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Numeral system

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Notation for expressing numbers
This article is about expressing numbers with symbols. For different kinds of numbers, seeNumber system. For expressing numbers with words, seeNumeral (linguistics).
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Numeral systems
List of numeral systems
Numbers written in different numeral systems

Anumeral system is a writing system for expressingnumbers; that is, amathematical notation for representing numbers of a given set, usingdigits or other symbols in a consistent manner.

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

The number the numeral represents is called itsvalue. Additionally, not all number systems can represent the same set of numbers; for example,Roman,Greek, andEgyptian numerals don't have a representation of the numberzero.

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 thealgebraic andarithmetic structure of the numbers.

For example, the usualdecimal representation gives every nonzeronatural number a unique representation as a finitesequence of 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 number systems, the system ofp-adic numbers, etc. Such systems are, however, not the topic of this article.

History

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See also:Numerical digit § History
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This sectionneeds expansion. You can help byadding to it.(July 2024)
Western Arabic0123456789
Eastern Arabic٠١٢٣٤٥٦٧٨٩
Persian۰۱۲۳۴۵۶۷۸۹
Devanagari०१२३४५६७८९

Early numeral systems varied across civilizations, with the Babylonians using a base-60 system, the Egyptians developing hieroglyphic numerals, and the Chinese employing rod numerals. The Mayans independently created a vigesimal (base-20) system that included a symbol forzero. Indian mathematicians, such as Brahmagupta in the 7th century, played a crucial role in formalizing arithmetic rules and the concept of zero, which was later refined by scholars like Al-Khwarizmi in the Islamic world. As these numeral systems evolved, the efficiency of positional notation and the inclusion of zero helped shape modern numerical representation, influencing global commerce, science, and technology. The first true writtenpositional numeral system is 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 digit zero had 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 theglyphs used to represent digits.[1]

The digits of the Maya numeral system

By the 13th century,Western Arabic numerals were accepted in European mathematical circles (Fibonacci used them in hisLiber Abaci). Initially met with resistance, Hindu–Arabic numerals gained wider acceptance in Europe due to their efficiency in arithmetic operations, particularly in banking and trade. The invention of the printing press in the 15th century helped standardize their use, as printed mathematical texts favored this system over Roman numerals. 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. By the 17th century, the system had become dominant in scientific works, influencing mathematical advancements by figures like Isaac Newton and René Descartes. In the 19th and 20th centuries, the widespread adoption of Arabic numerals facilitated global finance, engineering, and technological developments, forming the foundation for modern computing and digital data representation.

Other historical numeral systems using digits

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The exact age of theMaya numerals is 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. TheMayas had no equivalent of the moderndecimal separator, so their system could not represent fractions.[citation needed]

TheThai numeral system is identical to theHindu–Arabic numeral system except for the symbols used to represent digits. The use of these digits is less common inThailand than it once was, but they are still used alongside Arabic numerals.[4]

The rod numerals, the written forms ofcounting rods once used byChinese andJapanese mathematicians, are a decimal positional system used for performing decimal calculations. Rods were placed on a counting board and slid forwards or backwards to change the decimal place. TheSūnzĭ Suànjīng, a mathematical treatise dated to between the 3rd and 5th centuries AD, provides detailed instructions for the system, which is thought to have been in use since at least the 4th century BC.[5] Zero was not initially treated as a number, but as a vacant position.[6] Later sources introduced conventions for the expression of zero and negative numbers. The use of a round symbol〇 for zero is first attested in theMathematical Treatise in Nine Sections of 1247 AD.[7] The origin of this symbol is unknown; it may have been produced by modifying a square symbol.[8] TheSuzhou numerals, a descendant of rod numerals, are still used today for some commercial purposes.[citation needed]

Rod numerals (vertical)
0123456789
−0−1−2−3−4−5−6−7−8−9

Main numeral systems

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Main article:List of numeral systems

The most commonly used system of numerals isdecimal.Indian mathematicians are credited with developing the integer version, theHindu–Arabic numeral system.[9]Aryabhata ofKusumapura developed theplace-value notation in the 5th century and a century laterBrahmagupta introduced 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-Uqlidisi in 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 number is represented by a corresponding number of symbols. If the symbol/ is chosen, for example, then the number seven would be represented by///////.Tally marks represent 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 system was of this type, and theRoman numeral system was 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 alphabet 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 thebase of the system). This system is used when writingChinese numerals and 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 Address representing "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 theirbase orradix, which is the number of symbols calleddigits used 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 + 4×1 or 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).[10]

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 232 or 264 (grouping binary digits by 32 or 64, the length of themachine word) are used, as, for example, inGMP.

In certain biological systems, theunary coding system is employed. Unary numerals used in theneural circuits responsible forbirdsong production.[11] 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 thearithmetic numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and thegeometric numerals (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 usesboth arithmetic and geometric numerals.

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

Positional systems in detail

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See also:Positional notation

In a positional baseb numeral system (withb anatural number greater than 1 known as theradix orbase of the system),b basic symbols (or digits) corresponding to the firstb natural 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 thedecimal system (base 10), the numeral 4327 means(4×103) + (3×102) + (2×101) + (7×100), noting that100 = 1.

In general, ifb is the base, one writes a number in the numeral system of baseb by expressing it in the formanbn +an − 1bn − 1 +an − 2bn − 2 + ... +a0b0 and writing the enumerated digitsanan − 1an − 2 ...a0 in descending order. The digits are natural numbers between 0 andb − 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: numberbase. 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 denotes1×21 + 0×20 + 1×2−1 + 1×2−2 = 2.75.

In general, numbers in the baseb system are of the form:

(anan−1⋯a1a0.c1c2c3⋯)b=∑k=0nakbk+∑k=1∞ckb−k.{\displaystyle (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 numbersbk andb−k are theweights of the corresponding digits. The positionk is thelogarithm of the corresponding weightw, that isk=logb⁡w=logb⁡bk{\displaystyle k=\log _{b}w=\log _{b}b^{k}}. The highest used position is close to theorder of magnitude of the number.

The number oftally marks required in theunary numeral system fordescribing the weight would have beenw. In the positional system, the number of digits required to describe it is onlyk+1=logb⁡w+1{\displaystyle k+1=\log _{b}w+1}, fork ≥ 0. For example, to describe the weight 1000 then four digits are needed becauselog10⁡1000+1=3+1{\displaystyle \log _{10}1000+1=3+1}. The number of digits required todescribe the position islogb⁡k+1=logb⁡logb⁡w+1{\displaystyle \log _{b}k+1=\log _{b}\log _{b}w+1} (in positions 1, 10, 100,... only for simplicity in the decimal example).

Position3210−1−2⋯Weightb3b2b1b0b−1b−2⋯Digita3a2a1a0c1c2⋯Decimal example weight10001001010.10.01⋯Decimal example digit432700⋯{\displaystyle {\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 if it isrational; this does not depend on the base. A number that terminates in one base may repeat in another (thus0.310 = 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... = 1.27   or   321.3217878787878... = 321.32178.

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

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

Generalized variable-length integers

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Main article:Punycode

More general is using amixed radix notation (here writtenlittle-endian) likea0a1a2{\displaystyle a_{0}a_{1}a_{2}} fora0+a1b1+a2b1b2{\displaystyle 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 (t0,t1,…{\displaystyle t_{0},t_{1},\ldots }) which are fixed for every position in the number. A digitai{\displaystyle a_{i}} (in a given position in the number) that is lower than its corresponding threshold valueti{\displaystyle 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 weightb1 is 35 instead of 36. More generally, iftn is the threshold for then-th digit, it is easy to show thatbn+1=36−tn{\displaystyle 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 is36−t0=35{\displaystyle 36-t_{0}=35}. And the weight of the third symbol is35(36−t1)=35⋅34=1190{\displaystyle 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 9b where 9 andb each represent 35; yet the representation is unique becauseac andaca are not allowed – the firsta would 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.

See also

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References

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  1. ^abO'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".www.theclassroom.com. Retrieved2020-07-22.
  4. ^"Wissanu rejects dumping Thai numerals".Bangkok Post. 31 May 2022. Retrieved27 November 2024.
  5. ^O'Connor, John J.;Robertson, Edmund F. (January 2004),"Chinese numerals",MacTutor History of Mathematics Archive,University of St Andrews
  6. ^Shen Kanshen Crossley, John N.; Lun, Anthony W.-C. (1999).The Nine Chapters on the Mathematical Art: Companion and Commentary. Oxford University Press. p. 35.ISBN 978-0-19-853936-0.zero was regarded as a number in India ... whereas the Chinese employed a vacant position
  7. ^"Mathematics in the Near and Far East"(PDF).grmath4.phpnet.us. p. 262.Archived(PDF) from the original on 4 November 2013. Retrieved7 June 2012.
  8. ^Martzloff, Jean-Claude (2007).A History of Chinese Mathematics. Translated by Wilson, Stephen S. Springer. p. 208.ISBN 978-3-540-33783-6.
  9. ^David Eugene Smith; Louis Charles Karpinski (1911).The Hindu–Arabic numerals. Ginn and Company.
  10. ^Chowdhury, Arnab.Design of an Efficient Multiplier using DBNS. GIAP Journals.ISBN 978-93-83006-18-2.
  11. ^ 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

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External links

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Look upnumeration ornumeral in Wiktionary, the free dictionary.
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