008 at 04:09:00 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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