Al-Karaji

Last updated
Abū Bakr al-Karajī
Muhammad al karaji 01.jpg
Diagrams from Al-Karaji's work on "hidden waters"
Born953
Died1029 (aged 7576)
Nationality Persian
Main interests
Mathematics, Engineering

Abū Bakr Muḥammad ibn al Ḥasan al-Karajī (Persian : ابو بکر محمد بن الحسن الکرجی; c. 953 c. 1029) was a 10th-century Persian [1] [2] [3] mathematician and engineer who flourished at Baghdad. He was born in Karaj, a city near Tehran. His three principal surviving works are mathematical: Al-Badi' fi'l-hisab (Wonderful on calculation), Al-Fakhri fi'l-jabr wa'l-muqabala (Glorious on algebra), and Al-Kafi fi'l-hisab (Sufficient on calculation).

Contents

Work

Al-Karaji wrote on mathematics and engineering. Some consider him to be merely reworking the ideas of others (he was influenced by Diophantus) but most regard him as more original, [4] in particular for the beginnings of freeing algebra from geometry. Among historians, his most widely studied work is his algebra book al-fakhri fi al-jabr wa al-muqabala, which survives from the medieval era in at least four copies. [5]

In his book "Extraction of hidden waters" he has mentioned that earth is spherical in shape but considers it the centre of the universe long before Galileo Galilei, Johannes Kepler or Isaac Newton, but long after Aristotle and Ptolemy. He expounded the basic principles of hydrology [6] and this book reveals a profound knowledge of this science and has been described as the oldest extant text in this field. [7] [8] [9]

He systematically studied the algebra of exponents, and was the first to realise that the sequence x, x^2, x^3,... could be extended indefinitely; and the reciprocals 1/x, 1/x^2, 1/x^3,... . However, since for example the product of a square and a cube would be expressed, in words rather than in numbers, as a square-cube, the numerical property of adding exponents was not clear. [10]

His work on algebra and polynomials gave the rules for arithmetic operations for adding, subtracting and multiplying polynomials; though he was restricted to dividing polynomials by monomials.

F. Woepcke was the first historian to realise the importance of al-Karaji's work and later historians mostly agree with his interpretation. He praised Al-Karaji for being the first who introduced the theory of algebraic calculus. [5] [11]

Al-Karaji gave the first formulation of the binomial coefficients and the first description of Pascal's triangle. [12] [13] [14] He is also credited with the discovery of the binomial theorem. [15]

In a now lost work known only from subsequent quotation by al-Samaw'al Al-Karaji introduced the idea of argument by mathematical induction. [16] As Katz says

Another important idea introduced by al-Karaji and continued by al-Samaw'al and others was that of an inductive argument for dealing with certain arithmetic sequences. Thus al-Karaji used such an argument to prove the result on the sums of integral cubes already known to Aryabhata [...] Al-Karaji did not, however, state a general result for arbitrary n. He stated his theorem for the particular integer 10 [...] His proof, nevertheless, was clearly designed to be extendable to any other integer. [...] Al-Karaji's argument includes in essence the two basic components of a modern argument by induction, namely the truth of the statement for n = 1 (1 = 13) and the deriving of the truth for n = k from that of n = k - 1. Of course, this second component is not explicit since, in some sense, al-Karaji's argument is in reverse; this is, he starts from n = 10 and goes down to 1 rather than proceeding upward. Nevertheless, his argument in al-Fakhri is the earliest extant proof of the sum formula for integral cubes. [17]

See also

Notes

  1. "Muhammad Al-Karaji: A Mathematician Engineer from the Early 11th Century | Muslim Heritage". www.muslimheritage.com. Retrieved 2018-08-10. Of Persian origin, he spent an important part of his scientific life in Baghdad where he composed ground breaking mathematical books.
  2. Selin, Helaine (2008). Encyclopaedia of the history of science, technology, and medicine in non-western cultures. Berlin New York: Springer. p. 131. ISBN   9781402049606. Al-Karajī Abū Bakr Muh.ammad was a Persian mathematician and engineer.
  3. Meri, Josef W. (January 2006). Medieval Islamic Civilization, Volume 1 An Encyclopedia. Routledge. p. 32. ISBN   978-0-415-96691-7. During the tenth century CE, the Iranian mathematician al-Karaji (...)
  4. http://www-history.mcs.st-and.ac.uk/history/Biographies/Al-Karaji.html
  5. 1 2 O'Connor, John J.; Robertson, Edmund F., "Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji", MacTutor History of Mathematics archive , University of St Andrews .
  6. Robinson, M.; Ward, R. C. (2017-02-15). Hydrology: Principles and Processes. IWA Publishing. p. 19. ISBN   9781780407289.
  7. Muslim Heritage, Mohammed Abattouy " Al-Karaji is also the author of Inbat al-miyah al-khafiya (The Extraction of Hidden Waters), a technical treatise that reveals such a profound knowledge of hydrology that it should be celebrated as the oldest text of its kind in this field."
  8. Sorkhabi, Rasoul (2017-12-21). Tectonic Evolution, Collision, and Seismicity of Southwest Asia: In Honor of Manuel Berberian's Forty-Five Years of Research Contributions. Geological Society of America. p. 37. ISBN   9780813725253.
  9. Niazi, Kaveh (2016-01-01). "Karajī's Discourse on Hydrology". Oriens. 44 (1–2): 44–68. doi:10.1163/18778372-04401003. ISSN   0078-6527. The hydrological concepts presented in Inbāṭ al-miyāh al-khafīya, Muḥammad Karajī’s 11th century text on the construction of the qanāt, contain unexpected premises and theories that set this text apart from its contemporaries. Even when not straying far from the Aristotelian cosmology of the medieval world, Karajī’s hydrological discussions often represent a fresh take on the common scientific wisdom regarding the flow of water at and near the earth’s surface.
  10. Katz, History of Mathematics, first edition, p237
  11. "You Have Got to Know...Mathematics" "Page 26"
  12. Sidoli, Nathan; Brummelen, Glen Van (2013-10-30). From Alexandria, Through Baghdad: Surveys and Studies in the Ancient Greek and Medieval Islamic Mathematical Sciences in Honor of J.L. Berggren. Springer Science & Business Media. p. 54. ISBN   9783642367366.
  13. Selin, Helaine (2008-03-12). Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. Springer Science & Business Media. p. 132. ISBN   9781402045592.
  14. The Development of Arabic Mathematics Between Arithmetic and Algebra - R. Rashed "Page 63"
  15. "THE BINOMIAL THEOREM : A WIDESPREAD CONCEPT IN MEDIEVAL ISLAMIC MATHEMATICS" (PDF). core.ac.uk. p. 401. Retrieved 2019-01-08.
  16. Abattouy, Mohammed (2009). "Muhammad Al-Karaji: A Mathematician Engineer from the Early 11th Century". Muslim heritage. He was also the first to use the method of proof by mathematical induction to prove his results, which he also used to prove the sum formula for integral cubes, an important result in integral calculus.
  17. Katz (1998), p. 255

Related Research Articles

Abu Bakr First Muslim Caliph and close companion of the Islamic Prophet Muhammad

Abu Bakr Abdullah ibn Uthman was a companion and, through his daughter Aisha, a father-in-law of the Islamic prophet Muhammad, as well as the first of the Rashidun Caliphs.

Abu'l Hasan Ahmad ibn Ibrahim Al-Uqlidisi was a Muslim Arab mathematician, who was active in Damascus and Baghdad. He wrote the earliest surviving book on the positional use of the Arabic numerals, Kitab al-Fusul fi al-Hisab al-Hindi around 952. It is especially notable for its treatment of decimal fractions, and that it showed how to carry out calculations without deletions.

Muhammad ibn Musa al-Khwarizmi 9th century Persian mathematician, astronomer and geographer

Muḥammad ibn Mūsā al-Khwārizmī, Arabized as al-Khwarizmi and formerly Latinized as Algorithmi, was a Persian polymath who produced vastly influential works in mathematics, astronomy, and geography. Around 820 CE he was appointed as the astronomer and head of the library of the House of Wisdom in Baghdad.

Abu al-Wafa Buzjani Persian mathematician and astronomer (940–998)

Abū al-Wafāʾ, Muḥammad ibn Muḥammad ibn Yaḥyā ibn Ismāʿīl ibn al-ʿAbbās al-Būzjānī or Abū al-Wafā Būzhjānī was a Persian mathematician and astronomer who worked in Baghdad. He made important innovations in spherical trigonometry, and his work on arithmetics for businessmen contains the first instance of using negative numbers in a medieval Islamic text.

<i>The Compendious Book on Calculation by Completion and Balancing</i> Arabic Mathematical treatise of Algebra

The Compendious Book on Calculation by Completion and Balancing, also known as Al-Jabr (ٱلْجَبْر), is an Arabic mathematical treatise on algebra written by the Polymath Muḥammad ibn Mūsā al-Khwārizmī around 820 CE while he was in the Abbasid capital of Baghdad, modern-day Iraq. Al-Jabr was a landmark work in the history of mathematics, establishing algebra as an independent discipline, and with the term "algebra" itself derived from Al-Jabr.

Sharaf al-Dīn al-Muẓaffar ibn Muḥammad ibn al-Muẓaffar al-Ṭūsī was an Iranian mathematician and astronomer of the Islamic Golden Age.

This timeline of science and engineering in the Muslim world covers the time period from the eighth century AD to the introduction of European science to the Muslim world in the nineteenth century. All year dates are given according to the Gregorian calendar except where noted.

Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad Ibn Shujāʿ was an Egyptian mathematician during the Islamic Golden Age. He is considered the first mathematician to systematically use and accept irrational numbers as solutions and coefficients to equations. His mathematical techniques were later adopted by Fibonacci, thus allowing Abu Kamil an important part in introducing algebra to Europe.

Mathematics in medieval Islam

Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics and Indian mathematics. Important progress was made, such as full development of the decimal place-value system to include decimal fractions, the first systematised study of algebra, and advances in geometry and trigonometry.

Al-Samawal al-Maghribi

Al-Samawʾal ibn Yaḥyā al-Maghribī, commonly known as Samau'al al-Maghribi, was a mathematician, astronomer and physician. Born to a Jewish family, he concealed his conversion to Islam for many years in fear of offending his father, then openly embraced Islam in 1163 after he had a dream telling him to do so. His father was a Rabbi from Morocco.

Ibn al‐Bannāʾ al‐Marrākushī, also known as Abu'l-Abbas Ahmad ibn Muhammad ibn Uthman al-Azdi (29 December 1256 – c. 1321), was a Moroccan-Arab mathematician, astronomer, Islamic scholar, Sufi, and a one-time astrologer.

Abū al-Ḥasan ibn ʿAlī ibn Muḥammad ibn ʿAlī al-Qurashī al-Qalaṣādī was a Muslim Arab mathematician from Al-Andalus specializing in Islamic inheritance jurisprudence. Franz Woepcke stated that al-Qalaṣādī was known as one of the most influential voices in algebraic notation for taking "the first steps toward the introduction of algebraic symbolism''. He wrote numerous books on arithmetic and algebra, including al-Tabsira fi'lm al-hisab.

Algebra can essentially be considered as doing computations similar to those of arithmetic but with non-numerical mathematical objects. However, until the 19th century, algebra consisted essentially of the theory of equations. For example, the fundamental theorem of algebra belongs to the theory of equations and is not, nowadays, considered as belonging to algebra.

A timeline of key algebraic developments are as follows:

This is a timeline of pure and applied mathematics history. It is divided here into three stages, corresponding to stages in the development of mathematical notation: a "rhetorical" stage in which calculations are described purely by words, a "syncopated" stage in which quantities and common algebraic operations are beginning to be represented by symbolic abbreviations, and finally a "symbolic" stage, in which comprehensive notational systems for formulas are the norm.

The mathematical field of combinatorics was studied to varying degrees in numerous ancient societies. Its study in Europe dates to the work of Leonardo Fibonacci in the 13th century AD, which introduced Arabian and Indian ideas to the continent. It has continued to be studied in the modern era.

The al-Madhara'i were a family of officials from Iraq who served as and virtually monopolized the posts of director of finances (‘āmil) of Egypt and Syria for the Tulunid dynasty, the Abbasid Caliphate, and the Ikhshidid dynasty, between 879 and 946. In this role, they amassed "one of the largest personal fortunes in the medieval Arab east".

Abū al‐Qāsim Aṣbagh ibn Muḥammad ibn al‐Samḥ al‐Gharnāṭī al-Mahri, also known as Ibn al‐Samḥ, was an Arab mathematician and astronomer in Al-Andalus. He worked at the school founded by Al-Majriti in Córdoba, until political unrest forced him to move to Granada, where he was employed by Ḥabbūs ibn Māksan. He is known for treatises on the construction and use of the astrolabe, as well as the first known work on the planetary equatorium. Furthermore, in mathematics he is remembered for a commentary on Euclid and for contributions to early algebra, among other works. He is one of several writers referred to in Latin texts as "Abulcasim."