View source for Positional notation

{{Short description|Method for representing or encoding numbers}}{{Redirects|Positional system|the voting rule|positional voting}}[[File:Positional notation glossary-en.svg|thumb|300px|Glossary of terms used in the positional numeral systems]]
{{numeral systems}}

'''Positional notation''', also known as '''place-value notation''', '''positional numeral system''', or simply '''place value''', usually denotes the extension to any [[radix|base]] of the [[Hindu–Arabic numeral system]] (or [[decimal|decimal system]]). More generally, a positional system is a [[numeral system]] in which the contribution of a digit to the value of a number is the value of the digit multiplied by a factor determined by the position of the digit. In early [[numeral system]]s, such as [[Roman numerals]], a digit has only one value: I means one, X means ten and C a hundred (however, the values may be modified when combined). In modern positional systems, such as the [[decimal|decimal system]], the position of the digit means that its value must be multiplied by some value: in 555, the three identical symbols represent five hundreds, five tens, and five units, respectively, due to their different positions in the digit string.

The [[Babylonian Numerals|Babylonian numeral system]], base 60, was the first positional system to be developed, and its influence is present today in the way time and angles are counted in tallies related to 60, such as 60 minutes in an hour and 360 degrees in a circle. Today, the Hindu–Arabic numeral system ([[base ten]]) is the most commonly used system globally. However, the [[binary numeral system]] (base two) is used in almost all [[computer]]s and [[electronic device]]s because it is easier to implement efficiently in [[electronic circuit]]s.

Systems with negative base, [[complex number|complex]] base or negative digits have been described. Most of them do not require a minus sign for designating negative numbers.

The use of a [[radix point]] (decimal point in base ten), extends to include [[fraction (mathematics)|fractions]] and allows the representation of any [[real number]] with arbitrary accuracy. With positional notation, [[arithmetic|arithmetical computations]] are much simpler than with any older numeral system; this led to the rapid spread of the notation when it was introduced in western Europe.

== History ==
[[File:abacus 6.png|thumb|[[Suanpan]] (the number represented in the picture is 6,302,715,408)]]
Today, the base-10 ([[decimal]]) system, which is presumably motivated by counting with the ten [[finger]]s, is ubiquitous. Other bases have been used in the past, and some continue to be used today. For example, the [[Babylonian numerals|Babylonian numeral system]], credited as the first positional numeral system, was [[base-60]]. However, it lacked a real [[zero]]. Initially inferred only from context, later, by about 700&nbsp;BC, zero came to be indicated by a "space" or a "punctuation symbol" (such as two slanted wedges) between numerals.<ref name="multiref1">{{cite book | last = Kaplan | first = Robert | year = 2000 | title = The Nothing That Is: A Natural History of Zero | url = https://archive.org/details/nothingthatisnat00kapl | url-access = registration |via=archive.org | location = Oxford | publisher = Oxford University Press |pages=11–12 }}</ref> It was a [[variable (mathematics)|placeholder]] rather than a true zero because it was not used alone or at the end of a number. Numbers like 2 and 120 (2×60) looked the same because the larger number lacked a final placeholder. Only context could differentiate them.

The polymath [[Archimedes]] (ca. 287–212 BC) invented a decimal positional system based on 10<sup>8</sup> in his [[The Sand Reckoner|Sand Reckoner]];<ref name="Greek numerals">{{Cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Greek_numbers.html |title=Greek numerals |access-date=31 May 2016 |archive-url=https://web.archive.org/web/20161126013536/http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Greek_numbers.html |archive-date=26 November 2016 |url-status=dead }}</ref> 19th century German mathematician [[Carl Friedrich Gauss|Carl Gauss]] lamented how science might have progressed had Archimedes only made the leap to something akin to the modern decimal system.<ref>[[Karl Menninger (mathematics)|Menninger, Karl]]: ''Zahlwort und Ziffer. Eine Kulturgeschichte der Zahl'', Vandenhoeck und Ruprecht, 3rd. ed., 1979, {{isbn|3-525-40725-4}}, pp. 150–153</ref> [[Hellenistic period|Hellenistic]] and [[Roman Empire|Roman]] astronomers used a base-60 system based on the Babylonian model (see {{slink|Greek numerals|Zero}}).

Before positional notation became standard, simple additive systems ([[sign-value notation]]) such as [[Roman numerals]] or [[Chinese numerals]] were used, and accountants in the past used the [[abacus]] or stone counters to do arithmetic until the introduction of positional notation.<ref>Ifrah, page 187</ref>

[[File:Chounumerals.svg|thumb|right|280px|Chinese [[rod numerals]]; Upper row vertical form{{br}} Lower row horizontal form]]
[[Counting rods]] and most abacuses have been used to represent numbers in a positional numeral system. 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.

The oldest extant positional notation system is either that of Chinese [[rod numerals]], used from at least the early 8th century, or perhaps [[Khmer numerals]], showing possible usages of positional-numbers in the 7th century. Khmer numerals and other [[Indian numerals]] originate with the [[Brahmi numerals]] of about the 3rd century BC, which symbols were, at the time, not used positionally. Medieval Indian numerals are positional, as are the derived [[Arabic numerals]], recorded from the 10th century.

After the [[French Revolution]] (1789–1799), the new French government promoted the extension of the decimal system.<ref>L. F. Menabrea.
Translated by Ada Augusta, Countess of Lovelace.
[http://www.fourmilab.ch/babbage/sketch.html "Sketch of The Analytical Engine Invented by Charles Babbage"] {{Webarchive|url=https://web.archive.org/web/20080915134651/http://www.fourmilab.ch/babbage/sketch.html |date=15 September 2008 }}.
1842.</ref> Some of those pro-decimal efforts—such as [[decimal time]] and the [[decimal calendar]]—were unsuccessful. Other French pro-decimal efforts—currency [[decimalisation]] and the [[metrication]] of weights and measures—spread widely out of France to almost the whole world.

=== History of positional fractions ===
{{Main|Decimal}}

Decimal fractions were first developed and used by the Chinese in the form of [[Rod calculus#Fractions|rod calculus]] in the 1st century BC, and then spread to the rest of the world. <ref>[[Lam Lay Yong]], "The Development of Hindu-Arabic and Traditional Chinese Arithmetic", ''Chinese Science'', 1996 p38, Kurt Vogel notation</ref><ref name=jnfractn1>{{Cite book | author=[[Joseph Needham]] | chapter = Decimal System | title = [[Science and Civilisation in China|Science and Civilisation in China, Volume III, Mathematics and the Sciences of the Heavens and the Earth]] | year = 1959 | publisher = Cambridge University Press}}</ref> J. Lennart Berggren notes that positional decimal fractions were first used in the Arab by mathematician [[Abu'l-Hasan al-Uqlidisi]] as early as the 10th century.<ref name=Berggren>{{cite book | first=J. Lennart | last=Berggren | title=The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook | chapter=Mathematics in Medieval Islam | publisher=Princeton University Press | year=2007 | isbn=978-0-691-11485-9 | page=518 }}</ref> The Jewish mathematician [[Immanuel Bonfils]] used decimal fractions around 1350, but did not develop any notation to represent them.<ref>[[Solomon Gandz|Gandz, S.]]: The invention of the decimal fractions and the application of the exponential calculus by Immanuel Bonfils of Tarascon (c. 1350), Isis 25 (1936), 16–45.</ref> The Persian mathematician [[Jamshīd al-Kāshī]] made the same discovery of decimal fractions in the 15th century.<ref name=Berggren /> [[Al Khwarizmi]] introduced fractions to Islamic countries in the early 9th century; his fraction presentation was similar to the traditional Chinese mathematical fractions from [[Sunzi Suanjing]].<ref name=Lam>[[Lam Lay Yong]], "The Development of Hindu-Arabic and Traditional Chinese Arithmetic", ''Chinese Science'', 1996, p. 38, Kurt Vogel notation</ref> This form of fraction with numerator on top and denominator at bottom without a horizontal bar was also used by 10th century [[Abu'l-Hasan al-Uqlidisi]] and 15th century [[Jamshīd al-Kāshī]]'s work "Arithmetic Key".<ref name=Lam/><ref>{{cite journal | last1 = Lay Yong | first1 = Lam | author-link = Lam Lay Yong | title = A Chinese Genesis, Rewriting the history of our numeral system | journal = Archive for History of Exact Sciences | volume = 38 | pages = 101–108 }}</ref>
{| class="wikitable floatright"
! style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: right; " class="table-rh" | Number
|184.54290
|-
! style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: right; " class="table-rh" | Simon Stevin's notation
|184⓪5①4②2③9④0
|}
The adoption of the [[decimal representation]] of numbers less than one, a [[fraction (mathematics)|fraction]], is often credited to [[Simon Stevin]] through his textbook [[De Thiende]];<ref name=van>{{Cite book | author = B. L. van der Waerden | author-link = Bartel Leendert van der Waerden | year = 1985 | title = A History of Algebra. From Khwarizmi to Emmy Noether | url = https://archive.org/details/historyofalgebra0000waer| url-access = registration| publisher = Springer-Verlag | place = Berlin}}</ref> but both Stevin and [[E. J. Dijksterhuis]] indicate that [[Regiomontanus]] contributed to the European adoption of general [[decimal]]s:<ref name=EJD>[[E. J. Dijksterhuis]] (1970) ''Simon Stevin: Science in the Netherlands around 1600'', [[Martinus Nijhoff Publishers]], Dutch original 1943</ref>
: European mathematicians, when taking over from the Hindus, ''via'' the Arabs, the idea of positional value for integers, neglected to extend this idea to fractions. For some centuries they confined themselves to using common and [[sexagesimal]] fractions&nbsp;... This half-heartedness has never been completely overcome, and sexagesimal fractions still form the basis of our trigonometry, astronomy and measurement of time. ¶ ... Mathematicians sought to avoid fractions by taking the radius ''R'' equal to a number of units of length of the form 10<sup>''n''</sup> and then assuming for ''n'' so great an integral value that all occurring quantities could be expressed with sufficient accuracy by integers. ¶ The first to apply this method was the German astronomer Regiomontanus. To the extent that he expressed goniometrical line-segments in a unit ''R''/10<sup>''n''</sup>, Regiomontanus may be called an anticipator of the doctrine of decimal positional fractions.<ref name=EJD/>{{rp|17,18}}

In the estimation of Dijksterhuis, "after the publication of [[De Thiende]] only a small advance was required to establish the complete system of decimal positional fractions, and this step was taken promptly by a number of writers ... next to Stevin the most important figure in this development was Regiomontanus." Dijksterhuis noted that [Stevin] "gives full credit to Regiomontanus for his prior contribution, saying that the trigonometric tables of the German astronomer actually contain the whole theory of 'numbers of the tenth progress'."<ref name=EJD/>{{rp|19}}

== Mathematics ==
{{unreferenced section|date=March 2013}}

=== Base of the numeral system ===
In [[numeral system|mathematical numeral systems]] the [[radix]] {{mvar|r}} is usually the number of unique [[Numerical digit|digits]], including zero, that a positional numeral system uses to represent numbers. In some cases, such as with a [[negative base]], the radix is the [[absolute value]] <math>r=|b|</math> of the base {{mvar|b}}. For example, for the decimal system the radix (and base) is ten, because it uses the ten 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 two, 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 radix of that numeral system. The standard positional numeral systems differ from one another only in the base they use.

The radix is an integer that is greater than 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 more than <math>|b| </math> unique digits, numbers may have many different possible representations.

It is important that the radix is finite, from which follows that the number of digits is quite low. Otherwise, the length of a numeral would not necessarily be [[logarithm]]ic in its size.

(In certain [[non-standard positional numeral systems]], including [[bijective numeration]], the definition of the base or the allowed digits deviates from the above.)

In standard base-ten ([[decimal]]) positional notation, there are ten [[decimal digit]]s and the number
: <math>5305_{\mathrm{dec}} = (5 \times 10^3) + (3 \times 10^2) + (0 \times 10^1) + (5 \times 10^0)</math>.
In standard base-sixteen ([[hexadecimal]]), there are the sixteen hexadecimal digits (0–9 and A–F) and the number
: <math>14\mathrm{B}9_{\mathrm{hex}} = (1 \times 16^3) + (4 \times 16^2) + (\mathrm{B} \times 16^1) + (9 \times 16^0) \qquad (= 5305_{\mathrm{dec}}) ,</math>
where B represents the number eleven as a single symbol.

In general, in base-''b'', there are ''b'' digits <math>\{d_1,d_2,\dotsb,d_b\} =:D</math> and the number
:<math>(a_3 a_2 a_1 a_0)_b = (a_3 \times b^3) + (a_2 \times b^2) + (a_1 \times b^1) + (a_0 \times b^0) </math>
has <math>\forall k \colon a_k \in D .</math>
Note that <math>a_3 a_2 a_1 a_0</math> represents a sequence of digits, not [[multiplication]].

=== Notation ===
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 (this notation is used in this article). 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>.

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.

To a given radix ''b'' the set of digits {0, 1, ..., ''b''−2, ''b''−1} is called the standard set of digits. Thus, binary numbers have digits {0, 1}; decimal numbers have digits {{nowrap|{0, 1, 2, ..., 8, 9};}} and so on. Therefore, 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.)

=== Exponentiation ===
Positional numeral 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.

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:
: <math>4\times b^2 + 6\times b^1 + 5\times b^0</math>

If the number 465 was in base-10, then it would equal:
: <math>465_{10} = 4\times 10^2 + 6\times 10^1 + 5\times 10^0 = 4\times 100 + 6\times 10 + 5\times 1 = 465_{10}</math>

If however, the number were in base 7, then it would equal:
: <math>465_{7} = 4\times 7^2 + 6\times 7^1 + 5\times 7^0 = 4\times 49 + 6\times 7 + 5\times 1 = 243_{10}</math>

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.

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<sup>2</sup> (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<sup>2</sup> (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 a symbol that is used for positional notation, and a ''numeral'' consists of one or more digits used for representing a [[number]] with positional notation. Today's most common digits are the [[Arabic 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.

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> For example, the base-8 numeral 23<sub>8</sub> contains two digits, "2" and "3", and with a base number (subscripted) "8". When converted to base-10, the 23<sub>8</sub> is equivalent to 19<sub>10</sub>, i.e. 23<sub>8</sub> = 19<sub>10</sub>. 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, from base-4 up. In base-4, the "23" means 11<sub>10</sub>, i.e. 23<sub>4</sub> = 11<sub>10</sub>. In base-60, the "23" means the number 123<sub>10</sub>, i.e. 23<sub>60</sub> = 123<sub>10</sub>. The numeral "23" then, in this case, corresponds to the set of base-10 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" means "three of".

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>
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.) In general, the number of possible values that can be represented by a <math>d</math> digit number in base <math>r</math> is <math>r^d</math>.

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".

=== Radix point ===
{{main|Radix point}}
The notation can be extended into the negative exponents of the base ''b''. Thereby the so-called radix point, mostly ».«, is used as separator of the positions with non-negative from those with negative exponent.

Numbers that are not [[integer]]s use places beyond the [[radix point]]. For every position behind this point (and thus after the units digit), the exponent ''n'' of the power ''b''<sup>''n''</sup> decreases by 1 and the power approaches 0. For example, the number 2.35 is equal to:
:<math>2\times 10^0 + 3\times 10^{-1} + 5\times 10^{-2}</math>

=== Sign ===
{{main|Sign (mathematics)}}
If the base and all the digits in the set of digits are non-negative, negative numbers cannot be expressed. To overcome this, a [[Negative number|minus sign]], here −, is added to the numeral system. In the usual notation it is prepended to the string of digits representing the otherwise non-negative number.

=== Base conversion ===
<!-- This section is the target of a redirect -->
{{expand section|date=March 2017}}
The conversion to a base <math>b_2</math> of an integer {{math|''n''}} represented in base <math>b_1</math> can be done by a succession of [[Euclidean division]]s by <math>b_2:</math> the right-most digit in base <math>b_2</math> is the remainder of the division of {{math|''n''}} by <math>b_2;</math> the second right-most digit is the remainder of the division of the quotient by <math>b_2,</math> and so on. The left-most digit is the last quotient. In general, the {{math|''k''}}th digit from the right is the remainder of the division by <math>b_2</math> of the  {{math|(''k''−1)}}th quotient.

For example: converting A10B<sub>Hex</sub> to decimal (41227):
 0xA10B/10 = Q: 0x101A R: 7 (ones place)
 0x101A/10 = Q: 0x19C  R: 2 (tens place)
  0x19C/10 = Q: 0x29   R: 2 (hundreds place)
   0x29/10 = Q: 0x4    R: 1  ...
                          4

When converting to a larger base (such as from binary to decimal), the remainder represents <math>b_2</math> as a single digit, using digits from <math>b_1</math>. For example: converting 0b11111001 (binary) to 249 (decimal):
 0b11111001/10 = Q: 0b11000 R: 0b1001 (0b1001 = "9" for ones place)
    0b11000/10 = Q: 0b10    R: 0b100  (0b100 =  "4" for tens)
       0b10/10 = Q: 0b0     R: 0b10   (0b10 =   "2" for hundreds)

For the [[Fraction (mathematics)|fractional]] part, conversion can be done by taking digits after the radix point (the numerator), and [[Long division|dividing]] it by the [[Fraction (mathematics)#Decimal fractions and percentages|implied denominator]] in the target radix. Approximation may be needed due to a possibility of [[Repeating decimal#Extension to other bases|non-terminating digits]] if the [[Irreducible fraction|reduced]] fraction's denominator has a prime factor other than any of the base's prime factor(s) to convert to. For example, 0.1 in decimal (1/10) is 0b1/0b1010 in binary, by dividing this in that radix, the result is 0b0.0<span style="text-decoration: overline;">0011</span> (because one of the prime factors of 10 is 5). For more general fractions and bases see the [[Repeating decimal#Algorithm for positive bases|algorithm for positive bases]].

Alternatively, [[Horner's method]] can be used for base conversion using repeated multiplications, with the same computational complexity as repeated divisions.<ref>
{{cite book
 | last1 = Collins | first1 = G. E.
 | last2 = Mignotte | first2 = M.
 | last3 = Winkler | first3 = F.
 | editor1-last = Buchberger | editor1-first = Bruno
 | editor2-last = Collins | editor2-first = George Edwin
 | editor3-last = Loos | editor3-first = Rüdiger
 | editor4-last = Albrecht | editor4-first = Rudolf
 | contribution = Arithmetic in basic algebraic domains
 | contribution-url = https://www3.risc.jku.at/publications/download/risc_229/paper_55.pdf
 | doi = 10.1007/978-3-7091-7551-4_13
 | isbn = 3-211-81776-X
 | mr = 728973
 | pages = 189–220
 | publisher = Springer | location = Vienna
 | series = Computing Supplementa
 | title = Computer Algebra: Symbolic and Algebraic Computation
 | volume = 4
 | year = 1983
}}</ref>  A number in positional notation can be thought of as a polynomial, where each digit is a coefficient. Coefficients can be larger than one digit, so an efficient way to convert bases is to convert each digit, then evaluate the polynomial via Horner's method within the target base. Converting each digit is a simple [[lookup table]], removing the need for expensive division or modulus operations; and multiplication by x becomes right-shifting. However, other polynomial evaluation algorithms would work as well, like [[Exponentiation by squaring|repeated squaring]] for single or sparse digits. Example:
 Convert 0xA10B to 41227
  A10B = (10*16^3) + (1*16^2) + (0*16^1) + (11*16^0)

  Lookup table:
   0x0 = 0
   0x1 = 1
   ...
   0x9 = 9
   0xA = 10
   0xB = 11
   0xC = 12
   0xD = 13
   0xE = 14
   0xF = 15
  Therefore 0xA10B's decimal digits are 10, 1, 0, and 11.

  Lay out the digits out like this. The most significant digit (10) is "dropped":
   10 1   0    11 <- Digits of 0xA10B

   ---------------
   10
  Then we multiply the bottom number from the source base (16), the product is placed under the next digit of the source value, and then add:
   10 1   0    11
      160
   ---------------
   10 161

  Repeat until the final addition is performed:
   10 1   0    11
      160 2576 41216
   ---------------
   10 161 2576 41227

  and that is 41227 in decimal.

 Convert 0b11111001 to 249
  Lookup table:
   0b0 = 0
   0b1 = 1

 Result:
  1  1  1  1  1  0  0   1 <- Digits of 0b11111001
     2  6  14 30 62 124 248
  -------------------------
  1  3  7  15 31 62 124 249

=== Terminating fractions ===
The numbers which have a finite representation form the [[semiring]]
: <math>\frac{\N_0}{b^{\N_0}} := \left\{mb^{-\nu}\mid m\in \N_0 \wedge \nu\in \N_0 \right\} .</math>
More explicitly, if <math>p_1^{\nu_1} \cdot \ldots \cdot p_n^{\nu_n} := b</math> is a [[factorization]] of <math>b</math> into the primes <math>p_1, \ldots ,p_n \in \mathbb P</math> with exponents {{nowrap|<math>\nu_1, \ldots ,\nu_n \in \N</math>,<ref>The exact size of the <math>\nu_1, \ldots ,\nu_n</math> does not matter. They only have to be ≥ 1.</ref>}} then with the non-empty set of denominators <math> S := \{ p_1, \ldots, p_n \} </math>
we have
: <math> \Z_S := \left\{x \in \Q \left | \, \exists \mu_i \in \Z : x \prod_{i=1}^n {p_i}^{\mu_i} \in \Z \right . \right\} = b^{\Z} \, \Z = {\langle S\rangle}^{-1}\Z </math>
where <math>\langle S\rangle</math> is the group generated by the <math>p\in S</math> and <math> {\langle S\rangle}^{-1}\Z </math> is the so-called [[Localization (algebra)#Localization of a ring|localization]] of <math>\Z</math> with respect to {{nowrap|<math>S</math>.}}

The [[Fraction (mathematics)|denominator]] of an element of <math> \Z_S </math> contains if reduced to lowest terms only prime factors out of <math>S</math>.
This [[Ring (mathematics)|ring]] of all terminating fractions to base <math>b</math> is [[Dense set|dense]] in the field of [[rational number]]s <math>\Q</math>. Its [[Complete metric space|completion]] for the usual (Archimedean) metric is the same as for <math>\Q</math>, namely the real numbers <math>\R</math>. So, if <math> S = \{ p\} </math> then <math> \Z_{\{ p\}} </math> has not to be confused with <math>\Z_{(p)} </math>, the [[discrete valuation ring]] for the [[prime number|prime]] <math>p</math>, which is equal to <math>\Z_{T} </math> with <math> T = \mathbb P \setminus \{ p\} </math>.

If <math>b</math> divides <math>c</math>, we have <math> b^{\Z} \, \Z \subseteq c^{\Z} \, \Z.</math>

=== Infinite representations ===
==== Rational numbers ====
The representation of non-integers can be extended to allow an infinite string of digits beyond the point. For example, 1.12112111211112&nbsp;... base-3 represents the sum of the infinite [[series (mathematics)|series]]:
:<math>\begin{array}{l}
1\times 3^{0\,\,\,} + {}\\
1\times 3^{-1\,\,} + 2\times 3^{-2\,\,\,} + {}\\
1\times 3^{-3\,\,} + 1\times 3^{-4\,\,\,} + 2\times 3^{-5\,\,\,} + {}\\
1\times 3^{-6\,\,} + 1\times 3^{-7\,\,\,} + 1\times 3^{-8\,\,\,} + 2\times 3^{-9\,\,\,} + {}\\
1\times 3^{-10} + 1\times 3^{-11} + 1\times 3^{-12} + 1\times 3^{-13} + 2\times 3^{-14} + \cdots
\end{array}</math>

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:<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Vinculum |url=https://mathworld.wolfram.com/Vinculum.html |access-date=2024-08-22 |website=mathworld.wolfram.com |language=en}}</ref>
: <math>2.42\overline{314}_5 = 2.42314314314314314\dots_5</math>

This is the [[Repeating decimal#Notation|repeating decimal notation]] (to which there does not exist a single universally accepted notation or phrasing).
For base 10 it is called a repeating decimal or recurring decimal.

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:
: <math>0.1_3</math>
: <math>0.\overline3_{10} = 0.3333333\dots_{10}</math>
:: or, with the base implied:
:: <math>0.\overline3 = 0.3333333\dots</math> (see also [[0.999...]])
: <math>0.\overline{01}_2 = 0.010101\dots_2</math>
: <math>0.2_6</math>

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''.

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:
# A finite or infinite number of zeroes can be appended:
#: <math>3.46_7 = 3.460_7 = 3.460000_7 = 3.46\overline0_7</math>
# 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):
#: <math>3.46_7 = 3.45\overline6_7</math>
#: <math>1_{10} = 0.\overline9_{10}\qquad</math> (see also [[0.999...]])
#: <math>220_5 = 214.\overline4_5</math>

==== Irrational numbers ====
{{main|irrational number}}
A (real) irrational number has an infinite non-repeating representation in all integer bases.<ref>{{Cite web |date=2024-04-10 |title=Irrational Numbers: Definition, Examples and Properties |url=https://flamath.com/en/irrational-numbers#:~:text=The%20main%20characteristic%20of%20irrational%20numbers%20is%20that,number%20where%20a%20group%20of%20digits%20repeats%20constantly. |access-date=2024-08-22 |website=flamath.com |language=en-US}}</ref>

Examples are the non-solvable [[nth root|''n''th roots]]
: <math>y = \sqrt[n]{x} </math>
with <math>y^n = x</math> and {{math|''y'' ∉ '''Q'''}}, numbers which are called [[algebraic number|algebraic]], or numbers like
:<math>\pi,e</math>
which are [[transcendental number|transcendental]]. The number of transcendentals is [[uncountable]] and the sole way to write them down with a finite number of symbols is to give them a symbol or a finite sequence of symbols.

== Applications ==

=== Decimal system ===
{{Main|Decimal representation}}
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.

[[Decimal|Fraction]]al values are indicated by a [[Decimal separator|separator]], which can vary in different locations. 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.

As an example, the number 2674 in a base-10 numeral system is:
: (2 × 10<sup>3</sup>) + (6 × 10<sup>2</sup>) + (7 × 10<sup>1</sup>) + (4 × 10<sup>0</sup>)
or
: (2 × 1000) + (6 × 100) + (7 × 10) + (4 × 1).

=== Sexagesimal system ===
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.

Modern time separates each position by a colon or a [[Prime (symbol)|prime symbol]]. 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 {{nowrap|10°{{px2}}25′{{px2}}59″}} (10 [[degree (angle)|degree]]s 25 [[minute (angle)|minute]]s 59 [[second (angle)|second]]s). 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.{{citation needed|date=January 2019}} This contrasts with the numbers used by Hellenistic and [[Renaissance]] astronomers, who used [[third (angle)|third]]s, [[fourth (angle)|fourth]]s, etc. for finer increments. Where we might write {{nowrap|10°{{px2}}25′{{px2}}59.392″}}, they would have written {{nowrap|10°{{px2}}25′{{px2}}59′′{{px2}}23′′′{{px2}}31′′′′{{px2}}12′′′′′}} or {{nowrap|10°{{px2}}25<sup>{{smallcaps|i}}</sup>{{px2}}59<sup>{{smallcaps|ii}}</sup>{{px2}}23<sup>{{smallcaps|iii}}</sup>{{px2}}31<sup>{{smallcaps|iv}}</sup>{{px2}}12<sup>{{smallcaps|v}}</sup>}}.

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.<ref>{{Citation | last1 = Neugebauer | first1 = Otto | author-link = Otto Neugebauer | last2 = Sachs | first2 = Abraham Joseph | author2-link = Abraham Sachs | last3 = Götze | first3 = Albrecht | title = Mathematical Cuneiform Texts | place = New Haven | publisher = American Oriental Society and the American Schools of Oriental Research | series = American Oriental Series | volume = 29 | year = 1945 | page = 2 | isbn = 9780940490291 | url = https://books.google.com/books?id=i-juAAAAMAAJ&q=%22sexagesimal+notation%22&pg=PA1 | access-date = 18 September 2019 | archive-url = https://web.archive.org/web/20161001222131/https://books.google.com/books?hl=en&lr=&id=i-juAAAAMAAJ&oi=fnd&pg=PA1&dq=%22sexagesimal+notation%22&ots=nkmW8KTnQ_&sig=U_K4iOoKy5Xf70UrbjoTyS3hN2A#v=onepage&q=%22sexagesimal%20notation%22&f=false | archive-date = 1 October 2016 | url-status = live }}</ref> 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 numeral system|binary]] (base-2), octal (base-8) and [[hexadecimal]] (base-16) bases are most commonly 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.

Hexadecimal, decimal, octal, and a wide variety of other bases have been used for [[binary-to-text encoding]], implementations of [[arbitrary-precision arithmetic]], and other applications.

''For a list of bases and their applications, see [[list of numeral systems]].''

=== Other bases in human language ===
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]].

The [[Maya numerals|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]].

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. 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.

In English the same base-20 counting appears in the use of "[[20 (number)|scores]]".  Although mostly historical, it is occasionally used colloquially.  Verse 10 of Psalm 90 in the King James Version of the Bible starts: "The days of our years are threescore years and ten; and if by reason of strength they be fourscore years, yet is their strength labour and sorrow".  The Gettysburg Address starts: "Four score and seven years ago".

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''.

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.

The [[Inuit languages]] use a [[base-20]] counting system. Students from [[Kaktovik, Alaska]] invented a [[Kaktovik numerals|base-20 numeral system]] in 1994<ref name="kakt">{{cite journal |last=Bartley |first=Wm. Clark |date=January–February 1997 |title=Making the Old Way Count |url=http://www.ankn.uaf.edu/sop/SOPv2i1.pdf |journal=Sharing Our Pathways |volume=2 |issue=1 |pages=12–13 |access-date=27 February 2017 |archive-url=https://web.archive.org/web/20130625225547/http://ankn.uaf.edu/SOP/SOPv2i1.pdf |archive-date=25 June 2013 |url-status=live }}</ref>

[[Danish language#Numerals|Danish numerals]] display a similar [[vigesimal|base-20]] structure.

The [[Māori 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").

[[Binary numeral system|The binary system]] was used in the Egyptian Old Kingdom, 3000&nbsp;BC to 2050&nbsp;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#Mathematics|Eye of Horus]]).

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''.

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.

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.

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|url-access=registration|url=https://archive.org/details/piinskycounting00barr}}.</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>

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.

The [[Telefol language]], spoken in [[Papua New Guinea]], is notable for possessing a base-27 numeral system.

== Non-standard positional numeral systems ==
{{Main|Non-standard 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]]<ref>[[#Knuth|Knuth]], pages 195–213</ref> uses a base of 3 but the digit set is {{mset|{{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|&nbsp;&nbsp;}} 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.
: {{math|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.}}

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.

{|class="wikitable" style="text-align:center;" border="1"
!align="left" |Decimal equivalents
|width="6%" |−3
|width="6%" |−2
|width="6%" |−1
|width="6%" |0
|width="6%" |1
|width="6%" |2
|width="6%" |3
|width="6%" |4
|width="6%" |5
|width="8%" |6
|width="8%" |7
|width="8%" |8
|-
!align="left" |Balanced base 3
|{{overline|1}}0
|{{overline|1}}1
|{{overline|1}}
|0
|1
|1{{overline|1}}
|10
|11
|1{{overline|1}}{{overline|1}}
|1{{overline|1}}0
|1{{overline|1}}1
|10{{overline|1}}
|-
!align="left" |Base −2
|1101
|10
|11
|0
|1
|110
|111
|100
|101
|11010
|11011
|11000
|-
!align="left" |Factoroid
| || || ||0 ||10 ||100 ||110 ||200 ||210 ||1000 ||1010 ||1100
|}

== Non-positional positions ==
Each position does not need to be positional itself. [[Babylonian cuneiform numerals|Babylonian sexagesimal numerals]] were positional, but in each position were groups of two kinds of wedges representing ones and tens (a narrow vertical wedge | for the one and an open left pointing wedge ⟨ for the ten) — up to 5+9=14 symbols per position (i.e. 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>

== See also ==

Examples:
* [[List of numeral systems]]
* [[:Category: Positional numeral systems]]

Related topics:
* [[Algorism]]
* [[Hindu–Arabic numeral system]]
* [[Mixed radix]]
* [[Non-standard positional numeral systems]]
* [[Scientific notation]]

Other:
* [[Significant figures]]

== Notes ==
{{reflist}}

== References ==
* {{cite web
 |last1=O'Connor |first1=John
 |last2=Robertson |first2=Edmund
 |title=Babylonian Numerals
 |date=December 2000
 |url=http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_numerals.html
 |access-date=21 August 2010
 |archive-date=11 September 2014
 |archive-url=https://web.archive.org/web/20140911192557/http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_numerals.html
 |url-status=dead
}}
* {{cite journal
 |last=Kadvany |first=John
 |title=Positional Value and Linguistic Recursion
 |journal=Journal of Indian Philosophy
 |date=December 2007|volume=35
 |issue=5–6
 |pages=487–520
 |doi=10.1007/s10781-007-9025-5
 |s2cid=52885600
}}
* {{anchor|Knuth}}{{cite book
 |last=Knuth |first=Donald |author-link=Donald Knuth
 |title=The art of Computer Programming
 |volume=2
 |pages=195–213
 |publisher=Addison-Wesley
 |year=1997
 |isbn=0-201-89684-2
}}
* {{cite book
 |last=Ifrah |first=George
 |title=The Universal History of Numbers: From Prehistory to the Invention of the Computer
 |url=https://archive.org/details/unset0000unse_w3q2
 |url-access=registration
 |publisher=Wiley
 |year=2000
 |isbn=0-471-37568-3}}
* {{cite book
 |last=Kroeber |first=Alfred |author-link=Alfred Kroeber
 |title=Handbook of the Indians of California
 |publisher=Courier Dover Publications
 |year=1976
 |orig-year=1925
 |page=176
 |url=https://books.google.com/books?id=Plqf_OTz4ukC
 |isbn=9780486233680
}}

== External links ==

{{Commons category|Positional numeral systems}}
* [https://web.archive.org/web/20170204004954/http://ultrastudio.org/en/MechengburakalkanApplet-1.7.zip Accurate Base Conversion]
* [https://web.archive.org/web/20160310032143/http://ibrarian.net/navon/paper/the_development_of_hindu_arabic_and_traditional_c.pdf?paperid=1247217 The Development of Hindu Arabic and Traditional Chinese Arithmetics]
* [http://www.cut-the-knot.org/recurrence/conversion.shtml Implementation of Base Conversion] at [[cut-the-knot]]
* [http://www.intuitor.com/counting/ Learn to count other bases on your fingers]
* [https://web.archive.org/web/20161109022004/http://thedevtoolkit.com/tools/base_conversion Online Arbitrary Precision Base Converter]
{{Use dmy dates|date=August 2019}}

{{DEFAULTSORT:Positional Notation}}
[[Category:Positional numeral systems| ]]
[[Category:Mathematical notation]]
[[Category:Articles containing proofs]]