Numeral system

:Occasionally the term "number system" is used for this concept, but that is also the name of an altogether different concept; see number system.

History

: See also History of natural numbers and the status of zero.

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Tallies carved from wood and stone have been used since prehistoric times. Stone age cultures, including the American Indians, used tallies for gambling with horses, slaves, personal services and trade-goods.

Related Topics:
Tallies - American Indian

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The earliest known written tallies appear in the ruins of the Sumerian empire, using clay tablets impressed with a sharp stick and baked. The Sumerians had quite an exotic system based on counts to 60, used in astronomical and other calculations. This system was imported and used by every Mediterranean nation that used astronomy, including the Greeks, Romans and Egyptians. We still use it to count time (minutes per hour), and angle (degrees).

Related Topics:
Sumer - Mediterranean - Astronomy

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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.

Related Topics:
Prime number - Modular arithmetic - Digital signal processing

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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 system remained in common use in Europe until positional notation came into common use in the 1500s.

Related Topics:
Roman system - 1500s

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The Maya of Central America used a base 20/base 18 system, possibly inherited from the Olmec, including advanced features such as positional notation and a zero. They used this to do advanced astronomical calculations, including highly accurate calculations of the length of the solar year and the orbit of Venus.

Related Topics:
Maya - Olmec - Zero - Venus

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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.

Related Topics:
Quipu - Spanish - Conquistadors - 16th century - Andean

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Some authorities believe that positional arithmetic began with the wide use of the abacus in China. The earliest written positional records seem to be tallies of abacus results in China around 400. In particular, zero was correctly described by Chinese mathematicians around 932, and seems to have originated as a circle of a place empty of beads.

Related Topics:
Abacus - China - 400 - 932

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In India, recognizably modern positional numeral systems, passed to the Arabians, probably along with the astronomical tables, was brought to Baghdad by an Indian ambassador around 773 A.D.. For greater discussion of numeral systems from India, see Arabic numerals and Indian numerals.

Related Topics:
773 - Arabic numerals - Indian numerals

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From India, the thriving trade between Islamic Moguls and Africa carried the concept to Cairo. Arabic mathematicians extended the system to decimal fractions, and al-Khwarizmi wrote an important work about it in the 9th century. The system was introduced to Europe with the translation of this work in the 12th century in Spain and Leonardo of Pisas Liber Abaci of 1201. In Europe, the complete Indian system with the zero was derived from the Arabs in the 12th century.

Related Topics:
Al-Khwarizmi - 9th century - 12th century - Leonardo of Pisa - 1201

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The binary system (base 2), propagated in the 17th century by Gottfried Leibniz who had heard about it from China, came in common use in the 20th century because of computer applications.

Related Topics:
Binary system - 17th century - Gottfried Leibniz - 20th century

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