# Dictionary Definition

positional adj : of or relating to or determined
by position

# User Contributed Dictionary

## English

### Adjective

- Relating to the position of something.

# Extensive Definition

A positional notation or place-value notation
system is a numeral
system in which each position is related to the next by a
constant multiplier,
a
common ratio, called the base or radix of that numeral system. Each
position may be represented by a unique symbol or by a limited set
of symbols. The resultant value of each position is the value of
its symbol or symbols multiplied by a power of the base. The total
value of a positional number is the total of the resultant values
of all positions. The decimal system uses ten unique
symbols, whereas the sexagesimal system usually
uses a pseudo-decimal system for each position and separates each
position from the next by punctuation. Modern computers use
binary, octal, and
hexadecimal numbers,
the last using decimal numerals (0–9) plus the letters
A–F to provide the sixteen possible symbols in each
position.

Before positional notation became standard,
simple additive systems (sign-value
notation) were used such as Roman
Numerals. Roman numerals did not support arithmetic operations,
but were used for writing down numbers. That is why accountants in
ancient Rome and during the Middle Ages used the abacus or stone counters to do
arithmetic. With an abacus to perform arithmetic operations, the
writing of the starting, intermediate and final values of a
calculation could easily be done with a simple additive system in
each position or column. This approach required no memorization of
tables (as does positional notation) and could produce practical
results quickly. For four centuries (13th - 16th) there was strong
disagreement between those who believed in adopting the positional
system in writing numbers and those who wanted to stay with the
additive-system-plus-abacus. Although electronic calculators have
largely replaced the abacus, the latter continues to be used in
Japan and other Asian countries.

A key argument against the positional system was
its susceptibility to easy fraud by simply putting a number at the
beginning or end of a quantity, thereby changing (e.g.) 100 into
5100, or 100 into 1000. Modern bank cheques require a natural
language spelling of an amount, as well as the amount itself, to
prevent such fraud.

Generalising the positional system to infinite
sequences of digits yields an intuitive
description of the real line.

## History

Georges Ifrah concludes in his Universal History of Numbers:Thus it would seem highly
probable under the circumstances that the discovery of zero and the
place-value system were inventions unique to the Indian civilization.
As the Brahmi notation of
the first nine whole numbers (incontestably the graphical origin of
our present-day numerals and of all the decimal numeral systems in
use in India, Southeast and Central Asia and the Near East) was
autochthonous and free of any outside influence, there can be no
doubt that our decimal place-value system was born in India and was the
product of Indian civilization alone.|

Aryabhatta
stated "Stanam Stanam Dasa Gunam" meaning "Place to place ten times
in value". His system lacked zero. The zero was added by
Brahmagupta.
Brahmagupta
also was responsible for developing four fundamental operations
(addition, subtraction, multiplication and division). Indian
mathematicians and astronomers also developed Sanskrit positional
number words to describe astronomical facts or algorithms using
poetic sutras.

## Decimal system

In the decimal (base-10) Hindu-Arabic numeral system, each position starting from the right is a higher power of 10. The first position represents 100 (1), the second position 101 (10), the third position 102 (10 × 10 or 100), the fourth position 103 (10 × 10 × 10 or 1000), and so on.Fractional values
are indicated by a separator,
which varies by locale.
Usually this separator is a period or full stop, or a
comma.
Digits to the right of it are multiplied by 10 raised to a negative
power or exponent. The first position to the right of the separator
indicates 10-1 (0.1), the
second position 10-2 (0.01), and so
on for each successive position.

As an example, the number 2674 in a base 10
numeral system is :

- ( 2 × 103 ) + ( 6 × 102 ) + ( 7 × 101 ) + ( 4 × 100 )

or

- ( 2 × 1000 ) + ( 6 × 100 ) + ( 7 × 10 ) + ( 4 × 1 ).

## Digits and numerals

For a positional system up to ten the ubiquitous
digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are used, for octal only
eight digits up to 7 and for binary only two digits 0 and 1 are
needed. For bases above 10, extra digits are needed. For
hexadecimal the first six letters of the alphabet A, B, C, D, E,
and F are commonly used for decimal values 10 to 15. The alphabet
can cover numeral systems with a base up to 10 + 26 = 36. However,
some uppercase letters can be confused with 'existing' digits such
as an I with a 1 and O with 0. When these are omitted it can reach
34. Adding lowercase letters (none of them can be confused with
'existing' digits, except l in some fonts) extends the digit set to
62 (or 60 when uppercase I and O are omitted). For a base 60 system
a 'mixed' base with 10 as 'secondary' base is commonly used, please
see below.

## Sexagesimal system

The sexagesimal or base sixty
system was used for the integral and fractional portions of
Babylonian
numerals and other mesopotamian systems, by Hellenistic
astronomers using Greek
numerals for the fractional portion only, and is still used for
modern time and angles, but only for minutes and seconds. However,
not all of these uses were positional.

Modern time separates each position by a colon or
point. For example, the time might be 10:25:59 (10 hours 25 minutes
59 seconds). Angles use similar notation. For example, an angle
might be 10°25'59" (10 degrees 25 minutes 59 seconds). In both
cases, only minutes and seconds use sexagesimal notation
— angular degrees can be larger than 59 (one rotation
around a circle is 360°, two rotations are 720°, etc.), and both
time and angles use decimal fractions of a second. This contrasts
with the numbers used by Hellenistic and Renaissance
astronomers, who used thirds, fourths, etc. for finer increments.
Where we might write 10°25'59.392", they would have written
10°25′59″23‴3112' or 10°25I59II23III31IV12V.

Using a digit set of digits with upper and
lowercase letters allows short notation for sexagesimal numbers,
e.g. 10:25:59 becomes 'ARz' (by omitting I and O, but not i and o),
which be useful for use in URLs, etc., but it is not very
intelligible to humans. In the 1930s, Otto
Neugebauer introduced a modern notational system for Babylonian
and Hellenistic numbers that substitutes modern decimal notation
from 0 to 59 in each position, while using a semicolon (;) to
separate the integral and fractional portions of the number and
using a comma (,) to separate the positions within each portion.
For example, the mean synodic
month used by both Babylonian and Hellenistic astronomers and
still used in the Hebrew
calendar is 29;31,50,8,20 days, and the angle used in the
example above would be written 10;25,59,23,31,12 degrees.

## Non-positional positions

Each position does not need to be positional
itself. Babylonian sexagesimal numerals were positional, but in
each position were groups of two kinds of wedges representing ones
and tens (a narrow vertical wedge ( | ) and an open left pointing
wedge (<)) — up to 14 symbols per position (5 tens
(<<<<<) and 9 ones ( ||||||||| ) grouped into one or
two near squares containing up to three tiers of symbols, or a
place holder (\\) for the lack of a position). Hellenistic
astronomers used one or two alphabetic Greek numerals for each
position (one chosen from 5 letters representing 10–50
and/or one chosen from 9 letters representing 1–9, or a
zero symbol).

## See also

## External links

- Online Converter for Different Numeral Systems (Base 2-36, JavaScript, GPL)
- Implementation of Base Conversion at cut-the-knot

## References

- Donald Knuth. The Art of Computer Programming, Volume 2: Seminumerical Algorithms, Third Edition. Addison-Wesley, 1997. ISBN 0-201-89684-2. Section 4.1: Positional Number Systems, pp.195–213.
- Georges Ifrah. The Universal History of Numbers: From Prehistory to the Invention of the Computer, Wiley, 2000. ISBN 0-471-37568-3.
- John Kadvany. Positional Value and Linguistic Recursion. Journal of Indian Philosophy, December 2007.

positional in Czech: Poziční číselná
soustava

positional in Danish: Positionstalsystem

positional in German: Stellenwertsystem

positional in Estonian: Positsiooniline
arvusüsteem

positional in Modern Greek (1453-): Θεσιακό
σύστημα

positional in French: Notation
positionnelle

positional in Japanese: 位取り記数法

positional in Dutch: Positioneel
getalsysteem

positional in Norwegian: Posisjonssystem

positional in Polish: Systemy pozycyjne

positional in Portuguese: Notação
posicional

positional in Swedish: Positionssystem

positional in Chinese: 进位制