- Homeworks 4.1/2 and 4.3 are still not graded yet. I'll work hard to get them done by Monday.
- Reminder: Daniel Otero will be here on Friday, to speak on The History of Determinants. I hope that some of you will be able to make it!
- Lessons from China in 5.5
- Commercial from Tom; Paul's next; who's up after that (and after Thanksgiving!)? Who still needs to do one?
- Lessons from 6.1/6.2
- Arabic Mathematics
- A new empire, ruled by Caliphs, and based on the Islamic faith (founded by Muhammad (c. 570- 632)), swept the old civilizations away:
- 635: Damascus fell
- 637: Jerusalem
- c. 640: Egypt falls
- 711: cross the straits of Gibraltar
- A map of the Islamic empire (circa 750)
- Capital moved to Baghdad, circa 762 (population 800,000 in the 9th century -- larger than Constantinople)
- Caliph al-Ma'mun (786-833) sets up the "House of Wisdom", where extant bodies of knowledge were translated into Arabic ("...practically the whole extant body of Greek scientific ... writings had been recorded in the Arabic language." -- p. 240).
- Mathematicians:
- "most illustrious": Mohammed ibn Musa al-Khowarizmi (c. 780-850) (al khwahr 'iz mee)
- Friend of Caliph al-Ma'mun, associated with the House of Wisdom
- Through his two works (on arithmetic and algebra) Europe became aware of Hindu numerals and algebraic approach. (p. 240)
- Book of addition and subtraction according to the Hindu Calculation explained the Hindu decimal system of numerals, including making use of zero (p. 241).
- Interestingly, no copies of the Arabic remain, but it came to Europe in a Latin version from the 12th century.
- As the translation was from Arabic, the numerals aquired the name "Arabic" numerals. And so it goes....
- Europeans also became acquainted with algebra through al-Khowarizmi's work, Hisab al-jabr w'al muqabalah -- "The Science of reunion and reduction" -- translated as Liber Algebrae et Almucabola.
- reunion: moving negative terms to one side or the other, to make them positive
- reduction: cancellation of like terms
- al-Khowarizmi lives on in the word "algorithm", a Latin corruption of his name. Algorism "...meant the art of computing with Hindu-Arabic numerals." (p. 241)
- al-Khowarizmi is credited by our author with preserving the methods of an old Persian school -- he was not the inventor.
- His algebra suffered from a "complete lack of mathematical symbols" (p. 241) -- still using "flowery prose" instead of notation such as Diophantus attempted.
- "...the Arabs, inspired by Euclid's Elements, seemed to believe that an argument had to be geometric to be convincing." (p. 242).
- Three fundamental types of quadratic equations:
(NB: Which equation is missing?!)
One way of thinking about this is that Arabic mathematicians are reflecting more of the Greek tradition than the Indian.
-
- Example of completing the square: p. 243
- Moving from the geometric, to the algorithmic (even if the notation has not arrived): read the passage on p. 245, beginning "If you wish to subtract the root of 4 from the root of 9...."
"The old, ingenious Babylonian tricks and devices for solving individual problems are finally seen as part of al-Khowarizmi's systematic reduction of the quadratics to their standard types...."
- According to our author, irrational roots of quadratics were acceptable, but they did not perceive the reality of negative solutions to an equation.
- Abu Kamil (c. 850-930) -- "The Reckoner from Egypt"
- Commented and elaborated on al-Khowarizmi's work in his Book of Algebra, which Fibonacci would draw heavily on for his Liber Abaci.
- Expands on al-Khowarizmi's work, nonetheless. Adds methods of his own. Let's look at his problem on p. 244:
- Abu Kamil's major advance: irrational coefficients in indeterminate equations
- He'd developed a "calculus of radicals", based on the identity
- Thabit ibn Qurra (c. 836-901)
- Excellent translator of Greek works
- "Thabit's thorough revision of [the Elements] is the most faithful rendition into Arabic..., which in turn came to the West through Gerard of Ceremona's Latin adaptation in the twelfth century." (p. 246)
- His Book on the Determination of Amicable Numbers is considered "...the first [known] completely original mathematics written in Arabic." (p. 246)
- Nicomachus had mentioned them, but didn't give any theory.
- Thabit's result: if
,
, and
are all prime, then
and
are "amicable numbers".
- Try
:
- Generalized the Pythagorean theorem, and devised novel proofs of the Pythagorean theorem.
- An astronomer in Baghdad, he drew attention to errors in Ptolemy's Almagest.
- Abu Bakr al-Karaji (d. 1029)
- "The monomials are infinite", he declared.
- Developed an algebra of polynomials, as well as
- Several results on binomial expansions, including discovering Pascal's triangle a few years before Pascal (1623-1662).
- al-Samaw'al (c. 1180) took his formula for the sum of cubes in an interesting geometrical direction: let's look at p. 249, for an early example of recursion.
- "most illustrious": Mohammed ibn Musa al-Khowarizmi (c. 780-850) (al khwahr 'iz mee)
- Omar
Khayyam (1048-1123)
- The Eastern Caliphate was overrun by the Seljuk Turks (Sunni Muslims)
- The Western Caliphate, centered in Spain continued, but the Eastern Caliphate was done for.
- Famous as a Poet for The Rubayyat of Omar Khayyam
- Omar was Persian, from Nishapur
- Mathematically:
- He constructed the solutions of "all" the
"sensible" cubics (those without negative
coefficients, or repeated roots).
According to our author, "his mastery of the geometrical theory of third-degree equations... may be regarded as the most successful accomplishment of an Arabic mathematician."
Let's look at a rather protracted example of how to solve one of these cubics: #8, p. 266.
- He knew of, and loved, Pascal's Triangle (calling the numbers "Khayyam's beautiful numbers")
- He accepted the fifth postulate of Euclid, but "viewed it as being less obvious"; he attempted to derive the parallel postulate from another -- that converging lines intersect (shades of Lewis Carroll).
- Advances in Astronomy:
- Astronomy important for orienting Mosques towards Mecca; local times for the five daily prayers (sunrise, midday, afternoon, sunset, and evening)
- Adjusted the Persian calendar so that feasts and fasts might be kept at the proper times (the Jalalian calendar, more accurate than our Gregorian).
- There is some fascinating info on the
progress in astronomy under Al-Ma'mun's astronomers, Al-Tusi
(1201-1274, and a rogue!), and Al-Kashi (d. 1429, who knew
to 16 decimal places!), at Ulugh Beg's observatory (Beg, 1347-1449).
On my computer, with my favorite software, I type
Al-Kashi has me beat:
- He constructed the solutions of "all" the
"sensible" cubics (those without negative
coefficients, or repeated roots).
- A new empire, ruled by Caliphs, and based on the Islamic faith (founded by Muhammad (c. 570- 632)), swept the old civilizations away:
- Chinese Mathematics
- Ancient:
- Arithmetic Classic of the Gnomon and the Circular
Paths of Heaven (circa 300 BC)
- We've already seen the "behold" proof of the Pythagorean Theorem (reproduced by Bhaskara in India -- independently?!).
- Nine Chapters on the Mathematical Art
- "Oldest textbook on arithmetic in existence" (p. 255)
- No copies of the text extant (among those burned in the "Burning of the Books" of 213 BC).
- Surviving is a commentary by Liu Hui (AD 263)
- "In its influence on Chinese mathematical thought, Liu Hui's commentary ... is the most important of all ancient works, studied by generation after generation for more than a thousand years." (p. 255)
- Decreed the standard textbook in the 7th century.
- Printed in 1084!
- Not intended as a theoretical work, but rather a practical handbook.
- Sample problem: #15, p. 267
- One chapter is called "The Way of Calculating
Arrays", basically providing an introduction to Linear Algebra. See
"Matrices and Determinants at St. Andrews.
- They used red for negative numbers, and black for positive ("in the red?").
- They also could represent zero, by leaving a space on the board.
- Arithmetic Classic of the Gnomon and the Circular
Paths of Heaven (circa 300 BC)
- The Chinese also worked to improve the value of
- Square roots were obtained by Liu Hui in an interesting way, digit
by digit (p. 258): find the
.
- Later works:
- According to our author, Chinese mathematics peaked in the 13th century (which will also be the "Age of Fibonacci" in the West).
- Ch'in
Chu-shao (c. 1202-1261) (I put this link in, just in case you've forgotten!)
- Mathematical Treatise in Nine Section (1247)
- "oldest extant Chineses mathematical text to contain a round symbol for zero"
- "first in which numerical equations of degree higher than 3 occur" (p. 262)
- An "unsavory character" -- interesting read! (p. 262)
- Li Ye (1192-1279)
- Governer of a northern province when the Mongols overran northern China (1234, an interesting year number!).
- Asked to work for Kublai Khan, in 1260. Only could stand it for a year.
- His "original contribution to Chinese mathematical notation was to indicate negative quantities by drawing a diagonal stroke through the last digit of the number in question." (see p. 263)
- Chu Shih-chieh (c. 1280-1303)
- Introduction to Mathematical Studies (1299)
- Precious Mirror of the Four Elements (1303)
- Studied series ("piles of balls"), without any theoretical proofs
- Rediscovered/studies triangular numbers.
- Jesuits brought European mathematics to China in the late 1500s.
- Ancient:
- Section 6.1: The Decline and Revival of Learning
- Begins with a recap of the destruction of Roman culture,
and the Dark Ages (the depth reached in the 7th century), while
the Islamic culture was arising, and Brahmagupta was working
industriously away in India.
- The Battle of Tours (732) saw the Muslim army turned back
to Spain by Charles Martel, grandfather of Charlemagne.
- The Christian Church and the monastic schools are keeping the lights on, but just barely: their aim was producing ecclesiastical leaders, not scholars. There was suspicion of the pagans, like Archimedes....
- Our author admirably gives a shout-out to the scribes, who are diligently copying the texts (and keeping "the ancient culture from extinction" (p. 272), although other civilizations aren't going to let that happen).
- Charlemagne
(742-814) crowned Holy Roman Emperor on Christmas Day, 800.
- His empire
- Alcuin of York
(c. 740-804) Charlemagne's mathematician
- Englishman, renowned scholar
- Wrote textbooks for all the liberal arts
- Searcher after books
- Introduced the Carolingian_minuscule script, "the source of our printed alphabet today" (p. 272).
- Unfortunately, the principal sort of problem studied was how to calculate the ecclesiastical calendar....
- The jealous husbands problem
- Gave an "impulse" to education, which endured after his death.
- Brief Renaissance, followed by invasions from the north (Norsemen -- Go Norse!), Magyars from the east, Saracens from the south, ....
- The good news was that no one culture make a permanent takeover of Christian Europe, and the monasteries continued to function. Charlemagne's empire broke up into what would become France, Germany, and Italy today.
- Eventually the monastic schools were displaced in importance by the "cathedral schools" -- Cologne, Tours, Liege, Chartres, Reims, and Paris, which became the great universities of the 13th and 14th centuries.
- Greek wisdom would be reintroduced to Europe via Arabic,
and through Spain. Cordoba, in Spain, in the 7th century, had
- 600 mosques
- 600,000 volumes in their library
- and was an intellectual center of the western caliphate
- Burton credits the Arabic translations of the classic Greek works as "Islam's great and enduring contribution to the advancement of knowledge" (p. 275).
- Christians recovered Toledo, Spain, in 1085. It too had been an important center. It became important to translate these Arabic works.
- The crusadors of the 11th century, began to bring back Greek knowledge to Europe. Translators like Adelard_of_Bath (1080-1152) and Gerard_of_Cremona (1114-1187) translated the classics from Arabic into Latin (the lingua franca of Europe). The work could be dangerous: there's the interesting tale of Gerard in the disguise of an Islamic student taking in lectures in Cordoba (he would have been killed if discovered).
- The first printed copy of Campanus's Euclid's Elements comes out in 1482. Translated directly from Greek, it was an improvement on the translations derived from Arabic, which came out via a process of "telephone" (p. 276).
- "By the middle of the 1200s... Western scholars stood once again on the solid foundation of Hellenistic thought." (p. 279)
- Begins with a recap of the destruction of Roman culture,
and the Dark Ages (the depth reached in the 7th century), while
the Islamic culture was arising, and Brahmagupta was working
industriously away in India.
- Section 6.2: The Liber_Abaci and Liber Quadratorum
- "The greatest mathematician of the Middle Ages was Leonardo of Pisa, better known as Fibonacci.... [T]he mathematical renaissance of the West dates from him." (c. 1170-1250)
- Read the opening sentences of his Liber Abaci (p. 280): what a start! Arabic numerals introduced, via his "Book of Counting" (his father ran a customshouse). The arabic numbers were first introduced in the Codex Vigilanus around 976 -- p. 282). By the beginning of printed books, c. 1450, our arabic numerals had taken the form in which we know them today.
- Remember that Fibonacci borrowed from Abu Kamil commentaries on al-Khowarizmi's work.
- "The Liber Abaci embodied virtually all the arithmetical knowledge of Fibonacci's time...." (p. 280)
- Even so, it was not translated into English until 2002, on its 800th anniversary!
- Let's do one problem from the Liber Abaci: #3, p. 287
