Positional notation or placevalue notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations (such as Roman numerals) for its use of the same symbol for the different orders of magnitude (for example, the "ones place", "tens place", "hundreds place"). This greatly simplified arithmetic and led to the quick spread of the notation across the world.
With the use of a radix point (decimal point), the notation can be extended to include fractions and the numeric expansions of real numbers. The Hindu–Arabic numeral system is an example for a positional notation, based on the number 10.
History
Today, the base10 (decimal) system, which is likely motivated by counting with the ten fingers, is ubiquitous. Other bases have been used in the past however, and some continue to be used today. For example, the Babylonian numeral system, credited as the first positional number system, was base60. Counting rods and most abacuses have been used to represent numbers in a positional numeral system, but it lacked a real 0 value. 0 was indicated by a space between sexagesimal numerals. By 300 BC, a punctuation symbol (two slanted wedges) was coopted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish (dating from about 700 BC), the scribe Bêlbânaplu wrote his zeros with three hooks, rather than two slanted wedges.^{}
The Babylonian placeholder was not a true zero because it was not used
alone. Nor was it used at the end of a number. Thus numbers like 2 and
120 (2×60), 3 and 180 (3×60), 4 and 240 (4×60), looked the same because
the larger numbers lacked a final sexagesimal placeholder. Only context
could differentiate them.
Before positional notation became standard, simple additive systems (signvalue notation) such as Roman Numerals were used, and accountants in ancient Rome and during the Middle Ages used the abacus or stone counters to do arithmetic.^{}
With counting rods or 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 (from the 13th to the 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
additivesystemplusabacus. Although electronic calculators have
largely replaced the abacus, the latter continues to be used in Japan
and other Asian countries.
Georges Ifrah concludes in his Universal History of Numbers:
This it would seem highly probable under the circumstances that the discovery of zero and the placevalue system were inventions unique to the Indian civilization. As the Brahmi notation of the first nine whole numbers (incontestably the graphical origin of our presentday 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 placevalue system was born in India and was the product of Indian civilization alone.^{}
Aryabhata stated "sthānam sthānam daśa guṇam"
meaning "From place to place, ten times in value". Indian
mathematicians and astronomers also developed Sanskrit positional number
words to describe astronomical facts or algorithms using poetic sutras.
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 cheques
require a natural language spelling of an amount, as well as the
decimal amount itself, to prevent such fraud. For the same reason the
Chinese also use natural language numerals, for example 100 is written
as 壹佰, which can never be forged into 壹仟(1000) or 伍仟壹佰(5100).
Base of the numeral system
In mathematical numeral systems, the base or radix is usually the number of unique digits,
including zero, that a positional numeral system uses to represent
numbers. For example, for the decimal system the radix is 10, because it
uses the 10 digits from 0 through 9. When a number "hits" 9, the next
number will not be another different symbol, but a "1" followed by a
"0". In binary, the radix is 2, since after it hits "1", instead of "2"
or another written symbol, it jumps straight to "10", followed by "11"
and "100".
The highest symbol of a positional numeral system usually has the
value one less than the value of the base of that numeral system. The
standard positional numeral systems differ from one another only in the
base they use.
The base is an integer that is greater than 1 (or less than negative
1), since a radix of zero would not have any digits, and a radix of 1
would only have the zero digit. Negative bases are rarely used. In a
system with a negative radix, numbers may have many different possible
representations.
(In certain nonstandard positional numeral systems, including bijective numeration, the definition of the base or the allowed digits deviates from the above.)
In base10 (decimal) positional notation, there are 10 decimal digits and the number
 .
In base16 (hexadecimal), there are 16 hexadecimal digits (0–9 and A–F) and the number
 (where B represents the number eleven as a single symbol)
In general, in baseb, there are b digits and the number
 (Note that represents a sequence of digits, not multiplication)
Notation
Sometimes the base number is written in subscript after the number represented. For example, 23_{8}
indicates that the number 23 is expressed in base 8 (and is therefore
equivalent in value to the decimal number 19). This notation will be
used in this article.
When describing base in mathematical notation, the letter b is generally used as a symbol for this concept, so, for a binary system, b equals 2. Another common way of expressing the base is writing it as a decimal subscript after the number that is being represented. 1111011_{2} implies that the number 1111011 is a base2 number, equal to 123_{10} (a decimal notation representation), 173_{8} (octal) and 7B_{16} (hexadecimal). When using the written abbreviations of number bases, the base is not printed: Binary 1111011 is the same as 1111011_{2}.
The base b may also be indicated by the phrase "baseb". So binary numbers are "base2"; octal numbers are "base8"; decimal numbers are "base10"; and so on.
Numbers of a given radix b have digits {0, 1, ..., b−2, b−1}.
Thus, binary numbers have digits {0, 1}; decimal numbers have digits
{0, 1, 2, ..., 8, 9}; and so on. Thus the following are notational
errors: 52_{2}, 2_{2}, 1A_{9}. (In all cases, one or more digits is not in the set of allowed digits for the given base.)
Exponentiation
Positional number systems work using exponentiation
of the base. A digit's value is the digit multiplied by the value of
its place. Place values are the number of the base raised to the nth power, where n is the number of other digits between a given digit and the radix point. If a given digit is on the left hand side of the radix point (i.e. its value is an integer) then n is positive or zero; if the digit is on the right hand side of the radix point (i.e., its value is fractional) then n is negative.
As an example of usage, the number 465 in its respective base b (which must be at least base 7 because the highest digit in it is 6) is equal to:
If the number 465 was in base10, then it would equal:
(465_{10} = 465_{10})
If however, the number were in base 7, then it would equal:
(465_{7} = 243_{10})
10_{b} = b for any base b, since 10_{b} = 1×b^{1} + 0×b^{0}. For example 10_{2} = 2; 10_{3} = 3; 10_{16} = 16_{10}. Note that the last "16" is indicated to be in base 10. The base makes no difference for onedigit numerals.
Numbers that are not integers use places beyond a radix point. For every position behind this point (and thus after the units digit), the power n decreases by 1. For example, the number 2.35 is equal to:
This concept can be demonstrated using a diagram. One object
represents one unit. When the number of objects is equal to or greater
than the base b, then a group of objects is created with b objects. When the number of these groups exceeds b, then a group of these groups of objects is created with b groups of b objects; and so on. Thus the same number in different bases will have different values:
241 in base 5: 2 groups of 5^{2} (25) 4 groups of 5 1 group of 1 ooooo ooooo ooooo ooooo ooooo ooooo ooooo ooooo + + o ooooo ooooo ooooo ooooo ooooo ooooo
241 in base 8: 2 groups of 8^{2} (64) 4 groups of 8 1 group of 1 oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo + + o oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo
The notation can be further augmented by allowing a leading minus
sign. This allows the representation of negative numbers. For a given
base, every representation corresponds to exactly one real number
and every real number has at least one representation. The
representations of rational numbers are those representations that are
finite, use the bar notation, or end with an infinitely repeating cycle
of digits.
Digits and numerals
A digit is what is used as a position in placevalue notation, and a numeral is one or more digits. Today's most common digits are the decimal digits
"0", "1", "2", "3", "4", "5", "6", "7", "8", and "9". The distinction
between a digit and a numeral is most pronounced in the context of a
number base.
A nonzero numeral with more than one digit position will mean a different number in a different number base, but in general, the digits will mean the same.^{} The base8 numeral 23_{8} contains two digits, "2" and "3", and with a base number (subscripted) "8", means 19. In our notation here, the subscript "_{8}" of the numeral 23_{8} is part of the numeral, but this may not always be the case. Imagine the numeral "23" as having an ambiguous base
number. Then "23" could likely be any base, base4 through base60. In
base4 "23" means 11, and in base60 it means the number 123. The
numeral "23" then, in this case, corresponds to the set of numbers {11,
13, 15, 17, 19, 21, 23, ..., 121, 123} while its digits "2" and "3" always retain their original meaning: the "2" means "two of", and the "3" three.
In certain applications when a numeral with a fixed number of
positions needs to represent a greater number, a higher numberbase with
more digits per position can be used. A threedigit, decimal numeral
can represent only up to 999. But if the numberbase is increased
to 11, say, by adding the digit "A", then the same three positions,
maximized to "AAA", can represent a number as great as 1330. We
could increase the number base again and assign "B" to 11, and so on
(but there is also a possible encryption between number and digit in the
numberdigitnumeral hierarchy). A threedigit numeral "ZZZ" in base60
could mean 215,999. If we use the entire collection of our alphanumerics we could ultimately serve a base62 numeral system, but we remove two digits, uppercase "I" and uppercase "O", to reduce confusion with digits "1" and "0".^{ }We are left with a base60, or sexagesimal numeral system utilizing 60 of the 62 standard alphanumerics. (But see Sexagesimal system below.)
The common numeral systems in computer science are binary (radix 2), octal (radix 8), and hexadecimal (radix 16). In binary only digits "0" and "1" are in the numerals. In the octal numerals, are the eight digits 0–7. Hex
is 0–9 A–F, where the ten numerics retain their usual meaning, and the
alphabetics correspond to values 10–15, for a total of sixteen digits.
The numeral "10" is binary numeral "2", octal numeral "8", or
hexadecimal numeral "16".
Base conversion
Bases can be converted between each other by drawing the diagram
above and rearranging the objects to conform to the new base, for
example:
241 in base 5: 2 groups of 5^{2} 4 groups of 5 1 group of 1 ooooo ooooo ooooo ooooo ooooo ooooo ooooo ooooo + + o ooooo ooooo ooooo ooooo ooooo ooooo is equal to 107 in base 8: 1 group of 8^{2} 0 groups of 8 7 groups of 1 oooooooo oooooooo oooooooo oooooooo + + ooooooo oooooooo oooooooo oooooooo oooooooo
There is, however, a shorter method which is basically the above
method calculated mathematically. Because we work in base10 normally,
it is easier to think of numbers in this way and therefore easier to
convert them to base10 first, though it is possible (but difficult if
one is not used to the base the conversion is being performed in) to
convert straight between nondecimal bases without using this
intermediate step. (However, conversion from bases like 8, 16 or 256 to
base2 can be achieved by writing each digit in binary notation, and
subsequently, conversion from base2 to e.g. base16 can be achieved by
writing each group of four binary digits as one hexadecimal digit.)
A number a_{n}a_{n}_{−1}...a_{2}a_{1}a_{0} where a_{0}, a_{1} ... a_{n} are all digits in a base b (note
that here, the subscript does not refer to the base number; it refers
to different objects), the number can be represented in any other base,
including decimal, by:
Thus, in the example above:
To convert from decimal to another base one must simply start
dividing by the value of the other base, then dividing the result of the
first division and overlooking the remainder, and so on until the base
is larger than the result (so the result of the division would be a
zero). Then the number in the desired base is the remainders, the most
significant value being the one corresponding to the last division and
the least significant value being the remainder of the first division.
Example #1 decimal to septal:
Example #2 decimal to octal:
The most common example is that of changing from decimal to binary.
Infinite representations
The representation of nonintegers can be extended to allow an
infinite string of digits beyond the point. For example
1.12112111211112 ... base3 represents the sum of the infinite series:
Since a complete infinite string of digits cannot be explicitly
written, the trailing ellipsis (...) designates the omitted digits,
which may or may not follow a pattern of some kind. One common pattern
is when a finite sequence of digits repeats infinitely. This is
designated by drawing a bar across the repeating block:
For base10 it is called a recurring decimal or repeating decimal.
An irrational number has an infinite nonrepeating representation in all integer bases. Whether a rational number
has a finite representation or requires an infinite repeating
representation depends on the base. For example, one third can be
represented by:

 or, with the base implied:
For integers p and q with gcd(p, q) = 1, the fraction p/q has a finite representation in base b if and only if each prime factor of q is also a prime factor of b.
For a given base, any number that can be represented by a finite
number of digits (without using the bar notation) will have multiple
representations, including one or two infinite representations:
 1. A finite or infinite number of zeroes can be appended:
 2. The last nonzero digit can be reduced by one and an infinite
string of digits, each corresponding to one less than the base, are
appended (or replace any following zero digits):
Applications
Decimal system
Main article: Decimal representation
In the decimal (base10) Hindu–Arabic numeral system, each position starting from the right is a higher power of 10. The first position represents 10^{0} (1), the second position 10^{1} (10), the third position 10^{2} (10 × 10 or 100), the fourth position 10^{3} (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 base10 numeral system is:
 (2 × 10^{3}) + (6 × 10^{2}) + (7 × 10^{1}) + (4 × 10^{0})
or
 (2 × 1000) + (6 × 100) + (7 × 10) + (4 × 1).
Sexagesimal system
The sexagesimal or base60 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‴31''''12''''' or 10°25^{I}59^{II}23^{III}31^{IV}12^{V}.
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 is 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.
Computing
In computing, the binary (base2) and hexadecimal
(base16) bases are used. Computers, at the most basic level, deal only
with sequences of conventional zeroes and ones, thus it is easier in
this sense to deal with powers of two. The hexadecimal system is used as
"shorthand" for binary—every 4 binary digits (bits) relate to one and
only one hexadecimal digit. In hexadecimal, the six digits after 9 are
denoted by A, B, C, D, E, and F (and sometimes a, b, c, d, e, and f).
The octal
numbering system is also used as another way to represent binary
numbers. In this case the base is 8 and therefore only digits 0, 1, 2,
3, 4, 5, 6, and 7 are used. When converting from binary to octal every 3
bits relate to one and only one octal digit.
Other bases in human language
Base12 systems (duodecimal
or dozenal) have been popular because multiplication and division are
easier than in base10, with addition and subtraction being just as
easy. Twelve is a useful base because it has many factors.
It is the smallest common multiple of one, two, three, four and six.
There is still a special word for "dozen" in English, and by analogy
with the word for 10^{2}, hundred, commerce developed a word for 12^{2}, gross.
The standard 12hour clock and common use of 12 in English units
emphasize the utility of the base. In addition, prior to its conversion
to decimal, the old British currency Pound Sterling (GBP) partially
used base12; there were 12 pence (d) in a shilling (s), 20 shillings
in a pound (£), and therefore 240 pence in a pound. Hence the term LSD
or, more properly, £sd.
The Maya civilization and other civilizations of preColumbian Mesoamerica used base20 (vigesimal),
as did several North American tribes (two being in southern
California). Evidence of base20 counting systems is also found in the
languages of central and western Africa.
Remnants of a Gaulish base20 system also exist in French, as seen today in the names of the numbers from 60 through 99. For example, sixtyfive is soixantecinq (literally, "sixty [and] five"), while seventyfive is soixantequinze
(literally, "sixty [and] fifteen"). Furthermore, for any number between
80 and 99, the "tenscolumn" number is expressed as a multiple of
twenty (somewhat similar to the archaic English manner of speaking of "scores", probably originating from the same underlying Celtic system). For example, eightytwo is quatrevingtdeux (literally, four twenty[s] [and] two), while ninetytwo is quatrevingtdouze
(literally, four twenty[s] [and] twelve). In Old French, forty was
expressed as two twenties and sixty was three twenties, so that
fiftythree was expressed as two twenties [and] thirteen, and so on.
The Irish language also used base20 in the past, twenty being fichid, forty dhá fhichid, sixty trí fhichid and eighty ceithre fhichid. A remnant of this system may be seen in the modern word for 40, daoichead.
The Welsh language continues to use a base20 counting system,
particularly for the age of people, dates and in common phrases. 15 is
also important, with 16–19 being "one on 15", "two on 15" etc. 18 is
normally "two nines". A decimal system is commonly used.
Danish numerals display a similar base20 structure.
The Maori language of New Zealand also has evidence of an underlying base20 system as seen in the terms Te Hokowhitu a Tu referring to a war party (literally "the seven 20s of Tu") and Tamahokotahi, referring to a great warrior ("the one man equal to 20").
The binary system was used in the Egyptian Old Kingdom, 3000 BC to 2050 BC. It was cursive by rounding off rational numbers smaller than 1 to 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64, with a 1/64 term thrown away (the system was called the Eye of Horus).
A number of Australian Aboriginal languages employ binary or binarylike counting systems. For example, in Kala Lagaw Ya, the numbers one through six are urapon, ukasar, ukasarurapon, ukasarukasar, ukasarukasarurapon, ukasarukasarukasar.
North and Central American natives used base4 (quaternary)
to represent the four cardinal directions. Mesoamericans tended to add a
second base5 system to create a modified base20 system.
A base5 system (quinary)
has been used in many cultures for counting. Plainly it is based on the
number of digits on a human hand. It may also be regarded as a subbase
of other bases, such as base10, base20, and base60.
A base8 system (octal) was devised by the Yuki tribe of Northern California, who used the spaces between the fingers to count, corresponding to the digits one through eight.^{[citation needed]} There is also linguistic evidence which suggests that the Bronze Age ProtoIndo Europeans
(from whom most European and Indic languages descend) might have
replaced a base8 system (or a system which could only count up to 8)
with a base10 system. The evidence is that the word for 9, newm, is suggested by some to derive from the word for "new", newo, suggesting that the number 9 had been recently invented and called the "new number".^{}
Many ancient counting systems use five as a primary base, almost
surely coming from the number of fingers on a person's hand. Often these
systems are supplemented with a secondary base, sometimes ten,
sometimes twenty. In some African languages the word for five is the same as "hand" or "fist" (Dyola language of GuineaBissau, Banda language of Central Africa).
Counting continues by adding 1, 2, 3, or 4 to combinations of 5, until
the secondary base is reached. In the case of twenty, this word often
means "man complete". This system is referred to as quinquavigesimal. It is found in many languages of the Sudan region.
The Telefol language, spoken in Papua New Guinea, is notable for possessing a base27 numeral system.
Nonstandard positional numeral systems
Interesting properties exist when the base is not fixed or positive
and when the digit symbol sets denote negative values. There are many
more variations. These systems are of practical and theoretic value to
computer scientists.
Balanced ternary uses a base of 3 but the digit set is {1,0,1} instead of {0,1,2}. The "1" has an equivalent value of −1. The negation of a number is easily formed by switching the on the 1s. This system can be used to solve the balance problem, which requires finding a minimal set of known counterweights to determine an unknown weight. Weights of 1, 3, 9, ... 3^{n} known units can be used to determine any unknown weight up to 1 + 3 + ... + 3^{n}
units. A weight can be used on either side of the balance or not at
all. Weights used on the balance pan with the unknown weight are
designated with 1, with 1 if used on the empty pan, and with 0 if not used. If an unknown weight W is balanced with 3 (3^{1}) on its pan and 1 and 27 (3^{0} and 3^{3}) on the other, then its weight in decimal is 25 or 1011 in balanced base3. (1011_{3} = 1 × 3^{3} + 0 × 3^{2} − 1 × 3^{1} + 1 × 3^{0} = 25).
The factorial number system uses a varying radix, giving factorials as place values; they are related to Chinese remainder theorem and Residue number system enumerations. This system effectively enumerates permutations. A derivative of this uses the Towers of Hanoi
puzzle configuration as a counting system. The configuration of the
towers can be put into 1to1 correspondence with the decimal count of
the step at which the configuration occurs and vice versa.
Decimal equivalents:  −3  −2  −1  0  1  2  3  4  5  6  7  8 
Balanced base3:  10  11  1  0  1  11  10  11  111  110  111  101 
Base −2:  1101  10  11  0  1  110  111  100  101  11010  11011  11000 
Factoroid:  0  10  100  110  200  210  1000  1010  1100 
Nonpositional 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).^{}
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