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numeral,

symbol denoting anumbernumber,
entity describing the magnitude or position of a mathematical object or extensions of these concepts. The Natural Numbers

Cardinal numbers describe the size of a collection of objects; two such collections have the same (cardinal) number of objects if their
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. The symbol is a member of a family of marks, such as letters, figures, or words, which alone or in a group represent the members of a numerationnumeration,
in mathematics, process of designating numbers according to any particular system; the number designations are in turn called numerals. In any place value system of numeration, a base number must be specified, and groupings are then made by powers of the base number.
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 system. The earliest numerals were undoubtedly marks used to make a tally of a count of a number of acts or objects, one mark per object. This would be a unary system. About 3000 B.C.the ancient Egyptians began to use a demotic (a simplified cursive style of hieroglyphics) system of numerals based on a decimal systemdecimal system
[Lat.,=of tenths], numeration system based on powers of 10. A number is written as a row of digits, with each position in the row corresponding to a certain power of 10.
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. The Egyptians formed numerals by putting basic symbols together. This system did not include a symbol for zero nor did it use the principle of place value. About a thousand years later, the Babylonians devised a system of wedge-shaped cuneiformcuneiform
[Lat.,=wedge-shaped], system of writing developed before the last centuries of the 4th millennium B.C. in the lower Tigris and Euphrates valley, probably by the Sumerians (see Sumer).
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 symbols in conjunction with a numeration system based on a sexigesimal (base 60) numeration system. The majority of ancient peoples, however, including the Chinese, the Greeks, the Romans, and the Hebrews, used the decimal system.

The earliest numerical notation used by the Greeks was the Attic system. It employed the vertical stroke for 1, and symbols for 5, 10, 100, 1,000, and 10,000. About 500 B.C. the Greeks borrowed the Egyptian demotic numeral system and devised an alphabetic decimal system. This Ionic, or Ionian, system was a little more sophisticated than the Egyptian. It used a 27-letter Greek alphabet (the current 24-letter Greek alphabet plus three no longer used letters). Like the Egyptians, there was neither a provision for place value or a symbol for zero; the first 9 letters represented the numbers 1 through 9, the next 9 letters represented groups of ten from 10 through 90, and the last 9 represented groups of one hundred from 100 through 900.

About the same time, the Romans also developed an alphabetic numeral system. The Romans used letters of the alphabet to represent numbers, and this system is still used infrequently for such things as page numbers, clock faces, and dates of movies. The letters used in Roman numbers are: I (1), V (5), X (10), L (50), C (100), D (500), M (1,000). In general, letters are placed in decreasing order of value, for example, CXVI = 116. Letters can be repeated one or two times to increase value, for example, XX = 20 and XXX = 30. Letters cannot be repeated three times, so XXXX is not used for 40; insteadt XL (50 minus 10) is. Like in the Greek system, there was neither a provision for place value or a symbol for zero.

The Arabic numeral system (also called the Hindu numeral system or Hindu-Arabic numeral system) is considered one of the most significant developments in mathematics. It was developed in the 4th and 3d cent. B.C. Most historians agree that it was first conceived of in India (the Arabs themselves call the numerals they use "Indian numerals") and was then transmitted to the Islamic world and thence, via North Africa and Spain, to Europe. A place value decimal system, it used symbols for each number from one to nine. The Indians gradually developed a way of eliminating place names, and invented the symbol sunya [empty], which we call zero. During the 7th cent. A.D. the Arabs learned Indian arithmetic from scientific writings of the Indians and the Greeks. In the 10th cent. A.D. Arab mathematicians extended the decimal numeral system to include fractions. Leonardo FibonacciFibonacci, Leonardo
, b. c.1170, d. after 1240, Italian mathematician, known also as Leonardo da Pisa. In Liber abaci (1202, 2d ed. 1228), for centuries a standard work on algebra and arithmetic, he advocated the adoption of Arabic notation.
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, an Italian mathematician who had studied in Algeria, promoted the Arabic numeral system in his Liber Abaci (1202). The system did not come into wide use in Europe, however, until the invention of printing.

Other ancient peoples also had numeral systems. The earliest written positional records seem to be tallies of abacus results in China around A.D. 400, and zero was correctly described by Chinese mathematicians around 932. Use of numerical values is not found in the Hebrew scriptures but is thought to have originated under Greek influence. Sometime during the Maccabean period (2d cent. B.C.), the Hebrews transcribed the Ionic numeral system into their alphabet. The system of Hebrew numerals is a quasidecimal alphabetic numeral system using the letters of the Hebrew alphabet. There is no notation for zero or provision for place value—the letters are simply added up to determine the value. The system requires 27 letters, so the 22-letter Hebrew alphabet is sometimes extended by using five final forms of the Hebrew letters. The Maya of Central America used a vigesimal (base 20) system, possibly inherited from the Olmec. Their notation included advanced features such as positional notation and they had a symbol for zero before A.D. 300. The numerals are made up of three symbols: zero (egg shape), one (a dot), and five (a horizontal bar). For example, 14 is written as four dots in a horizontal row above two stacked horizontal bars, while 19 has a third stacked bar.

The decimal system is believed to have originated in counting on the fingers, using both hands as the most convenient method. Both the Arabic and the Roman symbols are believed to be related to this method: 1 or I is one finger, 2 or II is two fingers, and 3 or III is three fingers. The word "digit" is from the Latin digitus, meaning "finger." Some of the symbols are less easily explained, but V seems to be the open hand, and X seems to be two open hands.

Bibliography

See G. Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer (1999); D. E. Smith and L. C. Karpinski, The Hindu-Arabic Numerals (2004).

Numeral

 

a conventional symbol denoting a number.

The earliest and most primitive way of writing numbers is the use of words. In a few cases this method was retained for a considerable time; for example, some mathematicians in Middle Asia and the Middle East made systematic use of a verbal notation in the tenth century and even later. As peoples developed socially and economically, the use of number words proved to be an inefficient form of notation. Better methods of symbolizing numbers were needed, and principles for representing numbers had to be developed (seeNUMERATION SYSTEM). As Figure 1 shows, different peoples developed different sets of numerals.

The oldest known numerals are those of the Babylonians and Egyptians. Babylonian numerals were used from the second millennium B.C. to the beginning of the Common Era. The numbers one, ten, and, sometimes, 100 were represented by particular cuneiform characters; other natural numbers were expressed by combinations of these basic symbols. The Egyptian hieroglyphic system of numerals developed between 3000 B.C. and 2500 B.C. It had a separate symbol for each power of ten from 100 to 107. Later, in addition to the hieroglyphic script, the Egyptians used the more cursive hieratic notation, which had more symbols—for example, symbols for multiples of 10 up to 100. In about the eighth century B.C the hieratic forms of the numerals were largely supplanted by demotic forms.

Similar to the Egyptian hieroglyphic system of numeration are the Phoenician, Syrian, Palmyrene, and Greek Attic (or Herodianic) systems. Attic numerals came into use in the sixth century B.C. In Attica they were used until the first century A.D., but in other Greek lands they gave way to the more convenient Ionian numerals, which are alphabetic. In the Ionian system a letter was assigned to each of the first nine integers, the first nine integral multiples of ten, and the first nine integral multiples of 100; the remaining numbers up to 999 were represented by combinations of these letters. Such numerals were first used in the fifth century B.C.

Other peoples employing alphabetic systems of numerals included the Arabs, Syrians, Hebrews, Georgians, and Armenians. The Old Russian system of numeration, which arose in about the tenth century and was still encountered in the 16th century, was also alphabetic; it used Cyrillic and, in rare instances, Glagolitic letters (seeSLAVIC NUMERALS).

Of the ancient numeral systems, the Roman lasted the longest. It originated circa 500 B.C. among the Etruscans and in some cases is still used today (seeROMAN NUMERALS).

The forerunners of the modern numerals, including a zero, appeared in India, probably not later than the fifth century A.D. Until then, Kharoshthi numerals had been used in India, along with a system of numerals similar to the letters of the Brahmi alphabet. Numeral forms from inscriptions found in the caves at Nasik are shown in Figure 1. Hindu numerals spread from India to other countries because of the convenience of representing numbers with such numerals in a decimal positional numeration system. Between the tenth and 13th centuries Hindu numerals were brought to Europe by the Arabs—hence the modern name “arabic numerals.” They came into general use in the second half of the 15th century. The appearance of Hindu-Arabic numerals has undergone substantial changes with the passage of time (see Figure 2). The early history of these numerals remains poorly investigated.


Numeral

 

a nominal, the general lexical meaning of which is a quantity of persons or things. Grammatically, numerals are marked for the category of case (in languages with a developed morphology) and for gender (in languages with grammatical gender, some numerals possess marked forms; for example, Russian dva, “two” [masculine and neuter], and dve, “two” [feminine]) and are not marked for the category of number.

In Russian, numerals may be definite (dva, “two,” desiat’, “ten”) or indefinite (mnogo, “many,” malo, “few”), depending on the nature of the quantitative expression. Collective numerals designate quantity as a group (dvoe, “two,” troe, “three,” platero, “five,” oba, “both”); they constitute a special group. The structure of Russian numerals may be simple (dva, “two,” tri, “three,” odinnadtsat’, “eleven”), compound (piat’desiat, “fifty,” sem’desiat, “seventy”), or complex (tridtsat’ shest’, “thirty-six,” sto desiat’, “one hundred ten”). Many scholars consider ordinals and the word odin (“one”) as adjectives; such words have forms marked for both number and grammatical gender. The words desiatok (“a group of ten”), sotnia (“a group of one hundred”), and tysiacha (“[a group of] one thousand”) are nouns, inasmuch as they have all the formal characteristics of a noun. In the development of the Slavic languages, some numerals originated from other parts of speech; for example, Russian piat’ (“five”) was classified as a noun. Numerals should be distinguished from other words with quantitative meaning.

REFERENCES

Suprun, A. E. Slavianskie chislitel’nye. Minsk, 1969.
Vinogradov, V. V. Russkii iazyk, 2nd ed. Moscow, 1972.

V. A. VINOGRADOV

numeral

[′nüm·rəl]
(mathematics)
A symbol used to denote a number.

numeral

a symbol or group of symbols used to express a number: for example, 6 (Arabic), VI (Roman), 110 (binary)