Numeral Systems
Various notational systems (number systems) have been or are being used in mathematics to represent the abstract quantities called numbers.
A number system is defined by the base it uses, the base being the number of different symbols required by the system to represent any of the infinite series of numbers. Thus, the decimal system in universal use today (except for computer application) requires ten different symbols, or digits, to represent numbers and is therefore a base-10 system.
Throughout history, many different number systems have been used; in fact, any whole number greater than 1 can be used as a base. Some cultures have used systems based on the numbers 3, 4, or 5.
A number system is defined by the base it uses, the base being the number of different symbols required by the system to represent any of the infinite series of numbers. Thus, the decimal system in universal use today (except for computer application) requires ten different symbols, or digits, to represent numbers and is therefore a base-10 system.
Throughout history, many different number systems have been used; in fact, any whole number greater than 1 can be used as a base. Some cultures have used systems based on the numbers 3, 4, or 5.
The Babylonians used the sexagesimal system, based on the number 60, and the Romans used (for some purposes) the duodecimal system, based on the number 12.
The Mayas used the vigesimal system, based on the number 20.
The binary system, based on the number 2, was used by some tribes and, together with the system based on 8, is used today in computer systems.
The earliest forms of number notation were simply groups of straight lines, either vertical or horizontal, each line corresponding to the number. Such a system is inconvenient when dealing with large numbers, and as early as 3400 BC in Egypt and 3000 BC in Mesopotamia a special symbol was adopted for the number 10.
The addition of this second number symbol made it possible to express the number 11 with 2 instead of 11 individual symbols and the number 99 with 18 instead of 99 individual symbols. Later numeral systems introduced extra symbols for a number between 1 and 10, usually either 4 or 5, and additional symbols for numbers greater than 10.
In Babylonian cuneiform notation the numeral used for 1 was also used for 60 and for powers of 60; the value of the numeral was indicated by its context. The Egyptian hieroglyphic system used special symbols for 10, 100, 1000, and 10,000.
The ancient Greeks had two parallel systems of numerals. The earlier of these was based on the initial letters of the names of numbers: The number 5 was indicated by the letter Pi; 10 by the letter Delta; 100 by the antique form of the letter H; 1000 by the letter Chi; and 10,000 by the letter Mu.
The later system, which was first introduced about the 3rd century BC, employed all the letters of the Greek alphabet plus three letters borrowed from the Phoenician alphabet as number symbols. The first nine letters of the alphabet were used for the numbers 1 to 9, the second nine letters for the tens from 10 to 90, and the last nine letters for the hundreds from 100 to 900. Thousands were indicated by placing a bar to the left of the appropriate numeral, and tens of thousands by placing the appropriate letter over the letter M.
The late Greek system had the advantage that large numbers could be expressed with a minimum of symbols, but it had the disadvantage of requiring the user to memorize a total of 27 symbols.
The decimal system is believed to have originated in counting on the fingers, using both hands as the most convenient method. Arabic numerals are 1, 2, 3, 4, etc.
Roman numerals are I, II, III, IV, etc. 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. The Roman system has no symbol for zero, and in the Arabic system zero is much more recent than the other symbols. The Maya, who were skilled in mathematics, had a symbol for zero.
Some languages show traces of reckoning by units other than ten, e.g., 8, 12, 20. In the case of 20, the toes as well as the fingers may have been used in counting.
The Mayas used the vigesimal system, based on the number 20.
The binary system, based on the number 2, was used by some tribes and, together with the system based on 8, is used today in computer systems.
The earliest forms of number notation were simply groups of straight lines, either vertical or horizontal, each line corresponding to the number. Such a system is inconvenient when dealing with large numbers, and as early as 3400 BC in Egypt and 3000 BC in Mesopotamia a special symbol was adopted for the number 10.
The addition of this second number symbol made it possible to express the number 11 with 2 instead of 11 individual symbols and the number 99 with 18 instead of 99 individual symbols. Later numeral systems introduced extra symbols for a number between 1 and 10, usually either 4 or 5, and additional symbols for numbers greater than 10.
In Babylonian cuneiform notation the numeral used for 1 was also used for 60 and for powers of 60; the value of the numeral was indicated by its context. The Egyptian hieroglyphic system used special symbols for 10, 100, 1000, and 10,000.
The ancient Greeks had two parallel systems of numerals. The earlier of these was based on the initial letters of the names of numbers: The number 5 was indicated by the letter Pi; 10 by the letter Delta; 100 by the antique form of the letter H; 1000 by the letter Chi; and 10,000 by the letter Mu.
The later system, which was first introduced about the 3rd century BC, employed all the letters of the Greek alphabet plus three letters borrowed from the Phoenician alphabet as number symbols. The first nine letters of the alphabet were used for the numbers 1 to 9, the second nine letters for the tens from 10 to 90, and the last nine letters for the hundreds from 100 to 900. Thousands were indicated by placing a bar to the left of the appropriate numeral, and tens of thousands by placing the appropriate letter over the letter M.
The late Greek system had the advantage that large numbers could be expressed with a minimum of symbols, but it had the disadvantage of requiring the user to memorize a total of 27 symbols.
The decimal system is believed to have originated in counting on the fingers, using both hands as the most convenient method. Arabic numerals are 1, 2, 3, 4, etc.
Roman numerals are I, II, III, IV, etc. 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. The Roman system has no symbol for zero, and in the Arabic system zero is much more recent than the other symbols. The Maya, who were skilled in mathematics, had a symbol for zero.
Some languages show traces of reckoning by units other than ten, e.g., 8, 12, 20. In the case of 20, the toes as well as the fingers may have been used in counting.
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