Numeral system

Anumeral systemis a writing system for expressing numbers; that is, amathematical notationfor representing numbers of a given set, usingdigitsor other symbols in a consistent manner.

The same sequence of symbols may represent different numbers in different numeral systems. For example, "11" represents the numberelevenin thedecimal or base-10numeral system (today, the most common system globally), the numberthreein thebinary or base-2numeral system (used in modern computers), and the numbertwoin theunary numeral system(used intallyingscores).

The number the numeral represents is called its value. Not all number systems can represent the same set of numbers; for example,Roman numeralscannot represent the number zero.

Ideally, a numeral system will:

Represent a useful set of numbers (e.g. allintegers,orrational numbers)
Give every number represented a unique representation (or at least a standard representation)
Reflect thealgebraicandarithmeticstructure of the numbers.

For example, the usualdecimal representationgives every nonzeronatural numbera unique representation as a finitesequenceof digits, beginning with a non-zero digit.

Numeral systems are sometimes callednumber systems,but that name is ambiguous, as it could refer to different systems of numbers, such as the system ofreal numbers,the system ofcomplex numbers,varioushypercomplex numbersystems, the system ofp-adic numbers,etc. Such systems are, however, not the topic of this article.

History

Western Arabic	0	1	2	3	4	5	6	7	8	9
Eastern Arabic	٠	١	٢	٣	٤	٥	٦	٧	٨	٩
Persian	۰	۱	۲	۳	۴	۵	۶	۷	۸	۹
Devanagari	०	१	२	३	४	५	६	७	८	९

The first true writtenpositional numeral systemis considered to be theHindu–Arabic numeral system.This system was established by the 7th century in India,^[1]but was not yet in its modern form because the use of the digitzerohad not yet been widely accepted. Instead of a zero sometimes the digits were marked with dots to indicate their significance, or a space was used as a placeholder. The first widely acknowledged use of zero was in 876.^[2]The original numerals were very similar to the modern ones, even down to theglyphsused to represent digits.^[1]

By the 13th century,Western Arabic numeralswere accepted in European mathematical circles (Fibonacciused them in hisLiber Abaci). They began to enter common use in the 15th century.^[3]By the end of the 20th century virtually all non-computerized calculations in the world were done with Arabic numerals, which have replaced native numeral systems in most cultures.

Other historical numeral systems using digits

The exact age of theMaya numeralsis unclear, but it is possible that it is older than the Hindu–Arabic system. The system wasvigesimal(base 20), so it has twenty digits. The Mayas used a shell symbol to represent zero. Numerals were written vertically, with the ones place at the bottom. TheMayashad no equivalent of the moderndecimal separator,so their system could not represent fractions.

TheThai numeral systemis identical to theHindu–Arabic numeral systemexcept for the symbols used to represent digits. The use of these digits is less common inThailandthan it once was, but they are still used alongside Arabic numerals.

The rod numerals, the written forms ofcounting rodsonce used byChineseandJapanesemathematicians, are a decimal positional system able to represent not only zero but also negative numbers. Counting rods themselves predate the Hindu–Arabic numeral system. TheSuzhou numeralsare variants of rod numerals.

Rod numerals (vertical)
0	1	2	3	4	5	6	7	8	9

–0	–1	–2	–3	–4	–5	–6	–7	–8	–9

Main numeral systems

The most commonly used system of numerals isdecimal.Indian mathematiciansare credited with developing the integer version, theHindu–Arabic numeral system.^[4]AryabhataofKusumapuradeveloped theplace-value notationin the 5th century and a century laterBrahmaguptaintroduced the symbol for zero. The system slowly spread to other surrounding regions like Arabia due to their commercial and military activities with India. Middle-Eastern mathematicians extended the system to include negative powers of 10 (fractions), as recorded in a treatise by Syrian mathematicianAbu'l-Hasan al-Uqlidisiin 952–953, and the decimal point notation was introduced^[when?]bySind ibn Ali,who also wrote the earliest treatise on Arabic numerals. The Hindu–Arabic numeral system then spread to Europe due to merchants trading, and the digits used in Europe are calledArabic numerals,as they learned them from the Arabs.

The simplest numeral system is theunary numeral system,in which everynatural numberis represented by a corresponding number of symbols. If the symbol/is chosen, for example, then the number seven would be represented by///////.Tally marksrepresent one such system still in common use. The unary system is only useful for small numbers, although it plays an important role intheoretical computer science.Elias gamma coding,which is commonly used indata compression,expresses arbitrary-sized numbers by using unary to indicate the length of a binary numeral.

The unary notation can be abbreviated by introducing different symbols for certain new values. Very commonly, these values are powers of 10; so for instance, if / stands for one, − for ten and + for 100, then the number 304 can be compactly represented as+++ ////and the number 123 as+ − − ///without any need for zero. This is calledsign-value notation.The ancientEgyptian numeral systemwas of this type, and theRoman numeral systemwas a modification of this idea.

More useful still are systems which employ special abbreviations for repetitions of symbols; for example, using the first nine letters of the Alpha bet for these abbreviations, with A standing for "one occurrence", B "two occurrences", and so on, one could then write C+ D/ for the number 304 (the number of these abbreviations is sometimes called thebaseof the system). This system is used when writingChinese numeralsand other East Asian numerals based on Chinese. The number system of the English language is of this type ( "three hundred [and] four" ), as are those of other spoken languages, regardless of what written systems they have adopted. However, many languages use mixtures of bases, and other features, for instance 79 in French issoixante dix-neuf(60 + 10 + 9) and in Welsh ispedwar ar bymtheg a thrigain(4 + (5 + 10) + (3 × 20)) or (somewhat archaic)pedwar ugain namyn un(4 × 20 − 1). In English, one could say "four score less one", as in the famousGettysburg Addressrepresenting "87 years ago" as "four score and seven years ago".

More elegant is apositional system,also known as place-value notation. The positional systems are classified by theirbaseorradix,which is the number of symbols calleddigitsused by the system. In base 10, ten different digits 0,..., 9 are used and the position of a digit is used to signify the power of ten that the digit is to be multiplied with, as in304 = 3×100 + 0×10 + 4×1or more precisely3×10²+ 0×10¹+ 4×10⁰.Zero, which is not needed in the other systems, is of crucial importance here, in order to be able to "skip" a power. The Hindu–Arabic numeral system, which originated in India and is now used throughout the world, is a positional base 10 system.

Arithmetic is much easier in positional systems than in the earlier additive ones; furthermore, additive systems need a large number of different symbols for the different powers of 10; a positional system needs only ten different symbols (assuming that it uses base 10).^[5]

The positional decimal system is presently universally used in human writing. The base 1000 is also used (albeit not universally), by grouping the digits and considering a sequence of three decimal digits as a single digit. This is the meaning of the common notation 1,000,234,567 used for very large numbers.

In computers, the main numeral systems are based on the positional system in base 2 (binary numeral system), with twobinary digits,0 and 1. Positional systems obtained by grouping binary digits by three (octal numeral system) or four (hexadecimal numeral system) are commonly used. For very large integers, bases 2³²or 2⁶⁴(grouping binary digits by 32 or 64, the length of themachine word) are used, as, for example, inGMP.

In certain biological systems, theunary codingsystem is employed. Unary numerals used in theneural circuitsresponsible forbirdsongproduction.^[6]The nucleus in the brain of the songbirds that plays a part in both the learning and the production of bird song is the HVC (high vocal center). The command signals for different notes in the birdsong emanate from different points in the HVC. This coding works as space coding which is an efficient strategy for biological circuits due to its inherent simplicity and robustness.

The numerals used when writing numbers with digits or symbols can be divided into two types that might be called thearithmeticnumerals (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and thegeometricnumerals (1, 10, 100, 1000, 10000...), respectively. The sign-value systems use only the geometric numerals and the positional systems use only the arithmetic numerals. A sign-value system does not need arithmetic numerals because they are made by repetition (except for theIonic system), and a positional system does not need geometric numerals because they are made by position. However, the spoken language usesbotharithmetic and geometric numerals.

In some areas of computer science, a modified basekpositional system is used, calledbijective numeration,with digits 1, 2,...,k(k≥ 1), and zero being represented by an empty string. This establishes abijectionbetween the set of all such digit-strings and the set of non-negative integers, avoiding the non-uniqueness caused by leading zeros. Bijective base-knumeration is also calledk-adic notation, not to be confused withp-adic numbers.Bijective base 1 is the same as unary.

Positional systems in detail

In a positional basebnumeral system (withbanatural numbergreater than 1 known as theradixorbaseof the system),bbasic symbols (or digits) corresponding to the firstbnatural numbers including zero are used. To generate the rest of the numerals, the position of the symbol in the figure is used. The symbol in the last position has its own value, and as it moves to the left its value is multiplied byb.

For example, in thedecimalsystem (base 10), the numeral 4327 means $(4 \times10 3) + (3 \times10 2) + (2 \times10 1) + (7 \times10 0)$ ,noting that $100 = 1$ .

In general, ifbis the base, one writes a number in the numeral system of basebby expressing it in the form $a n b n + a n - 1 b n - 1 + a n - 2 b n - 2 +... + a 0 b 0$ and writing the enumerated digits $a n a n - 1 a n - 2 ... a 0$ in descending order. The digits are natural numbers between 0 and $b - 1$ ,inclusive.

If a text (such as this one) discusses multiple bases, and if ambiguity exists, the base (itself represented in base 10) is added in subscript to the right of the number, like this: number_base.Unless specified by context, numbers without subscript are considered to be decimal.

By using a dot to divide the digits into two groups, one can also write fractions in the positional system. For example, the base 2 numeral 10.11 denotes $1\times2 1 + 0\times2 0 + 1\times2 -1 + 1\times2 -2 = 2.75$ .

In general, numbers in the basebsystem are of the form:

(a_{n}a_{n-1}\cdots a_{1}a_{0}.c_{1}c_{2}c_{3}\cdots )_{b}=\sum _{k=0}^{n}a_{k}b^{k}+\sum _{k=1}^{\infty }c_{k}b^{-k}.

The numbersb^kandb^−kare theweightsof the corresponding digits. The positionkis thelogarithmof the corresponding weightw,that is $k=\log _{b}w=\log _{b}b^{k}$ .The highest used position is close to theorder of magnitudeof the number.

The number oftally marksrequired in theunary numeral systemfordescribing the weightwould have beenw.In the positional system, the number of digits required to describe it is only $k+1=\log _{b}w+1$ ,fork≥ 0. For example, to describe the weight 1000 then four digits are needed because $\log _{10}1000+1=3+1$ .The number of digits required todescribe the positionis $\log _{b}k+1=\log _{b}\log _{b}w+1$ (in positions 1, 10, 100,... only for simplicity in the decimal example).

{\begin{array}{l|rrrrrrr}{\text{Position}}&3&2&1&0&-1&-2&\cdots \\\hline {\text{Weight}}&b^{3}&b^{2}&b^{1}&b^{0}&b^{-1}&b^{-2}&\cdots \\{\text{Digit}}&a_{3}&a_{2}&a_{1}&a_{0}&c_{1}&c_{2}&\cdots \\\hline {\text{Decimal example weight}}&1000&100&10&1&0.1&0.01&\cdots \\{\text{Decimal example digit}}&4&3&2&7&0&0&\cdots \end{array}}

A number has a terminating or repeating expansionif and only ifit isrational;this does not depend on the base. A number that terminates in one base may repeat in another (thus $0.3 10 = 0.0100110011001... 2$ ). An irrational number stays aperiodic (with an infinite number of non-repeating digits) in all integral bases. Thus, for example in base 2, $π = 3.1415926... 10$ can be written as the aperiodic 11.001001000011111...₂.

Puttingoverscores,n,or dots,ṅ,above the common digits is a convention used to represent repeating rational expansions. Thus:

14/11 = 1.272727272727... = 1.27or 321.3217878787878... = 321.32178.

Ifb=pis aprime number,one can define base-pnumerals whose expansion to the left never stops; these are called thep-adic numbers.

It is also possible to define a variation of basebin which digits may be positive or negative; this is called asigned-digit representation.

Generalized variable-length integers

More general is using amixed radixnotation (here writtenlittle-endian) like $a_{0}a_{1}a_{2}$ for $a_{0}+a_{1}b_{1}+a_{2}b_{1}b_{2}$ ,etc.

This is used inPunycode,one aspect of which is the representation of a sequence of non-negative integers of arbitrary size in the form of a sequence without delimiters, of "digits" from a collection of 36: a–z and 0–9, representing 0–25 and 26–35 respectively. There are also so-called threshold values ( $t_{0},t_{1},\ldots$ ) which are fixed for every position in the number. A digit $a_{i}$ (in a given position in the number) that is lower than its corresponding threshold value $t_{i}$ means that it is the most-significant digit, hence in the string this is the end of the number, and the next symbol (if present) is the least-significant digit of the next number.

For example, if the threshold value for the first digit isb(i.e. 1) thena(i.e. 0) marks the end of the number (it has just one digit), so in numbers of more than one digit, first-digit range is only b–9 (i.e. 1–35), therefore the weightb₁is 35 instead of 36. More generally, ift_nis the threshold for then-th digit, it is easy to show that $b_{n+1}=36-t_{n}$ . Suppose the threshold values for the second and third digits arec(i.e. 2), then the second-digit range is a–b (i.e. 0–1) with the second digit being most significant, while the range is c–9 (i.e. 2–35) in the presence of a third digit. Generally, for anyn,the weight of the (n+ 1)-th digit is the weight of the previous one times (36 − threshold of then-th digit). So the weight of the second symbol is $36-t_{0}=35$ .And the weight of the third symbol is $35(36-t_{1})=35\cdot 34=1190$ .

So we have the following sequence of the numbers with at most 3 digits:

a(0),ba(1),ca(2),..., 9a(35),bb(36),cb(37),..., 9b(70),bca(71),..., 99a(1260),bcb(1261),..., 99b(2450).

Unlike a regularn-based numeral system, there are numbers like 9bwhere 9 andbeach represent 35; yet the representation is unique becauseacandacaare not allowed – the firstawould terminate each of these numbers.

The flexibility in choosing threshold values allows optimization for number of digits depending on the frequency of occurrence of numbers of various sizes.

The case with all threshold values equal to 1 corresponds tobijective numeration,where the zeros correspond to separators of numbers with digits which are non-zero.

References

^^a ^bO'Connor, J. J. and Robertson, E. F.Arabic Numerals.January 2001. Retrieved on 2007-02-20.
^Bill Casselman(February 2007)."All for Nought".Feature Column.AMS.
^Bradley, Jeremy."How Arabic Numbers Were Invented".theclassroom.Retrieved2020-07-22.
^David Eugene Smith; Louis Charles Karpinski (1911).The Hindu–Arabic numerals.Ginn and Company.
^Chowdhury, Arnab.Design of an Efficient Multiplier using DBNS.GIAP Journals.ISBN 978-93-83006-18-2.
^Fiete, I. R.; Seung, H. S. (2007). "Neural network models of birdsong production, learning, and coding". In Squire, L.; Albright, T.; Bloom, F.; Gage, F.; Spitzer, N. New Encyclopedia of Neuroscience.

Sources

Georges Ifrah.The Universal History of Numbers: From Prehistory to the Invention of the Computer,Wiley, 1999.ISBN 0-471-37568-3.
D. Knuth.The Art of Computer Programming.Volume 2, 3rd Ed.Addison–Wesley.pp. 194–213, "Positional Number Systems".
A.L. Kroeber(Alfred Louis Kroeber) (1876–1960), Handbook of the Indians of California, Bulletin 78 of the Bureau of American Ethnology of the Smithsonian Institution (1919)
J.P. Mallory; D.Q. Adams,Encyclopedia of Indo-European Culture,Fitzroy Dearborn Publishers, London and Chicago, 1997.
Hans J. Nissen; Peter Damerow; Robert K. Englund (1993).Archaic Bookkeeping: Early Writing and Techniques of Economic Administration in the Ancient Near East.University of Chicago Press.ISBN 978-0-226-58659-5.
Schmandt-Besserat, Denise(1996).How Writing Came About.University of Texas Press.ISBN 978-0-292-77704-0.
Zaslavsky, Claudia(1999).Africa counts: number and pattern in African cultures.Chicago Review Press.ISBN 978-1-55652-350-2.

External links

Media related toNumeral systemsat Wikimedia Commons

[O'Connor_and_Robertson-1] O'Connor, J. J. and Robertson, E. F.Arabic Numerals.January 2001. Retrieved on 2007-02-20.

[2] Bill Casselman(February 2007)."All for Nought".Feature Column.AMS.

[3] Bradley, Jeremy."How Arabic Numbers Were Invented".theclassroom.Retrieved2020-07-22.

[4] David Eugene Smith; Louis Charles Karpinski (1911).The Hindu–Arabic numerals.Ginn and Company.

[5] Chowdhury, Arnab.Design of an Efficient Multiplier using DBNS.GIAP Journals.ISBN 978-93-83006-18-2.

[6] Fiete, I. R.; Seung, H. S. (2007). "Neural network models of birdsong production, learning, and coding". In Squire, L.; Albright, T.; Bloom, F.; Gage, F.; Spitzer, N. New Encyclopedia of Neuroscience.

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