Arabian Mathematics - University of Florida · Abu al-Wafa’ Buzjani (940-998) Douglas Pfe er...

The Arabic WorldPost-10th Century

Arabian Mathematics

Douglas Pfeffer

Douglas Pfeffer Arabian Mathematics


Table of contents

1 The Arabic World

2 Post-10th Century



Pre-10th CenturyHouse of WisdomAlgebra

Outline

1 The Arabic World

2 Post-10th Century




In the mid-600s (around the time of Brahmagupta), theArabian peninsula was in turmoil

A desert nomad, Muhammad, was born in Mecca around 570and journeyed the peninsula

He came into contact with Jews and Christians and eventuallycame to the belief that he was the apostle of GodEventually, he returned to Mecca to preach. In 622, thereexisted a threat on his lifeMuhammad was invited to Medina for safety – his acceptanceis known as the Hegira and marked the beginning of theMuhammad Era

By 632, Muhammad had established the Muhammaden Statecentered at Mecca

He was both the religious and military leaderJews and Christians, being monotheistic, were offeredprotection and freedom of worship




By 750, war had turned down.There was a schism between western Arabs in Morocco andeastern Arabs in Baghdad under caliph al-Mansur

His rule was a religious, economic one and not really apolitical one

The conquerors, instead of imposing a culture, sought toabsorb the conquered’s (much like Rome over Greece)In short time, various mathematical texts were translated intoArabic

775 - Siddhanta780 - Ptolemy’s Tetrabilos

‘




House of Wisdom

The first century of the Muslin Empire (650-750) wasgenerally devoid of scientific achievement

By 750, Baghdad had become the new Alexandria

Scholars from Syria, Iran, and Mesopotamia all came to studyThere 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 throughthe classic Arabian Nights

It was during his reign that Euclid’s Elements began gettingtranslated into Arabic

During the rule of al-Mamun, he established the House ofWisdom at Baghdad




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




During the caliphate of al-Mamun (809-833), he was visited ina dream by Aristotle

Motivated to finish translations of Elements and AlmagestThe Greek manuscripts were obtained via the uneasy peacewith the nearby Byzantine Empire

The House of Wisdom was a place for scholarly advancementmuch like the Library of Alexandria or Plato’s Academy

Housed all translationsHoused an observatoryAmong many scholars, it was a place of study for Muhammadibn 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




Al-Khwarizmi

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




Al-Khwarizmi

Wrote half a dozen astronomical/mathematical texts basedloosely on the Indian Sindhind

Notably, he wrote a book each on arithmetic and algebra

The only surviving copy of the arithmetic text is a latintranslation De numero indorum

Based on Brahmagupta’s work and also gave a thorough, fullaccount of Hindu numeralsWhile Al-Khwarizmi did not claim ownership of the numerals,many future (western) readers mistakenly attributed theirorigin 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 Hindunumeration

Later, the word ‘algorithm’ would develop




Al-Jabr

In his work Hisob al-jabr wa’l muquabalah (The CompendiousBook on Calculation by Completion and Balancing), we yieldthe word ‘algebra’

The treatise provided for the systematic solution of linear andquadratic equations.From a modern lens, this text is really more of an arithmetictext – it did not use symbolism or admit negative numbers

Giving elementary, straightforward solutions to quadratic andlinear equations

Differed heavily from the indeterminant analysis of Diaphantusand Brahmagupta that gave difficult answers to hard questions




Al-Jabr

In Arabian fashion, it used clear argumentation from premiseto conclusion and systematic organization

In the preface, he praises the prohpet Muhammad and wrotethat the caliph al-Mamun encouraged him to

“compose a short work on Calculating by Completion andReduction, confining it to what is easiest and most useful inarithmetic, such as men constantly require in cases ofinheritance, legacies, partitions, lawsuits, and trade, and in alltheir dealings with one another, or where the measuring oflands, the digging of canals, geometrical computation, andother objects of various sorts and kinda are concerned.”

Very clearly concerned with applications




Quadratic Equations

In investigating the solutions to ‘all’ quadratic equations, itseparated it into many cases over 6 chapters

Its ‘solutions’ were simply giving a few examples and providinga prescription for computing the roots

Chapter 1:

Problem: x2 = 5x Answer: x = 5Problem: x2

3 = 4x Answer: x = 12Problem: 5x2 = 10x Answer: x = 2

Notably, 0 is not considered a solution to these forms

Used full paragraphs to describe the equations – no symbolism




Quadratic Equations

In future chapters he essentially gave prescriptions forcompleting the square and the quadratic formula

Chapter 4:

Problem: x2 + 10x = 39Interestingly, only gave the positive root

Al-Khwarizmi addresses the fact that the discriminant mustbe positive:

“You ought to understand also that when you take the half ofthe roots in this form of equation and then multiply the half byitself; if that which proceeds or results from the multiplicationis less than the units above mentioned as accompanying thesquare, you have an equation.”

That is, given ax2 + bx + c , if(b2

)2< ac or, multiplying by 4,

if b2 − 4ac < 0




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 aone, that one must divide first by this coefficientHe then proceeds to essentially complete the square andprovides the algorithm arriving at the answer

x =(

32

)+

√(32

)2+ 4

Notably, still omitting the negative root




Quadratic Equations

To what culture do we owe the most influence overAl-Khwarizmi?

Strictly India is a possible explanation, but without anydiscussion of indeterminate analysis, it is unlikely that therewas a large amount of influenceDue to geographic location, Mesopotamian influence is likelyas wellWhat about Greek? One might think not much... however...

After the tedious arithmetic algorithms to deduce solutions toquadratic equations, he then writes:

“We have said enough so far as numbers are concerned, butthe six types of equations. Now, however, it is necessary thatwe should demonstrate geometrically the truth of the sameproblems which we have explained in numbers.”

The answer is probably a blend of all three cultures!




‘Abd Al-Hamid ibn-Turk

Al-Khwarizmi’s book on Algebra was not the only textbook onthe subject

Around the same time, ‘Abd Al-Hamid ibn-Turk wrote Al-jabrwa’l muqabalahThis suggests that the topic had been solved for a while beforethese texts were written

Despite this other book, Al-Khwarizmi’s would (much likeEuclid’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 oneUnfortunately, the Arabs never did this – the best they did wasreplace number words by number signs




Numerals

Due to a lack of cultural unification in the Arabian Empire, itis exceedingly difficult to pinpoint the origins of our numeralsystem

What can be said is that the rules of numeration (theimportant concept) was inherited from IndiaFor this reason, our number system is generally called theHindu-Arabic Numeral System

The reason the digits are more commonly known as ‘Arabicnumerals’ in Europe and the Americas is that they wereintroduced to Europe in the 10th century by Arabic-speakersof North Africa

Arabs, on the other hand, call the base-10 system (not justthese digits) ‘Hindu numerals’ referring to their origin in India



Omar KhayyamJamshid Al-KashiConclusion and Future Directions

Outline

1 The Arabic World

2 Post-10th Century




10th and 11th Century Highlights

Abu al-Wafa’ Buzjani (940-998)




Abu al-Wafa’

He took trigonometric ideas from Ptolemy’s Almagest andBrahmagupta and formalized them into clear, nice exposition

Due to his clean exposition, the Law of Sines is attributed tohimHe also produced the most detailed trig tables at the time

Unfortunately, most of his work was not recognized during theensuing medieval period

He is also well known for translating Diophantus’ Arithmeticafrom Greek to Arabic




10th and 11th Century Highlights

Omar Khayyam (1048-1131)




Omar Khayyam

He wrote a textbook on algebra as well – but this one wentfurther than Al-Khwarizmi’s, it contained cubic equations aswell

In his book, he gave arithmetic and geometric solutions toquadratics (as did his predecessors)

He claimed, however, that a general (arithmetic) formula forthe cubic was impossible

A false statement, as Tartaglia, Cardano, and Ferrari wouldshow 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




Omar Khayyam

The first page of the discussion about intersecting conics.




Omar Khayyam

He did address equations of degree ≥ 4:

“What is called square-square by algebraists in continuousmagnitude is a theoretical fact. It does not exist in reality inany way.”

He viewed algebra and geometry as heavily intertwined:

Whoever thinks algebra is a trick in obtaining unknowns hasthought it in vain. No attention should be paid to the factthat 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 previouslyset forth a rule for finding 5th, 6th, and higher powers of abinomial

A clear reference to Pascal’s Triangle (independentlydiscovered in China around the same time as him)




Post 12th Century

When Omar Khayyam died in 1131, Islamic science was in astate of decline – but did not stop entirely

Jamshid Al-Kashi (c. 1380-1429)




Jamshid Al-Kashi

He wrote a textbook on arithmetic, algebra, and theirapplications 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 theChinese estimate:

2π ≈ 6; 16, 59, 28, 34, 51, 46, 15, 60 = 6.2831853071795865Note that 2π = 6.28318530717958647692...He achieved this feat by considering the perimeter of a3 × 228-gonThis estimate was unrivaled until the 16th century




End of the Middle Ages

Arabic mathematics was on the decline after 1100

By the end of the Middle Ages (Medieval Period) in the 15thcentury, Arabic mathematics had all but died

Fortunately, European scholarship began to pick back uparound this time

Inheriting the intellectual legacy left by the ancient world,Western mathematics would flourish heavily in the comingcenturies and introduce many modern topics of today




Where Next?

The Middle Ages (c. 476-1453) were dominated by 5 maincultures:

Western (Roman Empire)Greek (Byzantine Empire)ChineseIndianArabic

In the beginning of the next semester we will take a look atwhat work the Western world did in the Middle Ages and thenturn out attention to the post-Middle Ages (starting c. 1450)


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