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| {{original research|date=April 2013}}
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| {{numeral systems}}
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| '''Positional notation''' or '''place-value notation''' is a method of representing or encoding [[number]]s. 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]] led to the quick spread of the notation across the world.
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| With the use of a [[radix point]] (decimal point), the notation can be extended to include [[fraction (mathematics)|fraction]]s and the numeric expansions of [[real number]]s. The [[Hindu–Arabic numeral system]] is an example for a positional notation, [[decimal representation|based on the number 10]].
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| ==History==
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| Today, the base-10 ([[decimal]]) system, which is likely motivated by counting with the ten [[finger]]s, is ubiquitous. Other bases have been used in the past however, and some continue to be used today. For example, the [[Babylonian numerals|Babylonian numeral system]], credited as the first positional number system, was [[base-60]]. [[Counting rods]] and most [[abacus]]es have been used to represent numbers in a positional numeral system, but it lacked a real 0 value. Zero was indicated by a ''space'' between sexagesimal numerals. By 300 BC, a punctuation symbol (two slanted wedges) was co-opted as a [[Free variables and bound variables|placeholder]] in the same [[Babylonian numerals|Babylonian system]]. In a tablet unearthed at [[Kish (Sumer)|Kish]] (dating from about 700 BC), the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges.<ref name="multiref1">Kaplan, Robert. (2000). ''The Nothing That Is: A Natural History of Zero''. Oxford: Oxford University Press.</ref> 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.
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| Before positional notation became standard, simple additive systems ([[sign-value 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.<ref>Ifrah, page 187</ref>
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| 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 additive-system-plus-abacus. Although electronic calculators have largely replaced the abacus, the latter continues to be used in Japan and other Asian countries.
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| [[Georges Ifrah]] concludes in his ''Universal History of Numbers'':
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| {{quote|Thus it would seem highly probable under the circumstances that the discovery of zero and the place-value system were inventions unique to the [[India]]n civilization. As the [[Brahmi numeral|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.<ref>Ifrah {{Page needed|date=September 2010}}</ref>}}
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| [[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 [[cheque]]s 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).
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| ==Mathematics==
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| {{unreferenced section|date=March 2013}}
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| ===Base of the numeral system===
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| In [[numeral system|mathematical numeral systems]], the base or radix is usually the number of unique [[Numerical digit|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".
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| 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.
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| 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.
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| (In certain [[non-standard positional numeral systems]], including [[bijective numeration]], the definition of the base or the allowed digits deviates from the above.)
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| In base-10 (decimal) positional notation, there are 10 [[decimal digit]]s and the number
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| :<math>2506 = 2 \times 10^3 + 5 \times 10^2 + 0 \times 10^1 + 6 \times 10^0</math>.
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| In base-16 ([[hexadecimal]]), there are 16 hexadecimal digits (0–9 and A–F) and the number
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| :<math>171\mathrm{B} = 1 \times 16^3 + 7 \times 16^2 + 1 \times 16^1 + \mathrm{B} \times 16^0 </math> (where B represents the number eleven as a single symbol)
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| In general, in base-''b'', there are ''b'' digits and the number
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| :<math>a_3 a_2 a_1 a_0 = a_3 \times b^3 + a_2 \times b^2 + a_1 \times b^1 + a_0 \times b^0 </math> (Note that <math>a_3 a_2 a_1 a_0</math> represents a sequence of digits, not [[multiplication]])
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| ===Notation===
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| Sometimes the base number is written in [[subscript]] after the number represented. For example, 23<sub>8</sub> 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.
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| When describing base in [[mathematical notation]], the letter ''b'' is generally used as a [[symbol]] for this concept, so, for a [[Binary numeral system|binary]] system, ''b'' [[equality (mathematics)|equals]] 2. Another common way of expressing the base is writing it as a '''decimal''' subscript after the number that is being represented. 1111011<sub>2</sub> implies that the number 1111011 is a base-2 number, equal to 123<sub>10</sub> (a [[decimal notation]] representation), 173<sub>8</sub> ([[octal]]) and 7B<sub>16</sub> ([[hexadecimal]]). In books and articles, when using initially the written abbreviations of number bases, the base is not subsequently printed: it is assumed that binary 1111011 is the same as 1111011<sub>2</sub>.
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| 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.
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| 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<sub>2</sub>, 2<sub>2</sub>, 1A<sub>9</sub>. (In all cases, one or more digits is not in the set of allowed digits for the given base.)
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| ===Exponentiation===
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| 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 ''n''th 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.
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| 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:
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| :<math>4\times b^2 + 6\times b^1 + 5\times b^0</math>
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| If the number 465 was in base-10, then it would equal:
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| :<math>4\times 10^2 + 6\times 10^1 + 5\times 10^0 = 4\times 100 + 6\times 10 + 5\times 1 = 465</math>
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| (465<sub>10</sub> = 465<sub>10</sub>)
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| If however, the number were in base 7, then it would equal:
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| :<math>4\times 7^2 + 6\times 7^1 + 5\times 7^0 = 4\times 49 + 6\times 7 + 5\times 1 = 243</math>
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| (465<sub>7</sub> = 243<sub>10</sub>)
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| 10<sub>''b''</sub> = ''b'' for any base ''b'', since 10<sub>''b''</sub> = 1×''b''<sup>1</sup> + 0×''b''<sup>0</sup>. For example 10<sub>2</sub> = 2; 10<sub>3</sub> = 3; 10<sub>16</sub> = 16<sub>10</sub>. Note that the last "16" is indicated to be in base 10. The base makes no difference for one-digit numerals.
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| Numbers that are not [[integer]]s 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:
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| :<math>2\times 10^0 + 3\times 10^{-1} + 5\times 10^{-2}</math>
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| 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:
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| 241 in base 5:
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| 2 groups of 5<sup>2</sup> (25) 4 groups of 5 1 group of 1
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| ooooo ooooo
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| ooooo ooooo ooooo ooooo
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| ooooo ooooo + + o
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| ooooo ooooo ooooo ooooo
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| ooooo ooooo
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| 241 in base 8:
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| 2 groups of 8<sup>2</sup> (64) 4 groups of 8 1 group of 1
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| oooooooo oooooooo
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| oooooooo oooooooo
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| oooooooo oooooooo oooooooo oooooooo
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| oooooooo oooooooo + + o
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| oooooooo oooooooo
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| oooooooo oooooooo oooooooo oooooooo
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| oooooooo oooooooo
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| oooooooo oooooooo
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| 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.
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| ===Digits and numerals===
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| A ''digit'' is what is used as a position in place-value notation, and a ''numeral'' is one or more digits. Today's most common digits are the [[Hindu numerals|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.
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| A non-zero ''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.<ref>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 science]]s there is virtually only one positional-notation 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.</ref> The base-8 numeral 23<sub>8</sub> contains two digits, "2" and "3", and with a base number (subscripted) "8", means 19. In our notation here, the subscript "<sub>8</sub>" of the numeral 23<sub>8</sub> is part of the numeral, but this may not always be the case. Imagine the numeral "23" as having [[#Non-standard positional numeral systems|an ambiguous base]] number. Then "23" could likely be any base, base-4 through base-60. In base-4 "23" means 11, and in base-60 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.
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| In certain applications when a numeral with a fixed number of positions needs to represent a greater number, a higher number-base with more digits per position can be used. A three-digit, decimal numeral can represent only up to '''999'''. But if the number-base 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 number-digit-numeral hierarchy). A three-digit numeral "ZZZ" in base-60 could mean '''{{val|215999}}'''. If we use the entire collection of our [[alphanumerics]] we could ultimately serve a base-''62'' numeral system, but we remove two digits, uppercase "I" and uppercase "O", to reduce confusion with digits "1" and "0".<ref>We do ''not'' usually remove the ''lowercase'' digits "l" and lowercase "o", for in most fonts they are discernible from the digits "1" and "0".</ref>
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| We are left with a base-60, or sexagesimal numeral system utilizing 60 of the 62 standard alphanumerics. (But see ''[[#Sexagesimal system|Sexagesimal system]]'' below.)
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| The common numeral systems in computer science are binary (radix 2), octal (radix 8), and hexadecimal (radix 16). In [[Binary numeral system|binary]] only digits "0" and "1" are in the numerals. In the [[octal]] numerals, are the eight digits 0–7. [[Hexadecimal|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".
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| ===Base conversion===<!--This section is linked from [[Duodecimal]]-->
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| {{unreferenced section|date=April 2012}}
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| Bases can be converted between each other by drawing the diagram above and rearranging the objects to conform to the new base, for example:
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| 241 in base 5:
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| 2 groups of 5<sup>2</sup> 4 groups of 5 1 group of 1
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| ooooo ooooo
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| ooooo ooooo ooooo ooooo
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| ooooo ooooo + + o
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| ooooo ooooo ooooo ooooo
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| ooooo ooooo
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| is equal to 107 in base 8:
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| 1 group of 8<sup>2</sup> 0 groups of 8 7 groups of 1
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| oooooooo
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| oooooooo
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| oooooooo
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| oooooooo + + ooooooo
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| oooooooo
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| oooooooo
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| oooooooo
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| oooooooo
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| There is, however, a shorter method which is basically the above method calculated mathematically. Because we work in base-10 normally, it is easier to think of numbers in this way and therefore easier to convert them to base-10 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 non-decimal bases without using this intermediate step. (However, conversion from bases like 8, 16 or 256 to base-2 can be achieved by writing each digit in binary notation, and subsequently, conversion from base-2 to e.g. base-16 can be achieved by writing each group of four binary digits as one hexadecimal digit.)
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| A number ''a<sub>n</sub>a<sub>n''−1</sub>...''a''<sub>2</sub>''a''<sub>1</sub>''a''<sub>0</sub> where ''a''<sub>0</sub>, ''a''<sub>1</sub> ... ''a<sub>n''</sub> 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:
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| :<math>\sum_{i=0}^n \left( a_i\times b^i \right)</math>
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| Thus, in the example above:
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| :<math>241_5 = 2\times 5^2 + 4\times 5^1 + 1\times 5^0 = 50 + 20 + 1 = 71_{10}</math>
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| 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.
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| Example #1 decimal to septal:
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| :<math>\begin{align}123_{10} = 123 / 7 = 17\text{ with a remainder of }(4)\\
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| 17 / 7 = 2\text{ with a remainder of }(3)\\
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| 2 / 7 = 0\text{ with a remainder of }(2)\\
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| &= 234_7\end{align}</math>
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| Example #2 decimal to octal:
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| :<math>\begin{align}456_{10} = 456 / 8 = 57\text{ with a remainder of }(0)\\
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| 57 / 8 = 7\text{ with a remainder of }(1)\\
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| 7 / 8 = 0\text{ with a remainder of }(7)\\
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| &= 710_8\end{align}</math>
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| The most common example is that of changing from [[Binary numeral system#Conversion to and from other numeral systems|decimal to binary]].
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| ===Infinite representations===
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| {{unreferenced section|date=January 2013}}
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| The representation of non-integers 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 (mathematics)|series]]:
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| :<math>1\times 3^{0\,\,\,} + {}</math>
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| :<math>1\times 3^{-1\,\,} + 2\times 3^{-2\,\,\,} + {}</math>
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| :<math>1\times 3^{-3\,\,} + 1\times 3^{-4\,\,\,} + 2\times 3^{-5\,\,\,} + {}</math>
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| :<math>1\times 3^{-6\,\,} + 1\times 3^{-7\,\,\,} + 1\times 3^{-8\,\,\,} + 2\times 3^{-9\,\,\,} + {}</math>
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| :<math>1\times 3^{-10} + 1\times 3^{-11} + 1\times 3^{-12} + 1\times 3^{-13} + 2\times 3^{-14} + \cdots</math>
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| 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 [[Vinculum (symbol)|vinculum]] across the repeating block:
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| :<math>2.42\overline{314}_5 = 2.42314314314314314\dots_5</math>
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| For base-10 it is called a [[recurring decimal]] or repeating decimal.
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| An [[irrational number]] has an infinite non-repeating 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:
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| :<math>0.1_3\,</math>
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| :<math>0.\overline3_{10} = 0.3333333\dots_{10}</math>
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| ::or, with the base implied:
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| ::<math>0.\overline3 = 0.3333333\dots</math>
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| :<math>0.\overline{01}_2 = 0.010101\dots_2</math>
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| :<math>0.2_6\,</math>
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| For integers ''p'' and ''q'' with [[greatest common divisor|''gcd'']](''p'', ''q'') = 1, the [[fraction (mathematics)|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''.
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| 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:
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| :1. A finite or infinite number of zeroes can be appended:
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| ::<math>3.46_7 = 3.460_7 = 3.460000_7 = 3.46\overline0_7</math>
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| :2. The last non-zero 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):
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| ::<math>3.46_7 = 3.45\overline6_7</math>
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| ::<math>1_{10} = 0.\overline9_{10}</math>
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| ::<math>220_5 = 214.\overline4_5</math>
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| ==Applications==
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| ===Decimal system===
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| {{Main|Decimal representation}}
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| 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 [[1 E0|10<sup>0</sup>]] (1), the second position [[1 E1|10<sup>1</sup>]] (10), the third position [[1 E2|10<sup>2</sup>]] ({{nowrap|10 × 10}} or 100), the fourth position [[1000 (number)|10<sup>3</sup>]] ({{nowrap|10 × 10 × 10}} or 1000), and so on.
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| [[Decimal|Fraction]]al values are indicated by a [[Decimal separator|separator]], which varies by [[locale]]. Usually this separator is a period or [[full stop]], or a [[comma (punctuation)|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 [[1 E-1|10<sup>−1</sup>]] (0.1), the second position [[1 E-2|10<sup>−2</sup>]] (0.01), and so on for each successive position.
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| As an example, the number 2674 in a base-10 numeral system is:
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| :(2 × 10<sup>3</sup>) + (6 × 10<sup>2</sup>) + (7 × 10<sup>1</sup>) + (4 × 10<sup>0</sup>)
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| or
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| :(2 × 1000) + (6 × 100) + (7 × 10) + (4 × 1).
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| ===Sexagesimal system===
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| The [[sexagesimal]] or base-60 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.
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| 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 <nowiki>10°25′59″23‴31''''12'''''</nowiki> or 10°25<sup>I</sup>59<sup>II</sup>23<sup>III</sup>31<sup>IV</sup>12<sup>V</sup>.
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| 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.
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| 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.
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| ===Computing===
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| In [[computing]], the [[Binary numeral system|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).
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| 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.
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| ===Other bases in human language===
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| Base-12 systems ([[duodecimal]] or dozenal) have been popular because multiplication and division are easier than in base-10, with addition and subtraction being just as easy. Twelve is a useful base because it has many [[divisor|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<sup>2</sup>, ''hundred'', commerce developed a word for 12<sup>2</sup>, ''gross''. The standard 12-hour 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 base-12; 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]].
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| The [[Maya civilization]] and other civilizations of [[pre-Columbian]] [[Mesoamerica]] used base-20 ([[vigesimal]]), as did several North American tribes (two being in southern California). Evidence of base-20 counting systems is also found in the languages of central and western [[Africa]].
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| Remnants of a [[Gaulish language|Gaulish]] base-20 system also exist in French, as seen today in the names of the numbers from 60 through 99. For example, sixty-five is ''soixante-cinq'' (literally, "sixty [and] five"), while seventy-five is ''soixante-quinze'' (literally, "sixty [and] fifteen"). Furthermore, for any number between 80 and 99, the "tens-column" number is expressed as a multiple of twenty (somewhat similar to the archaic English manner of speaking of "[[20 (number)|scores]]", probably originating from the same underlying Celtic system). For example, eighty-two is ''quatre-vingt-deux'' (literally, four twenty[s] [and] two), while ninety-two is ''quatre-vingt-douze'' (literally, four twenty[s] [and] twelve). In Old French, forty was expressed as two twenties and sixty was three twenties, so that fifty-three was expressed as two twenties [and] thirteen, and so on.
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| The [[Irish language]] also used base-20 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''.
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| The [[Welsh language]] continues to use a [[vigesimal|base-20]] [[Welsh language#Counting system|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.
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| [[Danish language#Numerals|Danish numerals]] display a similar [[vigesimal|base-20]] structure.
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| The Maori language of New Zealand also has evidence of an underlying base-20 system as seen in the terms ''Te Hokowhitu a Tu'' referring to a war party (literally "the seven 20s of Tu") and ''Tama-hokotahi'', referring to a great warrior ("the one man equal to 20").
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| [[Binary numeral system|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 {{nowrap|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#In arithmetic|Eye of Horus]]).
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| A number of [[Australian Aboriginal languages]] employ binary or binary-like counting systems. For example, in [[Kala Lagaw Ya]], the numbers one through six are ''urapon'', ''ukasar'', ''ukasar-urapon'', ''ukasar-ukasar'', ''ukasar-ukasar-urapon'', ''ukasar-ukasar-ukasar''.
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| North and Central American natives used base-4 ([[Quaternary numeral system|quaternary]]) to represent the four cardinal directions. Mesoamericans tended to add a second base-5 system to create a modified base-20 system.
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| A base-5 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 sub-base of other bases, such as base-10, base-20, and base-60.
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| A base-8 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.<ref>{{citation|page=38|title=Pi in the sky: counting, thinking, and being|first=John D.|last=Barrow|publisher=Clarendon Press|year=1992|isbn=9780198539568}}.</ref> There is also linguistic evidence which suggests that the Bronze Age [[Proto-Indo European]]s (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".<ref>(Mallory & Adams 1997) [[Encyclopedia of Indo-European Culture]]</ref>
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| 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 [[Guinea-Bissau]], [[Banda languages|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.
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| The [[Telefol language]], spoken in [[Papua New Guinea]], is notable for possessing a base-27 numeral system.
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| ==Non-standard positional numeral systems==
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| {{Main|Non-standard positional numeral systems}}
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| 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.
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| [[Balanced ternary]] uses a base of 3 but the digit set is {{{overline|1}},0,1} instead of {0,1,2}. The "{{overline|1}}" has an equivalent value of −1. The negation of a number is easily formed by switching the {{overline| }} on the 1s. This system can be used to solve the [[balance problem]], which requires finding a minimal set of known counter-weights to determine an unknown weight. Weights of 1, 3, 9, ... 3<sup>''n''</sup> known units can be used to determine any unknown weight up to 1 + 3 + ... + 3<sup>''n''</sup> 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 {{overline|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<sup>1</sup>) on its pan and 1 and 27 (3<sup>0</sup> and 3<sup>3</sup>) on the other, then its weight in decimal is 25 or 10{{overline|1}}1 in balanced base-3. {{nowrap|(10{{overline|1}}1<sub>3</sub> {{=}} 1 × 3<sup>3</sup> + 0 × 3<sup>2</sup> − 1 × 3<sup>1</sup> + 1 × 3<sup>0</sup> {{=}} 25).}}
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| The [[factorial number system]] uses a varying radix, giving [[factorial]]s 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.
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| {|class="wikitable" style="text-align:center;" border="1"
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| |align="left" |Decimal equivalents:
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| |width="6%" |−3
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| |width="6%" |−2
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| |width="6%" |−1
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| |width="6%" |0
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| |width="6%" |1
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| |width="6%" |2
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| |width="6%" |3
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| |width="6%" |4
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| |width="6%" |5
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| |width="8%" |6
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| |width="8%" |7
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| |width="8%" |8
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| |-
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| |align="left" |Balanced base-3:
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| |{{overline|1}}0
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| |{{overline|1}}1
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| |{{overline|1}}
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| |0
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| |1
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| |1{{overline|1}}
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| |10
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| |11
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| |1{{overline|1}}{{overline|1}}
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| |1{{overline|1}}0
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| |1{{overline|1}}1
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| |10{{overline|1}}
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| |-
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| |align="left" |Base −2:
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| |1101
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| |10
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| |11
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| |0
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| |1
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| |110
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| |111
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| |100
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| |101
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| |11010
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| |11011
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| |11000
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| |-
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| |align="left" |Factoroid: || || || ||0 ||10 ||100 ||110 ||200 ||210 ||1000 ||1010 ||1100
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| |}
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| ==Non-positional positions==
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| 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).<ref>Ifrah, pages 326, 379</ref> 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 [[Greek numerals#Hellenistic zero|zero symbol]]).<ref>Ifrah, pages 261-264</ref>
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| ==See also==
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| *[[Numeral system]]
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| *[[Hindu-Arabic numeral system]]
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| *[[Non-standard positional numeral systems]]
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| *[[:Category:Positional numeral systems]]
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| *[[Mixed radix]]
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| *[[Algorism]]
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| *[[Subtractive notation]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| *{{Cite web
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| |last=O'Connor
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| |first=John
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| |last2=Robertson
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| |first2=Edmund
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| |title=Babylonian Numerals
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| |date=December 2000
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| |url=http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_numerals.html Babylonian numerals
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| |accessdate=21 August 2010
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| |postscript=.}}
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| *{{Cite journal
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| |last=Kadvany
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| |first=John
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| |title=Positional Value and Linguistic Recursion
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| |journal=Journal of Indian Philosophy
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| |date=December 2007}}
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| *{{Cite book
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| |last=Knuth
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| |first=Donald
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| |authorlink=Donald Knuth
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| |title=The art of Computer Programming
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| |volume=2
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| |pages=195–213
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| |publisher=Addison-Wesley
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| |year=1997
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| |isbn=0-201-89684-2}}
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| *{{Cite book
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| |last=Ifrah
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| |first=George
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| |title=The Universal History of Numbers: From Prehistory to the Invention of the Computer
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| |publisher=Wiley
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| |year=2000
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| |isbn=0-471-37568-3}}
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| *{{Cite book
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| |last=Kroeber
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| |first=Alfred
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| |authorlink=Alfred Kroeber
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| |title=Handbook of the Indians of California
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| |publisher=Courier Dover Publications
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| |year=1976
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| |origyear=1925
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| |page=176
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| |url=http://books.google.com/books?id=Plqf_OTz4ukC
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| |isbn=0-486-23368-5, 9780486233680}}
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| ==External links==
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| *[http://ultrastudio.org/en/MechengburakalkanApplet-1.7.zip Accurate Base Conversion]
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| *[http://sciences.aum.edu/~sbrown/Hindu%20Arabic%20and%20Chinese.pdf The Development of Hindu Arabic and Traditional Chinese Arithmetics]
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| *[http://www.cut-the-knot.org/recurrence/conversion.shtml Implementation of Base Conversion] at [[cut-the-knot]]
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| *[http://www.intuitor.com/counting/ Learn to count other bases on your fingers]
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| {{Use dmy dates|date=September 2010}}
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| *[http://www.codeproject.com/Articles/350252/From-one-to-another-number-system/ From one to another number system]
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| {{DEFAULTSORT:Positional Notation}}
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| [[Category:Positional numeral systems]]
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| [[Category:Mathematical notation]]
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| [[Category:Articles containing proofs]]
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| {{Link GA|de}}
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