 Positional notation

Numeral systems by culture HinduArabic numerals Western Arabic (Hindu numerals)
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UrnfieldList of numeral system topics Positional systems by base Decimal (10) 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 20, 24, 30, 36, 60, 64 List of numeral systems 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, the notation can be extended to include fractions and the numeric expansions of real numbers. The HinduArabic numeral system is an example for a positional notation, based on the number 10.
Contents
History
Today, the base 10 (decimal) system is ubiquitous. It was likely motivated by counting with the ten fingers. However, other bases have been used. For example, the Babylonian numeral system, credited as the first positional number system, was base 60.
Counting rods and most abacuses in history represented numbers in a positional numeral system. Before positional notation became standard, simple additive systems (signvalue notation) were used such as Roman Numerals, and accountants in ancient Rome and during the Middle Ages used the abacus or stone counters to do arithmetic.^{[1]}
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 (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 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:
Thus 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.^{[2]}Aryabhata stated "sthānam sthānam daśa guṇam" meaning "From place to place, ten times in value". His system lacked zero^{[citation needed]}. The zero was added by Brahmagupta^{[citation needed]}. 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 when writing cheque, used natural language numerals, for instance 100 is written as 壹佰,which can never be forged into 壹仟(1000) or 伍仟壹佰(5100).
Mathematics
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 a_{3}a_{2}a_{1}a_{0} represents a sequence of digits, not multiplication)
Notation
Sometimes, a subscript notation is used where the base number is written in subscript after the number represented. For example, 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 base 2 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 "base b". So binary numbers are "base 2"; octal numbers are "base 8"; decimal numbers are "base 10"; and so on.
Numbers of a given radix b have digits {0, 1, ..., b2, b1}. 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 and do not make sense: 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 base 10, 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 nonzero numeral with more than one digit position will mean a different number in a different number base, but in general, the digits "0", "1", "2", "3", "4", "5", "6", "7", "8" , and "9" will mean the same. ^{[3]} 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. "23" could likely be any base, base4 through base60. "23" in base4 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 it's 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 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 215999. 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". ^{[4]} 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 5² 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 8² 0 groups of 8 7 groups of 1 oooooooo oooooooo o o oooooooo oooooooo + + o o o oooooooo oooooooo o o oooooooo oooooooo
There is, however, a shorter method which is basically the above method calculated mathematically. Because we work in base ten normally, it is easier to think of numbers in this way and therefore easier to convert them to base ten first, though it is possible (but difficult) to convert straight between nondecimal bases without using this intermediate step.
A number a_{n}a_{n1}...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 being the most significant value the one corresponding to the last division and the least significant value is 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 ... base 3 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 base 10 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 representationIn the decimal (base10) HinduArabic 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 base 10 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 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‴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 (base 2) and hexadecimal (base 16) 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.
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, 3,000 BCE to 2,050 BCE. 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 base 4 (quaternary) to represent the four cardinal directions. Mesoamericans tended to add a second base 5 system to create a modified base 20 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 base 10, base 20, and base 60.
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 base 8 system (or a system which could only count up to 8) with a base 10 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'.^{[5]}
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
Main article: Nonstandard positional numeral systemsInteresting 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 base 3. (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 1 to 1 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 base 3: 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).^{[6]} 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).^{[7]}
See also
 Numeral system
 HinduArabic numeral system
 Nonstandard positional numeral systems
 Category:Positional numeral systems
 Mixed radix
 Algorism
 Subtractive notation
Notes
 ^ Ifrah, page 187
 ^ Ifrah^{[page needed]}
 ^ The digit will retain its meaning in other number bases, in general, because a higher number base would normally be a notational extension of the lower number base in any systematic organization. In the mathematical sciences there is virtually only one positionalnotation numeral system for each base below 10, and this extends with few, if insignificant, variations on the choice of alphabetic digits for those bases above 10.
 ^ We don't usually remove the digits lowercase "l" and lowercase "o", for they are discernable from the digits "1" and "0" in most fonts.
 ^ (Mallory & Adams 1997) Encyclopedia of IndoEuropean Culture
 ^ Ifrah, pages 326, 379
 ^ Ifrah, pages 261264
References
 O'Connor, John; Robertson, Edmund (December 2000), Babylonian numerals Babylonian Numerals, http://wwwgroups.dcs.stand.ac.uk/~history/HistTopics/Babylonian_numerals.html Babylonian numerals, retrieved 21 August 2010
 Kadvany, John (December 2007). "Positional Value and Linguistic Recursion". Journal of Indian Philosophy.
 Knuth, Donald (1997). The art of Computer Programming. 2. AddisonWesley. pp. 195–213. ISBN 0201896842.
 Ifrah, George (2000). The Universal History of Numbers: From Prehistory to the Invention of the Computer. Wiley. ISBN 0471375683.
 Kroeber, Alfred (1976) [1925]. Handbook of the Indians of California. Courier Dover Publications. p. 176. ISBN 0486233685, 9780486233680. http://books.google.com/books?id=Plqf_OTz4ukC.
External links
Categories: Numeration
 Mathematical notation
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