Number names

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In linguistics, a number name, or numeral, is a symbol or group of symbols, or a word in a natural language that represents a number. Numerals differ from numbers just as words differ from the things they refer to. The symbols "11", "eleven" and "XI" are different numerals, all representing the same number. This article attempts to explain the various systems of numerals.

Contents 1 History 2 Counting base 3 Systems of numerals 4 See also 5 References

[edit] History

See also: Natural number

The numbers one through ten in different numeral systems

Indian	١	٢	٣	٤	٥	٦	٧	٨	٩	١٠
Devanagari	१	२	३	४	५	६	७	८	९	१०
Hebrew	א	ב	ג	ד	ה	ו	ז	ח	ט	י
Arabic	1	2	3	4	5	6	7	8	9	10
Malayalam	൧	൨	൩	൪	൫	൬	൭	൮	൯	൧൦
Chinese	一	二	三	四	五	六	七	八	九	十
Suzhou	〡	〢	〣	〤	〥	〦	〧	〨	〩	〡〇
Roman	I	II	III	IV	V	VI	VII	VIII	IX	X
Thai	๑	๒	๓	๔	๕	๖	๗	๘	๙	๑๐

Counting aids, especially the use of body parts (counting on fingers), were certainly used in prehistoric times as today. There are many variations. Besides counting 10 fingers, some cultures have counted knuckles, the space between fingers, and toes as well as fingers. The Oksapmin culture of New Guinea uses a system of 27 upper body locations to represent numbers.

To preserve numerical information, Tallies carved in wood, bone, and stone have been used since prehistoric times. Stone age cultures, including ancient American Indian groups, used tallies for gambling, personal services, and trade-goods.

A method of preserving numeric information in clay was invented by the Sumerians between 8000 and 3500 BC. This was done with small clay tokens of various shapes that were strung like beads on a string. Beginning about 3500 BC clay tokens were gradually replaced by number signs impressed with a round stylus at different angles in clay tablets (originally containers for tokens) which were then baked. About 3100 BC written numbers were dissociated from the things being counted and became abstract numerals.

Between 2700 BC and 2000 BC in Sumer, the round stylus was gradually replaced by a reed stylus that was used to press wedge-shaped cuneiform signs in clay. These cuneiform number signs resembled the round number signs they replaced and retained the additive sign-value notation of the round number signs. These systems gradually converged on a common sexagesimal number system; this was a place-value system consisting of only two impressed marks, the vertical wedge and the chevron, which could also represent fractions. This sexagesimal number system was fully developed at the beginning of the Old Babylonia period (about 1950 BC) and became standard in Babylonia.

Sexagesimal numerals were a mixed radix system that retained the alternating base 10 and base 6 in a sequence of cuneiform vertical wedges and chevrons. By 1950 BC this was a positional notation system. Sexagesimal numerals came to be widely used in commerce, but were also used in astronomical and other calculations. This system was exported from Babylonia and used throughout Mesopotamia, and by every Mediterranean nation that used standard Babylonian units of measure and counting, including the Greeks, Romans and Egyptians. Babylonian-style sexagesimal numeration is still used in modern societies to measure time (minutes per hour) and angles (degrees).

In China, armies and provisions were counted using modular tallies of prime numbers. Unique numbers of troops and measures of rice appear as unique combinations of these tallies. A great convenience of modular arithmetic is that it is easy to multiply, though quite difficult to add. This makes use of modular arithmetic for provisions especially attractive. Conventional tallies are quite difficult to multiply and divide. In modern times modular arithmetic is sometimes used in Digital signal processing.

Numeral systems by culture
Hindu-Arabic numerals
Indian Eastern Arabic Khmer	Indian family Brahmi Thai
East Asian numerals
Chinese Suzhou Counting rods	Japanese Korean
Alphabetic numerals
Abjad Armenian Cyrillic Ge'ez	Hebrew Greek (Ionian) Āryabhaṭa
Other systems
Attic Babylonian Egyptian Etruscan	Mayan Roman Urnfield
List of numeral system topics
Positional systems by base
Decimal (10)
2, 4, 8, 16, 32, 64
1, 3, 9, 12, 20, 24, 30, 36, 60, more…
v • d • e

The oldest Greek system was the that of the Attic numerals, but in the 4th century BC they began to use a quasidecimal alphabetic system (see Greek numerals). Jews began using a similar system (Hebrew numerals), with the oldest examples known being coins from around 100 BC.

The Roman empire used tallies written on wax, papyrus and stone, and roughly followed the Greek custom of assigning letters to various numbers. The Roman numerals system remained in common use in Europe until positional notation came into common use in the 1500s.

The Maya of Central America used a mixed base 18^{[citation needed]} and base 20 system, possibly inherited from the Olmec, including advanced features such as positional notation and a zero. They used this system to make advanced astronomical calculations, including highly accurate calculations of the length of the solar year and the orbit of Venus.

The Incan Empire ran a large command economy using quipu, tallies made by knotting colored fibers. Knowledge of the encodings of the knots and colors was suppressed by the Spanish conquistadors in the 16th century, and has not survived although simple quipu-like recording devices are still used in the Andean region.

Some authorities believe that positional arithmetic began with the wide use of counting rods in China. The earliest written positional records seem to be rod calculus results in China around 400. In particular, zero was correctly described by Chinese mathematicians around 932.

The modern positional Arabic numeral system was developed by mathematicians in India, and passed on to Muslim mathematicians, along with astronomical tables brought to Baghdad by an Indian ambassador around 773.

From India, the thriving trade between Islamic sultans and Africa carried the concept to Cairo. Arabic mathematicians extended the system to include decimal fractions, and Muḥammad ibn Mūsā al-Ḵwārizmī wrote an important work about it in the 9th century. The modern Arabic numerals were introduced to Europe with the translation of this work in the 12th century in Spain and Leonardo of Pisa's Liber Abaci of 1201. In Europe, the complete Indian system with the zero was derived from the Arabs in the 12th century.

The binary system (base 2), was propagated in the 17th century by Gottfried Leibniz. Leibniz had developed the concept early in his career, and had revisited it when he reviewed a copy of the I ching from China. Binary numbers came into common use in the 20th century because of computer applications.

[edit] Counting base

Although a majority of traditional number systems are based on the decimal numeral system, there are many regional variations even within decimal, including:

• Western system: based on thousands, with variants (see English-language numerals)
• Indian system: crore, lakh (see Indian numbering system. Indian numerals)
• East Asian system: based on ten-thousands (see below)

Duodecimal numbers have only been used consistently in a few cases. Among these, the Chepang language of Nepal, the Mahl language of Minicoy Island in India, and several languages of the Nigerian Middle Belt, such as Janji, Kahugu and the Nimbia dialect of Gwandara. However, duodecimal subdivisions offer practical advantages over decimal because of the better divisibility of twelve (which is a highly composite number), and as such they have been used extensively in many other cultures as well; for example, in time divisions (twelve months in a year, the twelve-hour clock), in the imperial system of units (twelve inches to the foot, twelve Troy ounces to the Troy pound), or in the former British monetary system (twelve pence to the shilling). As a result, languages such as English eventually borrowed or evolved terms such dozen, gross and great gross, which allow for a rudimentary duodecimal nomenclature (e.g., saying "two gross and six dozen" instead of "three hundred and sixty"). Ancient Romans used decimal for integers, but switched to duodecimal for fractions, and correspondingly Latin developed a rich vocabulary for duodecimal-based fractions (see Roman numerals). In fiction, J. R. R. Tolkien's Elvish languages used duodecimal along with decimal.

Vigesimal numbers were the standard among ancient Mesoamerican cultures, and are still in use in the modern indigenous languages of their present-day descendants, such as the Nahuatl and Mayan languages (see also Maya numerals). Vigesimal terminology is also found to some extent in some European languages (Basque, Celtic languages, French, Danish, Georgian). English has a remnant of vigesimal numeration in the word score (famously used in the opening of the Gettysburg Address).

Quinary is found in Inuit languages.

Octal is used in the Yuki language of California and in the Pamean languages of Mexico, because their speakers count using the spaces between their fingers rather than the fingers themselves.^[1]

For very large (and very small) numbers, traditional systems have been superseded by the use of scientific notation and the system of SI prefixes. Traditional systems continue to be used in everyday life.

[edit] Systems of numerals

• Arabic numeral system
• Armenian numerals
• Babylonian numerals
• Chinese numerals
• English-language numerals
• Greek numerals
• Hebrew numerals
• Indian numerals
• Japanese numerals
• Korean numerals
• Mayan numerals
• Quipu
• Rod numerals
• Roman numerals

[edit] See also

• Abacus
• History of large numbers
• List of numbers in various languages
• List of numeral system topics
• Long and short scales
• Myriad
• Names of large numbers
• Numeral system

[edit] References