The Arabic World Post-10th Century

Arabian Mathematics

Douglas Pfeffer

Douglas Pfeffer Arabian Mathematics The Arabic World Post-10th Century Table of contents

1 The Arabic World

2 Post-10th Century

Douglas Pfeffer Arabian Mathematics Pre-10th Century The Arabic World House of Wisdom Post-10th Century Algebra Outline

1 The Arabic World

2 Post-10th Century

Douglas Pfeffer Arabian Mathematics Pre-10th Century The Arabic World House of Wisdom Post-10th Century Algebra

In the mid-600s (around the time of Brahmagupta), the Arabian peninsula was in turmoil A desert nomad, Muhammad, was born in Mecca around 570 and journeyed the peninsula He came into contact with Jews and Christians and eventually came to the belief that he was the apostle of God Eventually, he returned to Mecca to preach. In 622, there existed a threat on his life Muhammad was invited to Medina for safety – his acceptance is known as the Hegira and marked the beginning of the Muhammad Era By 632, Muhammad had established the Muhammaden State centered at Mecca He was both the religious and military leader Jews and Christians, being monotheistic, were offered protection and freedom of worship

Douglas Pfeffer Arabian Mathematics Pre-10th Century The Arabic World House of Wisdom Post-10th Century Algebra

Douglas Pfeffer Arabian Mathematics Pre-10th Century The Arabic World House of Wisdom Post-10th Century Algebra

By 750, war had turned down. There was a schism between western Arabs in Morocco and eastern Arabs in Baghdad under caliph al-Mansur His rule was a religious, economic one and not really a political one The conquerors, instead of imposing a culture, sought to absorb the conquered’s (much like Rome over Greece) In short time, various mathematical texts were translated into Arabic 775 - Siddhanta 780 - Ptolemy’s Tetrabilos ‘

Douglas Pfeffer Arabian Mathematics Pre-10th Century The Arabic World House of Wisdom Post-10th Century Algebra House of Wisdom

The first century of the Muslin Empire (650-750) was generally devoid of scientific achievement By 750, Baghdad had become the new Alexandria Scholars from Syria, Iran, and Mesopotamia all came to study There were three great leaders that cared about academics: al-Mansur, Haroun al-Raschid, and al-Mamun We are familiar with the reign of Haroun al-Raschid through the classic Arabian Nights It was during his reign that Euclid’s Elements began getting translated into Arabic During the rule of al-Mamun, he established the House of Wisdom at Baghdad

Douglas Pfeffer Arabian Mathematics Pre-10th Century The Arabic World House of Wisdom Post-10th Century Algebra

Scholars at the House of Wisdom in Baghdad. Illustration by Yahy al-Wasiti, 1237

Douglas Pfeffer Arabian Mathematics Pre-10th Century The Arabic World House of Wisdom Post-10th Century Algebra

During the caliphate of al-Mamun (809-833), he was visited in a dream by Aristotle Motivated to finish translations of Elements and Almagest The Greek manuscripts were obtained via the uneasy peace with the nearby Byzantine Empire The House of Wisdom was a place for scholarly advancement much like the Library of Alexandria or Plato’s Academy Housed all translations Housed an observatory Among many scholars, it was a place of study for Muhammad ibn Musa al-Khwarizmi (780-850) Destroyed in a 13th century Mongol invasion of Baghdad The books were not burned, but instead thrown into the river

Douglas Pfeffer Arabian Mathematics Pre-10th Century The Arabic World House of Wisdom Post-10th Century Algebra Al-Khwarizmi

Muhammad ibn Musa al-Khwarizmi and his work Hisob al-jabr wa’l muquabalah

Douglas Pfeffer Arabian Mathematics Pre-10th Century The Arabic World House of Wisdom Post-10th Century Algebra Al-Khwarizmi

Wrote half a dozen astronomical/mathematical texts based loosely on the Indian Sindhind Notably, he wrote a book each on arithmetic and algebra The only surviving copy of the arithmetic text is a latin translation De numero indorum Based on Brahmagupta’s work and also gave a thorough, full account of Hindu numerals While Al-Khwarizmi did not claim ownership of the numerals, many future (western) readers mistakenly attributed their origin to him and not to the Hindu’s Of interest is that it is his name-sake that led to the work ‘algorismi’ for the scheme of numeration based on Hindu numeration Later, the word ‘algorithm’ would develop

Douglas Pfeffer Arabian Mathematics Pre-10th Century The Arabic World House of Wisdom Post-10th Century Algebra Al-Jabr

In his work Hisob al-jabr wa’l muquabalah (The Compendious Book on Calculation by Completion and Balancing), we yield the word ‘algebra’ The treatise provided for the systematic solution of linear and quadratic equations. From a modern lens, this text is really more of an arithmetic text – it did not use symbolism or admit negative numbers Giving elementary, straightforward solutions to quadratic and linear equations Differed heavily from the indeterminant analysis of Diaphantus and Brahmagupta that gave difficult answers to hard questions

Douglas Pfeffer Arabian Mathematics Pre-10th Century The Arabic World House of Wisdom Post-10th Century Algebra Al-Jabr

In Arabian fashion, it used clear argumentation from premise to conclusion and systematic organization In the preface, he praises the prohpet Muhammad and wrote that the caliph al-Mamun encouraged him to “compose a short work on Calculating by Completion and Reduction, confining it to what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partitions, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computation, and other objects of various sorts and kinda are concerned.” Very clearly concerned with applications

Douglas Pfeffer Arabian Mathematics Pre-10th Century The Arabic World House of Wisdom Post-10th Century Algebra Quadratic Equations

In investigating the solutions to ‘all’ quadratic equations, it separated it into many cases over 6 chapters Its ‘solutions’ were simply giving a few examples and providing a prescription for computing the roots Chapter 1: Problem: x 2 = 5x Answer: x = 5 x2 Problem: 3 = 4x Answer: x = 12 Problem: 5x 2 = 10x Answer: x = 2 Notably, 0 is not considered a solution to these forms Used full paragraphs to describe the equations – no symbolism

Douglas Pfeffer Arabian Mathematics Pre-10th Century The Arabic World House of Wisdom Post-10th Century Algebra Quadratic Equations

In future chapters he essentially gave prescriptions for completing the square and the quadratic formula Chapter 4: Problem: x 2 + 10x = 39 Interestingly, only gave the positive root Al-Khwarizmi addresses the fact that the discriminant must be positive: “You ought to understand also that when you take the half of the roots in this form of equation and then multiply the half by itself; if that which proceeds or results from the multiplication is less than the units above mentioned as accompanying the square, you have an equation.” 2 b
2 That is, given ax + bx + c, if 2 < ac or, multiplying by 4, if b2 − 4ac < 0

Douglas Pfeffer Arabian Mathematics Pre-10th Century The Arabic World House of Wisdom Post-10th Century Algebra Quadratic Equations

In Chapter 6, it uses only the single example x2 = 3x + 4 He reminds the reader that if the leading coefficient is not a one, that one must divide first by this coefficient He then proceeds to essentially complete the square and provides the algorithm arriving at the answer q 3
3
2 x = 2 + 2 + 4 Notably, still omitting the negative root

Douglas Pfeffer Arabian Mathematics Pre-10th Century The Arabic World House of Wisdom Post-10th Century Algebra Quadratic Equations

To what culture do we owe the most influence over Al-Khwarizmi? Strictly India is a possible explanation, but without any discussion of indeterminate analysis, it is unlikely that there was a large amount of influence Due to geographic location, Mesopotamian influence is likely as well What about Greek? One might think not much... however... After the tedious arithmetic algorithms to deduce solutions to quadratic equations, he then writes: “We have said enough so far as numbers are concerned, but the six types of equations. Now, however, it is necessary that we should demonstrate geometrically the truth of the same problems which we have explained in numbers.” The answer is probably a blend of all three cultures!

Douglas Pfeffer Arabian Mathematics Pre-10th Century The Arabic World House of Wisdom Post-10th Century Algebra ‘Abd Al-Hamid ibn-Turk

Al-Khwarizmi’s book on Algebra was not the only textbook on the subject Around the same time, ‘Abd Al-Hamid ibn-Turk wrote Al-jabr wa’l muqabalah This suggests that the topic had been solved for a while before these texts were written Despite this other book, Al-Khwarizmi’s would (much like Euclid’s Elements) be the ‘standard’ text Both texts on algebra still suffered from one serious flaw: A symbolic notation would have to replace the rhetorical one Unfortunately, the Arabs never did this – the best they did was replace number words by number signs

Douglas Pfeffer Arabian Mathematics Pre-10th Century The Arabic World House of Wisdom Post-10th Century Algebra Numerals

Due to a lack of cultural unification in the Arabian Empire, it is exceedingly difficult to pinpoint the origins of our numeral system What can be said is that the rules of numeration (the important concept) was inherited from India For this reason, our number system is generally called the Hindu-Arabic Numeral System The reason the digits are more commonly known as ‘Arabic numerals’ in Europe and the Americas is that they were introduced to Europe in the 10th century by Arabic-speakers of North Africa Arabs, on the other hand, call the base-10 system (not just these digits) ‘Hindu numerals’ referring to their origin in India

Douglas Pfeffer Arabian Mathematics Omar Khayyam The Arabic World Jamshid Al-Kashi Post-10th Century Conclusion and Future Directions Outline

1 The Arabic World

2 Post-10th Century

Douglas Pfeffer Arabian Mathematics Omar Khayyam The Arabic World Jamshid Al-Kashi Post-10th Century Conclusion and Future Directions 10th and 11th Century Highlights

Abu al-Wafa’ Buzjani (940-998)

Douglas Pfeffer Arabian Mathematics Omar Khayyam The Arabic World Jamshid Al-Kashi Post-10th Century Conclusion and Future Directions Abu al-Wafa’

He took trigonometric ideas from Ptolemy’s Almagest and Brahmagupta and formalized them into clear, nice exposition Due to his clean exposition, the Law of Sines is attributed to him He also produced the most detailed trig tables at the time Unfortunately, most of his work was not recognized during the ensuing medieval period He is also well known for translating Diophantus’ Arithmetica from Greek to Arabic

Douglas Pfeffer Arabian Mathematics Omar Khayyam The Arabic World Jamshid Al-Kashi Post-10th Century Conclusion and Future Directions 10th and 11th Century Highlights

Omar Khayyam (1048-1131)

Douglas Pfeffer Arabian Mathematics Omar Khayyam The Arabic World Jamshid Al-Kashi Post-10th Century Conclusion and Future Directions Omar Khayyam

He wrote a textbook on algebra as well – but this one went further than Al-Khwarizmi’s, it contained cubic equations as well In his book, he gave arithmetic and geometric solutions to quadratics (as did his predecessors) He claimed, however, that a general (arithmetic) formula for the cubic was impossible A false statement, as Tartaglia, Cardano, and Ferrari would show in the 16th century He did, however, give a geometric reasoning for finding the (positive) roots of a general cubic This solution involved intersecting conics

Douglas Pfeffer Arabian Mathematics Omar Khayyam The Arabic World Jamshid Al-Kashi Post-10th Century Conclusion and Future Directions Omar Khayyam

The first page of the discussion about intersecting conics.

Douglas Pfeffer Arabian Mathematics Omar Khayyam The Arabic World Jamshid Al-Kashi Post-10th Century Conclusion and Future Directions Omar Khayyam

He did address equations of degree ≥ 4: “What is called square-square by algebraists in continuous magnitude is a theoretical fact. It does not exist in reality in any way.” He viewed algebra and geometry as heavily intertwined: Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved.” In his textbook, he alludes to the fact that he had previously set forth a rule for finding 5th, 6th, and higher powers of a binomial A clear reference to Pascal’s Triangle (independently discovered in China around the same time as him)

Douglas Pfeffer Arabian Mathematics Omar Khayyam The Arabic World Jamshid Al-Kashi Post-10th Century Conclusion and Future Directions Post 12th Century

When Omar Khayyam died in 1131, Islamic science was in a state of decline – but did not stop entirely Jamshid Al-Kashi (c. 1380-1429)

Douglas Pfeffer Arabian Mathematics Omar Khayyam The Arabic World Jamshid Al-Kashi Post-10th Century Conclusion and Future Directions Jamshid Al-Kashi

He wrote a textbook on arithmetic, algebra, and their applications to architecture, surveying, and commerce Most well-known for his incredible computation skills Well-versed in sexagecimal and decimal fractions Regarded himself as the inventor of the decimal fraction (though he more than likely inherited it from China) Set the record for approximating π (via 2π), improving on the Chinese estimate: 2π ≈ 6; 16, 59, 28, 34, 51, 46, 15, 60 = 6.2831853071795865 Note that 2π = 6.28318530717958647692... He achieved this feat by considering the perimeter of a 3 × 228-gon This estimate was unrivaled until the 16th century

Douglas Pfeffer Arabian Mathematics Omar Khayyam The Arabic World Jamshid Al-Kashi Post-10th Century Conclusion and Future Directions End of the Middle Ages

Arabic mathematics was on the decline after 1100 By the end of the Middle Ages (Medieval Period) in the 15th century, Arabic mathematics had all but died Fortunately, European scholarship began to pick back up around this time Inheriting the intellectual legacy left by the ancient world, Western mathematics would flourish heavily in the coming centuries and introduce many modern topics of today

Douglas Pfeffer Arabian Mathematics Omar Khayyam The Arabic World Jamshid Al-Kashi Post-10th Century Conclusion and Future Directions Where Next?

The Middle Ages (c. 476-1453) were dominated by 5 main cultures: Western (Roman Empire) Greek (Byzantine Empire) Chinese Indian Arabic In the beginning of the next semester we will take a look at what work the Western world did in the Middle Ages and then turn out attention to the post-Middle Ages (starting c. 1450)

Douglas Pfeffer Arabian Mathematics