Many ancient cultures calculated from early on with numerals based on ten: Egyptian hieroglyphs, in evidence since around 3000 BC, used a purely decimal system,[9][10] just as the Cretan hieroglyphs (ca. 1625−1500 BC) of the Minoans whose numerals are closely based on the Egyptian model.[11][12] The decimal system was handed down to the consecutive Bronze Age cultures of Greece, including Linear A (ca. 18th century BC−1450 BC) and Linear B (ca. 1375−1200 BC) — the number system of classical Greece also used powers of ten, including, like the Roman numerals did, an intermediate base of 5.[13] Notably, the polymath Archimedes (c. 287–212 BC) invented a decimal positional system in his Sand Reckoner which was based on 108[13] and later led the German mathematician Carl Friedrich Gauss to lament what heights science would have already reached in his days if Archimedes had fully realized the potential of his ingenious discovery.[14] The Hittites hieroglyphs (since 15th century BC), just like the Egyptian and early numerals in Greece, was strictly decimal.[15]
The Egyptian hieratic numerals, the Greek alphabet numerals, the Roman numerals, the Chinese numerals and early Indian Brahmi numerals are all non-positional decimal systems, and required large numbers of symbols. For instance, Egyptian numerals used different symbols for 10, 20, to 90, 100, 200, to 900, 1000, 2000, 3000, 4000, to 10,000.[16]
History of decimal fractions[edit]
counting rod decimal fraction 1/7
According to Joseph Needham, decimal fractions were first developed and used by the Chinese in the 1st century BC, and then spread to the Middle East and from there to Europe.[17] The written Chinese decimal fractions were non-positional.[17] However, counting rod fractions were positional.
Qin Jiushao in his book Mathematical Treatise in Nine Sections (1247) denoted 0.96644 by
寸
Counting rod 0.pngCounting rod h9 num.pngCounting rod v6.pngCounting rod h6.pngCounting rod v4.pngCounting rod h4.png, meaning
寸
096644
[18]
The Jewish mathematician Immanuel Bonfils invented decimal fractions around 1350, anticipating Simon Stevin, but did not develop any notation to represent them.[19]
The Persian mathematician Jamshīd al-Kāshī claimed to have discovered decimal fractions himself in the 15th century, though J. Lennart Berggren notes that positional decimal fractions were used five centuries before him by Arab mathematician Abu'l-Hasan al-Uqlidisi as early as the 10th century.[20]
Khwarizmi introduced fractions to Islamic countries in the early 9th century. . This form of fraction with the numerator on top and the denominator on the bottom, without a horizontal bar, was also used in the 10th century by Abu'l-Hasan al-Uqlidisi and again in the 15th century work "Arithmetic Key" by Jamshīd al-Kāshī.[citation needed]
Stevin-decimal notation.svg
A forerunner of modern European decimal notation was introduced by Simon Stevin in the 16th century.[21]
Natural languages[edit]
Telugu language uses a straightforward decimal system. Other Dravidian languages such as Tamil and Malayalam have replaced the number nine tondu with 'onpattu' ("one to ten") during the early Middle Ages, while Telugu preserved the number nine as tommidi.
The Hungarian language also uses a straightforward decimal system. All numbers between 10 and 20 are formed regularly (e.g. 11 is expressed as "tízenegy" literally "one on ten"), as with those between 20-100 (23 as "huszonhárom" = "three on twenty").
A straightforward decimal rank system with a word for each order 10十,100百,1000千,10000万, and in which 11 is expressed as ten-one and 23 as two-ten-three, and 89345 is expressed as 8 (ten thousands) 万9 (thousand) 千3 (hundred) 百4 (tens) 十 5 is found in Chinese languages, and in Vietnamese with a few irregularities. Japanese, Korean, and Thai have imported the Chinese decimal system. Many other languages with a decimal system have special words for the numbers between 10 and 20, and decades. For example in English 11 is "eleven" not "ten-one".
Incan languages such as Quechua and Aymara have an almost straightforward decimal system, in which 11 is expressed as ten with one and 23 as two-ten with three.
Some psychologists suggest irregularities of the English names of numerals may hinder children's counting ability.[22]
Other bases[edit]
Some cultures do, or did, use other bases of numbers.
Pre-Columbian Mesoamerican cultures such as the Maya used a base-20 system (using all twenty fingers and toes).
The Yuki language in California and the Pamean languages[23] in Mexico have octal (base-8) systems because the speakers count using the spaces between their fingers rather than the fingers themselves.[24]
The existence of a non-decimal base in the earliest traces of the Germanic languages, is attested by the presence of words and glosses meaning that the count is in decimal (cognates to ten-count or tenty-wise), such would be expected if normal counting is not decimal, and unusual if it were.[improper synthesis?] Where this counting system is known, it is based on the long hundred of 120 in number, and a long thousand of 1200 in number. The descriptions like 'long' only appear after the small hundred of 100 in number appeared with the Christians. Gordon's Introduction to Old Norse p 293, gives number names that belong to this system. An expression cognate to 'one hundred and eighty' is translated to 200, and the cognate to 'two hundred' is translated at 240. Goodare details the use of the long hundred in Scotland in the Middle Ages, giving examples, calculations where the carry implies i C (i.e. one hundred) as 120, etc. That the general population were not alarmed to encounter such numbers suggests common enough use. It is also possible to avoid hundred-like numbers by using intermediate units, such as stones and pounds, rather than a long count of pounds. Goodare gives examples of numbers like vii score, where one avoids the hundred by using extended scores. There is also a paper by W.H. Stevenson, on 'Long Hundred and its uses in England'.[citation needed]
Many or all of the Chumashan languages originally used a base-4 counting system, in which the names for numbers were structured according to multiples of 4 and 16.[25]
Many languages[26] use quinary (base-5) number systems, including Gumatj, Nunggubuyu,[27] Kuurn Kopan Noot[28] and Saraveca. Of these, Gumatj is the only true 5–25 language known, in which 25 is the higher group of 5.
Some Nigerians use base-12 systems.[29]
The Huli language of Papua New Guinea is reported to have base-15 numbers.[30] Ngui means 15, ngui ki means 15×2 = 30, and ngui ngui means 15×15 = 225.
Umbu-Ungu, also known as Kakoli, is reported to have base-24 numbers.[31] Tokapu means 24, tokapu talu means 24×2 = 48, and tokapu tokapu means 24×24 = 576.
Ngiti is reported to have a base-32 number system with base-4 cycles.[26]
The Egyptian hieratic numerals, the Greek alphabet numerals, the Roman numerals, the Chinese numerals and early Indian Brahmi numerals are all non-positional decimal systems, and required large numbers of symbols. For instance, Egyptian numerals used different symbols for 10, 20, to 90, 100, 200, to 900, 1000, 2000, 3000, 4000, to 10,000.[16]
History of decimal fractions[edit]
counting rod decimal fraction 1/7
According to Joseph Needham, decimal fractions were first developed and used by the Chinese in the 1st century BC, and then spread to the Middle East and from there to Europe.[17] The written Chinese decimal fractions were non-positional.[17] However, counting rod fractions were positional.
Qin Jiushao in his book Mathematical Treatise in Nine Sections (1247) denoted 0.96644 by
寸
Counting rod 0.pngCounting rod h9 num.pngCounting rod v6.pngCounting rod h6.pngCounting rod v4.pngCounting rod h4.png, meaning
寸
096644
[18]
The Jewish mathematician Immanuel Bonfils invented decimal fractions around 1350, anticipating Simon Stevin, but did not develop any notation to represent them.[19]
The Persian mathematician Jamshīd al-Kāshī claimed to have discovered decimal fractions himself in the 15th century, though J. Lennart Berggren notes that positional decimal fractions were used five centuries before him by Arab mathematician Abu'l-Hasan al-Uqlidisi as early as the 10th century.[20]
Khwarizmi introduced fractions to Islamic countries in the early 9th century. . This form of fraction with the numerator on top and the denominator on the bottom, without a horizontal bar, was also used in the 10th century by Abu'l-Hasan al-Uqlidisi and again in the 15th century work "Arithmetic Key" by Jamshīd al-Kāshī.[citation needed]
Stevin-decimal notation.svg
A forerunner of modern European decimal notation was introduced by Simon Stevin in the 16th century.[21]
Natural languages[edit]
Telugu language uses a straightforward decimal system. Other Dravidian languages such as Tamil and Malayalam have replaced the number nine tondu with 'onpattu' ("one to ten") during the early Middle Ages, while Telugu preserved the number nine as tommidi.
The Hungarian language also uses a straightforward decimal system. All numbers between 10 and 20 are formed regularly (e.g. 11 is expressed as "tízenegy" literally "one on ten"), as with those between 20-100 (23 as "huszonhárom" = "three on twenty").
A straightforward decimal rank system with a word for each order 10十,100百,1000千,10000万, and in which 11 is expressed as ten-one and 23 as two-ten-three, and 89345 is expressed as 8 (ten thousands) 万9 (thousand) 千3 (hundred) 百4 (tens) 十 5 is found in Chinese languages, and in Vietnamese with a few irregularities. Japanese, Korean, and Thai have imported the Chinese decimal system. Many other languages with a decimal system have special words for the numbers between 10 and 20, and decades. For example in English 11 is "eleven" not "ten-one".
Incan languages such as Quechua and Aymara have an almost straightforward decimal system, in which 11 is expressed as ten with one and 23 as two-ten with three.
Some psychologists suggest irregularities of the English names of numerals may hinder children's counting ability.[22]
Other bases[edit]
Some cultures do, or did, use other bases of numbers.
Pre-Columbian Mesoamerican cultures such as the Maya used a base-20 system (using all twenty fingers and toes).
The Yuki language in California and the Pamean languages[23] in Mexico have octal (base-8) systems because the speakers count using the spaces between their fingers rather than the fingers themselves.[24]
The existence of a non-decimal base in the earliest traces of the Germanic languages, is attested by the presence of words and glosses meaning that the count is in decimal (cognates to ten-count or tenty-wise), such would be expected if normal counting is not decimal, and unusual if it were.[improper synthesis?] Where this counting system is known, it is based on the long hundred of 120 in number, and a long thousand of 1200 in number. The descriptions like 'long' only appear after the small hundred of 100 in number appeared with the Christians. Gordon's Introduction to Old Norse p 293, gives number names that belong to this system. An expression cognate to 'one hundred and eighty' is translated to 200, and the cognate to 'two hundred' is translated at 240. Goodare details the use of the long hundred in Scotland in the Middle Ages, giving examples, calculations where the carry implies i C (i.e. one hundred) as 120, etc. That the general population were not alarmed to encounter such numbers suggests common enough use. It is also possible to avoid hundred-like numbers by using intermediate units, such as stones and pounds, rather than a long count of pounds. Goodare gives examples of numbers like vii score, where one avoids the hundred by using extended scores. There is also a paper by W.H. Stevenson, on 'Long Hundred and its uses in England'.[citation needed]
Many or all of the Chumashan languages originally used a base-4 counting system, in which the names for numbers were structured according to multiples of 4 and 16.[25]
Many languages[26] use quinary (base-5) number systems, including Gumatj, Nunggubuyu,[27] Kuurn Kopan Noot[28] and Saraveca. Of these, Gumatj is the only true 5–25 language known, in which 25 is the higher group of 5.
Some Nigerians use base-12 systems.[29]
The Huli language of Papua New Guinea is reported to have base-15 numbers.[30] Ngui means 15, ngui ki means 15×2 = 30, and ngui ngui means 15×15 = 225.
Umbu-Ungu, also known as Kakoli, is reported to have base-24 numbers.[31] Tokapu means 24, tokapu talu means 24×2 = 48, and tokapu tokapu means 24×24 = 576.
Ngiti is reported to have a base-32 number system with base-4 cycles.[26]
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