Arabic Numerals

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Working Title: Arabic Numerals | Original Publisher: Materia Islamica | Publication Date: December 16th, 2016 | Last Updated: February 14th, 2020 | Written by: Canadian786 |

The evolution of the modern numerical system; taken from "Encyclopedia Britannica".[1] Note the sharp divergence between Gwailor (Indian) and West Arabic numerals (i.e. ghubar / modern numerals), East Arabic numerals and Devanagari numerals by the 11th century.

The ghubar numerals are sometimes erroneously called "Hindu-Arabic" numerals despite the obvious divergence in their appearance. Furthermore, the ghubar numerals weren't invented in the Middle-east but Muslim Spain (711—1492[2]), with the earliest evidence coming from Islamic manuscripts dated to the years 874 and 888.[3]
  • Ghubar Numerals—The modern numerical system which consists of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, also known as "ghubar numerals" ("ghubar" meaning "abacus" and not simply "dust" as is erroneously believed[3]), "Western Arabic numerals", "Arabic numerals" or "Islamic numerals" (not to be confused with the "Eastern Arabic numerals"—which are also confusingly known as "Hindu numerals"[3]—which are represented as ١,٢,٣,٤,٥,٦,٧,٨,٩;[4] despite the fact that these newer glyphs were invented by the Eastern Arabs and not Hindus; indeed the last true "Hindu" numerals date to the Gwailor period of India in the 9th century—see right-hand image[1]) were first invented in the 9th century in Muslim Spain.[3] The ghubar numerals evolved from the primitive numbers developed on the Indian subcontinent (much like how English branched off from earlier Indo-European languages).
    • The fact that this number system is sometimes also called "Hindu-Arabic" is erroneous. The ghubar numerals appear totally different to those from India, and shouldn't really be called "Hindu" at all. According to the "National Institute of Sciences of India" itself, the "Ghubar numerals resemble modern European numerals much more closely than do the Hindu numerals".[5] This is actually quite obvious when tracing their history; see right hand-image for example. This otherwise would be akin to erroneously calling the English language "German" just because it is directly related in the vast tree of the 448 Indo-European languages.[6]
    • The earliest documents in history which depict ghubar numerals are from two surviving manuscripts; one dated 874, and the other 888; and curiously the oldest European documents to depict these numerals is a Latin manuscript written by Gerbert (of Aurillac; 930/946—1003[3][7]) in 976 (however this latter source described them without reference to the number zero).[3]
      • Regarding the etymological origins of the name "ghubar", historians have noted "[t]he ghubar, originated in Spain, is not an original Arabic term alluding to the Hindu custom of writing on the sand, but it is an attempt to arabicise the Roman term abacus. When the Arabs arrived in Spain, they learned to know the abacus and the abacus numerals; thus they rendered the word abacus by ghubar, and the numerals found on the abacus were called huruif-al-ghubdr, the abacus numerals".[3]
      • Interestingly, Gerbert later became Pope, and was Christened with the name Pope Sylvester II[7] (reigning as head of the Catholic church from 999 to 1003[8]). As a student, prior to becoming pope, he was taught the numbers by his Arab teachers, when he was residing in Spain for his education.[7] The Muslim Arabs had originally conquered Spain in 711 and flourished until 1492.[9] Interestingly, Pope Sylvester II was a curious character himself; his death was said to have been particularly horrendous; caused as a result of him having "made a pact with the devil", for which he was punished by dismemberment; but oddly at his own request as payment for his repentance to God, which itself was said to have been "last-minute".[10]
The difference between "Eastern Arabic" (sometimes called Hindu numerals) and "Western Arabic numerals" (Ghubar numerals i.e. modern numerals). The earliest description of the modern numerals is 874.[3]
  • A Note on the History and Invention of 0 (Zero)—The worlds oldest representation of the number zero (i.e. the symbol for "0") is to be found in ancient Pakistan.[11][12] Evidence for this lies in an ancient manuscript called the "Bakhshali",[11] which was discovered by chance by a farmer tending to his fields in 1881 during the reign of the British Raj (1858—1947[13]), and subsequently brought to the attention of the British Indologist A. F. R. Hoernle (1841—1918[14]).[11] The name of the manuscript itself, "Bakhshali", is named after the village from where it was found, 80 km[15] from Peshawar.[11] It was then transferred over to the United Kingdom in 1902 for safekeeping, and has since lain dormant at the University of Oxford (or more specifically at it's prestigious Bodleian Library).[11][12]
    • Interestingly, the famous South Asian mathematician, Brahmagupta (598—668[16]), was born in (or near[17][18]) Multan, Punjab (modern-day Pakistan),[19][20][21][22][23] or Bhillamala, Sind (modern-day Sindh, Pakistan).[24][25][26] Others claim Abu, Rajasthan (modern-day India).[23][27]
      • In the history of mathematics however he is known as "Brahmagupta of Multan" or "Brahmagupta of Sind", who is known to have popularised the use of zero hundreds of years after its conception.[19][28] However, at some point he moved to Ujjain (modern-day Madhya Pradesh) where he lived and worked.[27] To the Chinese he was known as "Pi-lo-mo-lo".[24] His contributions themselves were popularised by al-Biruni (973—1048).[29]
    • Prior to the discovery of the manuscript, it was thought the earliest depiction of zero as a symbol was first developed in modern-day India, as the earliest use was found in a 9th century inscription on the wall of a temple in Gwalior, Madhya Pradesh.[11] According to the University of Oxford, "[t]he findings are highly significant for the study of the early history of mathematics" as the newer evidence predates the use of zero by at least five centuries. [11]
      • One of the reasons why this wasn't known until only recently is perhaps because the manuscripts used to depict zero were made out of birch bark—a natural biodegradable perishable material that was extensively used in the medieval age (especially in ancient Buddhist Pakistan) prior to the invention of paper in China and its subsequent spread by the Sogdians (Iranian nomads) and Arabs (Abbasids) in the 8th century, after the Battle of Talas (751).
    • The concept of zero however may be even older; the Babylonians "used a character for the absence of [a] number" and "they made use of a primitive kind of place value" but "they did not create a system of numeration in which the zero played any such part as it does in the one which we now use".[30]
Pen and paper algorithms can be seen here on the BBC website. They were first invented by al-Uqlidisi in 945—952.
  • Arithmetic, Pen and Paper Algorithms—When the Indian numerals gradually evolved into the newly invented Arabic numerals, a new method of calculation was also invented to make them even easier to use. It is important here then to distinguish between the dust-board algorithms that the old Indian system used versus the newly devised pen-and-paper algorithms developed by the Arabs. The dust board algorithms were "gradually replaced by algorithms performed with pen and paper. The earliest text to describe these new methods was al Uqlidisi's Arithmetic, written in Baghdad in 945 C.E.; in it the author argues for pen-and-paper techniques so that arithmetics could be distinguished from the dust board-wielding astrologers".[31]
    • The dust board algorithms from India were written down by al-Khwarizmi, but there was significant resistance to the adoption of the new numerical system largely because of how difficult people found it.[32] Indeed, "[c]ertainly the fact that the Indian system required a dust board had been one of the main obstacles to its acceptance".[32] Others at the time testified that "[o]fficial scribes nevertheless avoid using [the Indian system] because it requires equipment [like a dust board] and they consider that a system that requires nothing but the members of the body is more secure and more fitting to the dignity of a leader".[32] The crucial step forward was then developed by Abu'l Hasan Ahmad ibn Ibrahim al-Uqlidisi (920—980[33]) when he invented the pen-and-paper algorithms (indeed, it was he "who gave algorithms for use with pen and paper, as opposed to those of al-Khwarizmi which were for dust board").[34] Al-Uqlidisi was therefore the first to invent pen-and-paper algorithms which are still used today.[31][n. 1]
      • It should be noted that the dust board's physical appearance itself is not known (a modern statue of the Indian mathematician Bhaskara II (1114—1185[35]) depicts him holding a board on his knees but historians have noted it "would have been difficult to perform computations in this way" with other historians speculating the dust board could have been a "sort of laptop blackboard"); but what is known is that it's computation technique (it's algorithms) were not the same as those created by al-Uqlidisi since he clearly distinguishes the old techniques from the new in his book, "Kitab al-fusul fi al-hisab al-Hindi", where he directly states he presented the "airthmatic of the Indians that has been done on the takht [dust board], but here with no takht and no erasing; we carry it out on a sheet of paper, thus dispensing with the dust and the board".[36]
        • In India itself, the first computations without rubbing out appeared in the 15th century, several centuries after al-Uqlidisi.[36]

Astronomical MotionSystems & NumbersAccuracy & PrecisionTrigonometry (Functions & TablesIdentities) • Algebra & Algebraic Theory (Differential & Integral Calculus) • Geometry (Spherical TrianglesNumber Theory & Summation Series) • Navigational, Celestial & Planetary CalculatorsList of MathematiciansSee alsoSources • (FootnotesReferencesAcknowledgements) • External LinksTotal Inventions & Discoveries Listed: 52

The evolution of the modern numerical system; taken from "Encyclopedia Britannica".[1] Note the sharp divergence between Gwailor (Indian) and West Arabic numerals (i.e. ghubar / modern numerals), East Arabic numerals and Devanagari numerals by the 11th century.

The ghubar numerals are sometimes erroneously called "Hindu-Arabic" numerals despite the obvious divergence in their appearance. Furthermore, the ghubar numerals weren't invented in the Middle-east but Muslim Spain (711—1492[2]), with the earliest evidence coming from Islamic manuscripts dated to the years 874 and 888.[3]
  • Ghubar Numerals—The modern numerical system which consists of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, also known as "ghubar numerals" ("ghubar" meaning "abacus" and not simply "dust" as is erroneously believed[3]), "Western Arabic numerals", "Arabic numerals" or "Islamic numerals" (not to be confused with the "Eastern Arabic numerals"—which are also confusingly known as "Hindu numerals"[3]—which are represented as ١,٢,٣,٤,٥,٦,٧,٨,٩;[4] despite the fact that these newer glyphs were invented by the Eastern Arabs and not Hindus; indeed the last true "Hindu" numerals date to the Gwailor period of India in the 9th century—see right-hand image[1]) were first invented in the 9th century in Muslim Spain.[3] The ghubar numerals evolved from the primitive numbers developed on the Indian subcontinent (much like how English branched off from earlier Indo-European languages).
    • The fact that this number system is sometimes also called "Hindu-Arabic" is erroneous. The ghubar numerals appear totally different to those from India, and shouldn't really be called "Hindu" at all. According to the "National Institute of Sciences of India" itself, the "Ghubar numerals resemble modern European numerals much more closely than do the Hindu numerals".[5] This is actually quite obvious when tracing their history; see right hand-image for example. This otherwise would be akin to erroneously calling the English language "German" just because it is directly related in the vast tree of the 448 Indo-European languages.[6]
    • The earliest documents in history which depict ghubar numerals are from two surviving manuscripts; one dated 874, and the other 888; and curiously the oldest European documents to depict these numerals is a Latin manuscript written by Gerbert (of Aurillac; 930/946—1003[3][7]) in 976 (however this latter source described them without reference to the number zero).[3]
      • Regarding the etymological origins of the name "ghubar", historians have noted "[t]he ghubar, originated in Spain, is not an original Arabic term alluding to the Hindu custom of writing on the sand, but it is an attempt to arabicise the Roman term abacus. When the Arabs arrived in Spain, they learned to know the abacus and the abacus numerals; thus they rendered the word abacus by ghubar, and the numerals found on the abacus were called huruif-al-ghubdr, the abacus numerals".[3]
      • Interestingly, Gerbert later became Pope, and was Christened with the name Pope Sylvester II[7] (reigning as head of the Catholic church from 999 to 1003[8]). As a student, prior to becoming pope, he was taught the numbers by his Arab teachers, when he was residing in Spain for his education.[7] The Muslim Arabs had originally conquered Spain in 711 and flourished until 1492.[9] Interestingly, Pope Sylvester II was a curious character himself; his death was said to have been particularly horrendous; caused as a result of him having "made a pact with the devil", for which he was punished by dismemberment; but oddly at his own request as payment for his repentance to God, which itself was said to have been "last-minute".[10]
The difference between "Eastern Arabic" (sometimes called Hindu numerals) and "Western Arabic numerals" (Ghubar numerals i.e. modern numerals). The earliest description of the modern numerals is 874.[3]
  • A Note on the History and Invention of 0 (Zero)—The worlds oldest representation of the number zero (i.e. the symbol for "0") is to be found in ancient Pakistan.[11][12] Evidence for this lies in an ancient manuscript called the "Bakhshali",[11] which was discovered by chance by a farmer tending to his fields in 1881 during the reign of the British Raj (1858—1947[13]), and subsequently brought to the attention of the British Indologist A. F. R. Hoernle (1841—1918[14]).[11] The name of the manuscript itself, "Bakhshali", is named after the village from where it was found, 80 km[15] from Peshawar.[11] It was then transferred over to the United Kingdom in 1902 for safekeeping, and has since lain dormant at the University of Oxford (or more specifically at it's prestigious Bodleian Library).[11][12]
    • Interestingly, the famous South Asian mathematician, Brahmagupta (598—668[16]), was born in (or near[17][18]) Multan, Punjab (modern-day Pakistan),[19][20][21][22][23] or Bhillamala, Sind (modern-day Sindh, Pakistan).[24][25][26] Others claim Abu, Rajasthan (modern-day India).[23][27]
      • In the history of mathematics however he is known as "Brahmagupta of Multan" or "Brahmagupta of Sind", who is known to have popularised the use of zero hundreds of years after its conception.[19][28] However, at some point he moved to Ujjain (modern-day Madhya Pradesh) where he lived and worked.[27] To the Chinese he was known as "Pi-lo-mo-lo".[24] His contributions themselves were popularised by al-Biruni (973—1048).[29]
    • Prior to the discovery of the manuscript, it was thought the earliest depiction of zero as a symbol was first developed in modern-day India, as the earliest use was found in a 9th century inscription on the wall of a temple in Gwalior, Madhya Pradesh.[11] According to the University of Oxford, "[t]he findings are highly significant for the study of the early history of mathematics" as the newer evidence predates the use of zero by at least five centuries. [11]
      • One of the reasons why this wasn't known until only recently is perhaps because the manuscripts used to depict zero were made out of birch bark—a natural biodegradable perishable material that was extensively used in the medieval age (especially in ancient Buddhist Pakistan) prior to the invention of paper in China and its subsequent spread by the Sogdians (Iranian nomads) and Arabs (Abbasids) in the 8th century, after the Battle of Talas (751).
    • The concept of zero however may be even older; the Babylonians "used a character for the absence of [a] number" and "they made use of a primitive kind of place value" but "they did not create a system of numeration in which the zero played any such part as it does in the one which we now use".[30]
Pen and paper algorithms can be seen here on the BBC website. They were first invented by al-Uqlidisi in 945—952.
  • Arithmetic, Pen and Paper Algorithms—When the Indian numerals gradually evolved into the newly invented Arabic numerals, a new method of calculation was also invented to make them even easier to use. It is important here then to distinguish between the dust-board algorithms that the old Indian system used versus the newly devised pen-and-paper algorithms developed by the Arabs. The dust board algorithms were "gradually replaced by algorithms performed with pen and paper. The earliest text to describe these new methods was al Uqlidisi's Arithmetic, written in Baghdad in 945 C.E.; in it the author argues for pen-and-paper techniques so that arithmetics could be distinguished from the dust board-wielding astrologers".[31]
    • The dust board algorithms from India were written down by al-Khwarizmi, but there was significant resistance to the adoption of the new numerical system largely because of how difficult people found it.[32] Indeed, "[c]ertainly the fact that the Indian system required a dust board had been one of the main obstacles to its acceptance".[32] Others at the time testified that "[o]fficial scribes nevertheless avoid using [the Indian system] because it requires equipment [like a dust board] and they consider that a system that requires nothing but the members of the body is more secure and more fitting to the dignity of a leader".[32] The crucial step forward was then developed by Abu'l Hasan Ahmad ibn Ibrahim al-Uqlidisi (920—980[33]) when he invented the pen-and-paper algorithms (indeed, it was he "who gave algorithms for use with pen and paper, as opposed to those of al-Khwarizmi which were for dust board").[34] Al-Uqlidisi was therefore the first to invent pen-and-paper algorithms which are still used today.[31][n. 2]
      • It should be noted that the dust board's physical appearance itself is not known (a modern statue of the Indian mathematician Bhaskara II (1114—1185[35]) depicts him holding a board on his knees but historians have noted it "would have been difficult to perform computations in this way" with other historians speculating the dust board could have been a "sort of laptop blackboard"); but what is known is that it's computation technique (it's algorithms) were not the same as those created by al-Uqlidisi since he clearly distinguishes the old techniques from the new in his book, "Kitab al-fusul fi al-hisab al-Hindi", where he directly states he presented the "airthmatic of the Indians that has been done on the takht [dust board], but here with no takht and no erasing; we carry it out on a sheet of paper, thus dispensing with the dust and the board".[36]
        • In India itself, the first computations without rubbing out appeared in the 15th century, several centuries after al-Uqlidisi.[36]

See also

Sources

Footnotes

  1. ^ In 952, Abu'l Hasan Ahmad ibn Ibrahim al-Uqlidisi became the first person in the world to solve the Indian numerical system without having to use the dust board as the Indians had been doing for centuries (the Indians had never used pen-and-paper algorithms to solve their mathematical problems); this is proven by the following peer-reviewed sources on the history of the numerals;
    • Quote: "A step forward in the development of the decimal-place value system was due to Abu'l Hasan Ahmad ibn Ibrahim al-Uqlidisi (920-980)...who gave algorithms for use with pen and paper [in 1952], as opposed to those of al-Khwarizmi which were for dust board, and, more importantly introduced decimal fractions."
    1. Ethan D. Bloch (14 May 2011). The Real Numbers and Real Analysis. Springer Science & Business Media. p. 54. ISBN 978-0-387-72177-4.
    They were new algorithms, unique to him, because the Indians used physical equipment to solve their problems. Historains explicitly state the Indians did not use pen-and-paper style arithmetic until the Arabs had solved for it;
    The Indian numerical system never used pen-and-paper style algorithmic arithmetic:
    • Historians note that "[f]rom their origins in the late eighth and early ninth centuries, the numerals spread throughout the Islamic world, though not without resistance or confusion. Many conservative scribes and bookkeepers resisted the new numerals in favour of older calculation on the fingers and with numerical words".
    • Historians say "[t]he Arabs borrowed not only the Indian numerals, but also a host of computational techniques and devices, including the dust-board, a flat tablet strewn with sand into which figures could be written for undertaking computations...Other techniques available included complex Greek-derived system of finger reckoning and the use of shells; accordingly, the use of written pen-and-paper-arithmetic was apparently not part of the initial practice of Indian derived numeration".
    • After the work of al-Uqlidisi, only then did pen-and-paper-arithmetic began to be widely used; well after his algorithms on pen-and-paper arithmetic been published and disseminated across the world; "[y]et once the [treatise] had been established by the eleventh and twelfth centuries, Arabic mathematical texts began to advocate doing computations with paper and ink instead of the dust-board".
    1. Stephen Chrisomalis (2010). Numerical Notation: A Comparative History. Cambridge University Press. p. 215. ISBN 978-0-521-87818-0.
  2. ^ In 952, Abu'l Hasan Ahmad ibn Ibrahim al-Uqlidisi became the first person in the world to solve the Indian numerical system without having to use the dust board as the Indians had been doing for centuries (the Indians had never used pen-and-paper algorithms to solve their mathematical problems); this is proven by the following peer-reviewed sources on the history of the numerals;
    • Quote: "A step forward in the development of the decimal-place value system was due to Abu'l Hasan Ahmad ibn Ibrahim al-Uqlidisi (920-980)...who gave algorithms for use with pen and paper [in 1952], as opposed to those of al-Khwarizmi which were for dust board, and, more importantly introduced decimal fractions."
    1. Ethan D. Bloch (14 May 2011). The Real Numbers and Real Analysis. Springer Science & Business Media. p. 54. ISBN 978-0-387-72177-4.
    They were new algorithms, unique to him, because the Indians used physical equipment to solve their problems. Historains explicitly state the Indians did not use pen-and-paper style arithmetic until the Arabs had solved for it;
    The Indian numerical system never used pen-and-paper style algorithmic arithmetic:
    • Historians note that "[f]rom their origins in the late eighth and early ninth centuries, the numerals spread throughout the Islamic world, though not without resistance or confusion. Many conservative scribes and bookkeepers resisted the new numerals in favour of older calculation on the fingers and with numerical words".
    • Historians say "[t]he Arabs borrowed not only the Indian numerals, but also a host of computational techniques and devices, including the dust-board, a flat tablet strewn with sand into which figures could be written for undertaking computations...Other techniques available included complex Greek-derived system of finger reckoning and the use of shells; accordingly, the use of written pen-and-paper-arithmetic was apparently not part of the initial practice of Indian derived numeration".
    • After the work of al-Uqlidisi, only then did pen-and-paper-arithmetic began to be widely used; well after his algorithms on pen-and-paper arithmetic been published and disseminated across the world; "[y]et once the [treatise] had been established by the eleventh and twelfth centuries, Arabic mathematical texts began to advocate doing computations with paper and ink instead of the dust-board".
    1. Stephen Chrisomalis (2010). Numerical Notation: A Comparative History. Cambridge University Press. p. 215. ISBN 978-0-521-87818-0.

References

  1. ^ a b c d Evolution of Hindu-Arabic Numerals. 2006. Encyclopedia Britannica. WayBackMachine Link. Retrieved February 11th, 2020.
  2. ^ a b Mark K. Bauman (7 June 2011). Jewish American Chronology: Chronologies of the American Mosaic: Chronologies of the American Mosaic. ABC-CLIO. p. 1. ISBN 978-0-313-37605-4.
  3. ^ a b c d e f g h i j k l m n o p Gandz, Solomon (1931). "The Origin of the Ghubār Numerals, or the Arabian Abacus and the Articuli". Isis (A Journal of the History of Science Society). The University of Chicago Press Journals. 16 (2): 393–424. doi:10.1086/346615. ISSN 0021-1753.
  4. ^ a b Numbers in Arabic. Rocket Languages. WayBackMachine Link. Retrieved February 5th, 2020.
  5. ^ a b Bulletin of the National Institute of Sciences of India. National Institute of Sciences of India. 1962. pp. 27–28.
  6. ^ a b Indo-European. Ethnologue. WayBackMachine Link. Retrieved February 5th, 2020.
  7. ^ a b c d e f Ethan D. Bloch (14 May 2011). The Real Numbers and Real Analysis. Springer Science & Business Media. p. 54. ISBN 978-0-387-72177-4.
  8. ^ a b S. J. Tester (1987). A History of Western Astrology. Boydell & Brewer Ltd. p. 130. ISBN 978-0-85115-255-4.
  9. ^ a b Andrea L. Stanton (2012). Cultural Sociology of the Middle East, Asia, and Africa: An Encyclopedia. SAGE. p. 106. ISBN 978-1-4129-8176-7.
  10. ^ a b Ian Watt (13 February 1997). Myths of Modern Individualism: Faust, Don Quixote, Don Juan, Robinson Crusoe. Cambridge University Press. p. 26. ISBN 978-0-521-58564-4.
  11. ^ a b c d e f g h i j k l m n Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol 'zero'. 14th September 2017. University of Oxford. WayBackMachine Link. Retrieved September 9th, 2019.
  12. ^ a b c d Hannah Devlin (14th September 2017). Much ado about nothing: ancient Indian text contains earliest zero symbol. The Guardian. WayBackMachine Link. Retrieved September 9th, 2019.
  13. ^ a b Lionel Knight (2012). Britain in India, 1858-1947. Anthem Press. ISBN 978-0-85728-517-1.
  14. ^ a b Stuart Brown; Hugh Terence Bredin (August 2005). Dictionary of Twentieth-Century British Philosophers. A&C Black. p. 434. ISBN 978-1-84371-096-7.
  15. ^ a b Helaine Selin (12 March 2008). Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. Springer Science & Business Media. pp. 1–2. ISBN 978-1-4020-4559-2.
  16. ^ a b S. Frederick Starr (2 June 2015). Lost Enlightenment: Central Asia's Golden Age from the Arab Conquest to Tamerlane. Princeton University Press. p. 170. ISBN 978-0-691-16585-1.
  17. ^ a b Arthur Berriedale Keith (1928). The Development and History of Sanskrit Literature. Sanjay Prakashan. p. 522. ISBN 978-81-7453-059-2.
  18. ^ a b H. N. Verma (April 1992). 100 Great Indians Through the Ages. Gip Books. p. 103. ISBN 978-1-881155-00-3.
  19. ^ a b c d Carboni, Alessandro (2011). ABQ – From Quad to Zero Mathematical and choreographic processes –between number and not number. Performance Research. 12 (1): 50–56. doi:10.1080/13528160701398073. ISSN 1352-8165.
  20. ^ a b Bhowmik Subrata (2010). GREAT INDIAN MATHEMATICIANS OF POST-CHRISTIAN ERA. Bulletin of Tripura Mathematical Society. XXX. 11-24. WayBackMachine Link (Archive.is Link). Retrieved October 6th, 2019.
  21. ^ a b Rasik Vihari Joshi (1988). Studies in Indology: Prof. Rasik Vihari Joshi Felicitation Volume. Shree Publishing House. p. 110.
  22. ^ a b Akshaya Kumari Devi (1939). A History of Sanskrit Literature. Vijaya Krishna brothers. p. 124.
  23. ^ a b c d Govind Sadashiv Ghurye (1957). Vidȳas: A Homage to Comte and a Contribution to Sociology of Knowledge. Indian Sociological Society. p. 66.
  24. ^ a b c d B.S. Yadav; Man Mohan (20 January 2011). Ancient Indian Leaps into Mathematics. Springer Science & Business Media. p. 185. ISBN 978-0-8176-4695-0.
  25. ^ a b Gerard G. Emch; R. Sridharan; M. D. Srinivas (15 October 2005). Contributions to the History of Indian Mathematics. Hindustan Book Agency. p. 15. ISBN 978-93-86279-25-5.
  26. ^ a b Mainak Kumar Bose (1988). Late classical India. A. Mukherjee & Co. p. 326.
  27. ^ a b c d S. Devadas Pillai (1997). Indian Sociology Through Ghurye, a Dictionary. Popular Prakashan. p. 216. ISBN 978-81-7154-807-1.
  28. ^ a b Roger L. Cooke (14 February 2011). The History of Mathematics: A Brief Course. John Wiley & Sons. p. 25. ISBN 978-1-118-03024-0.
  29. ^ a b Francis Borceux (31 October 2013). An Axiomatic Approach to Geometry: Geometric Trilogy I. Springer Science & Business Media. p. 167. ISBN 978-3-319-01730-3.
  30. ^ a b David E. Smith (1 June 1958). History of Mathematics. Courier Corporation. p. 69. ISBN 978-0-486-20430-7.
  31. ^ a b c d Thomas F. Glick; Steven Livesey; Faith Wallis (27 January 2014). Medieval Science, Technology, and Medicine: An Encyclopedia. Routledge. p. 46. ISBN 978-1-135-45932-1.
  32. ^ a b c d e f J. J. O'Connor; E. F. Robertson (January 2001). The Arabic numeral system. Math History. St. Andrews University. WayBackMachine Link. Retrieved February 11th, 2020.
  33. ^ a b Clifford A. Pickover (2009). The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics. Sterling Publishing Company, Inc. p. 92. ISBN 978-1-4027-5796-9.
  34. ^ a b Ethan D. Bloch (14 May 2011) pen and paper. The Real Numbers and Real Analysis. Springer Science & Business Media. p. 54. ISBN 978-0-387-72177-4.
  35. ^ a b Donald Bindner; Martin J. Erickson; Joe Hemmeter (21 August 2014). Mathematics for the Liberal Arts. John Wiley & Sons. p. 88. ISBN 978-1-118-37174-9.
  36. ^ a b c d Alexei Volkov; Viktor Freiman (11 January 2019). Computations and Computing Devices in Mathematics Education Before the Advent of Electronic Calculators. Springer. pp. 111–112. ISBN 978-3-319-73396-8.

External Links

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Specific Islamic Golden Age Topics: Snell's Law · Arabic Numerals
Non-Golden Age Articles Inventions & Discoveries: Inventions of Ancient & Modern Pakistan (and Islamic India) (2019)
Modern Contributions & Scientific Achievements: Muslim Soldiers During World War II (Allied Side) (2018) · Pakistan's Atomic Bombs: Project 706/726 (2018) · Muslim Musicians in Western Music (2020)
Total References Used in All Articles Mentioned Within This Template: 2,379