numeral

The Columbia Encyclopedia, 6th ed.

numeral

numeral, symbol denoting anumber. 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 numeration 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 BCthe ancient Egyptians began to use a demotic (a simplified cursive style of hieroglyphics) system of numerals based on a decimal system. 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 cuneiform 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 BC 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. BC 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. AD the Arabs learned Indian arithmetic from scientific writings of the Indians and the Greeks. In the 10th cent. AD Arab mathematicians extended the decimal numeral system to include fractions. Leonardo Fibonacci, 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 AD 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. BC), 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 AD 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).

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