Lecture 17. Mathematics of Medieval Arabs

Transcription

1 Lecture 17. Mathematics of Medieval Arabs The Arabs The term Islam means resignation, i.e., resignation to the will of God as expressed in the Koran, the sacred book, which contains the revelations made to Islam s founder Muhammad ( ). Arabs were nomads occupying the region of modern Arabia. After some initial hostility, Islam soon attracted many followers, uniting peoples of different backgrounds in a common religious cause. In less than a century after Muhammad s death in 632, they had conquered lands from India to Spain, including North Africa and southern Italy. Figure 17.1 Ancient ruins of Palmyra in Syria. In 642, Islamic armies overthrew the Sassanian Empire in the Middle East. By 680 they had taken Armenia, North Africa, and Syria from the Byzantines, which was the final blow to the Alexandrian civilization. By 750, the Islamic Empire extended from southern Spain to Pakistan and Central Asia. 104

2 In 755 the Arab empire split into two independent kingdoms, the eastern part having its capital at Bagdad and the western one at Cordova in Spain. After their conquests were completed, the former nomads settled down to build a civilization and a culture. Soon the Arabs became interested in the arts and sciences. Bagdad became a commercial and intellectual center. It had an academy, a library, and an astronomical observatory. It attracted many scientists and supported their works; they invited Hindus to settle in Baghdad and they attracted Greek scholars, who were from Plato s Academy after the Academy was closed by Justinian in A.D Since Arabs did not oppose pagans, they attracted people from all over the world. They also established contacts with Greeks of the Independent Byzantine Empire. For sciences and mathematics, the Greek culture became part of the Arab scientific and mathematical world. Figure 17.2 This is page 14 from the Geometry (1412) of Qadi Zada al-rami ( ) At the top of the page is a discussion of Euclid s Proposition I-5. Preservation of classical Greek mathematics Fundamentally, what the Arabs possessed was Greek knowledge obtained directly from Greek manuscripts or from Syrian and Hebrew versions. All the major works became accessible to them. They obtained a copy of 105

3 Euclid s Elements from the Byzantines about 800 and translated it into Arabic. Ptolemy s Mathematical Syntaxis 1 was translated into Arabic in 827, and was known as the Almagest, meaning the greatest work. In the course of time, the works of Aristotle, Apollonius, Archimedes, Heron, Diophantus, and of the Hindus became accessible in Arabic. The Arabs then improved the translation and made corrections and commentaries. Absorbing and preserving Greek and Hindu mathematics, the Arab scientific effort was extensive. It is through these translations, some still extant, that the classical Greek mathematics, whose originals have been lost, became available to Europe. This is the significant contribution to mathematics that we owe to the Arabs. We cannot imagine what would happen for modern mathematics without these translated works. Origin of the word algebra The word algebra is a Latin variant of the Arabic word al-jabr. This came from a book written in 830 by the Arab astronomer and mathematician Mohammed ibn Musa al-khwārizmī, ebtitled Al-jabr w al-muqubala. Here the words al-jabr meant restoring. In this context, restoring the balance in an equation by placing on one side of an equation a term that has been removed from the other; for example, x 2 7 = 3 = x 2 = The word al-muqubalah meant simplification, (e.g. 3x + 4x = 7x). Al-muqubala also meant bonesetter, i.e., a restoration of a broken bone. When the Moors brought the word into Spain, it became algebrista and meant bonesetter. By the way, at one time, it was common in Spain to see a sign Algebrista y Sangrador (bonesetter and bloodletter) over the entrance to a barbershop because barbers administered these simple medical treatments at that time and even centuries later. Figure 17.3 Blood letting, by barber surgeons in a barber shop 1 Claudius Ptolemy (90-168) was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer and a poet of a single epigram in the Greek Anthology. 106

4 Mu sa al-khwa rizmi, , is the most famous Arab mathematician. He wrote two books, one was introduction of Hindus mathematics and another was the Al-jabr w almuqubala (algebra). Figure 17.4 Musa al-khwa rizmi, and his algebra text written about 825. In the Al-jabr w al-muqubala, several kinds of linear or quadratic equations and the solutions were discussed. For example, ax2 = bx, x2 = c, ax2 + bx = c, ax2 + c = bx. After detailed study, he solved the equation x2 + bx + c = 0 and gave the formula for its solutions p : x = 2b ± (b/2)2 c. All of his discussions used words, not mathematical notation; for each detailed solution, he gave a geometric proof. For example, considering the equation x2 + 10x = 39, Mu sa al-khwa rizmi made a square with side x whose area is x2. Then he extended this square to a big square whose area is x2 + 10x + 25 with the side x + 5. The big square contains the small one at the center position. Since x2 + 10x = 39, it implies the area of the bigger square is = 64 so that the side of the bigger square is 8. Then the side of the small square x = 8 5 =

5 Figure 17.5 Mathematical induction The earliest implicit traces of mathematical induction can be found in Euclid s proof that the number of primes is infinite. Here is the proof. Suppose that p1 = 2 < p2 = 3 <... < pr are all of the primes. Then P = p1 p2...pr + 1 must be a non prime number so that there is a prime p dividing P. Now we claim that p cannot be any of p1, p2,..., pr, otherwise p would divide the difference P p1 p2...pr = 1, which is impossible. So this prime p is still another prime, and p1, p2,..., pr would not be all of the primes. The earliest implicit proof by mathematical induction for arithmetic sequences was introduced in the al-fakhri written by al-karaji around 1000 AD, who used it to prove the binomial theorem, Pascal s triangle, and the sum formula for integral cubes. For the formula for integral cubes, he first proved the formula for n = 10: = ( )2. Then by assuming the formula holds for n, he showed that it holds for n 1. He proved n3 = ( n)2, 1 n 10. Nevertheless, his proof can be used to prove the above formula for any integer n > 0. Shortly afterwards, Ibn al-haytham ( ) used the inductive method to prove the sum of fourth powers: n X j=1 4 j = 1 1 n 1 + n n+ (n + 1)n

6 and by extension, the sum of any integral powers. He only stated it for particular integers, but his proof for those integers was by induction and generalizable. Another inductive argument, relating to the binomial theorem and the Pascal s triangle, n (a + b) = n X Ckn an k bk, k=0 is found in al-samaw al s Al-Ba hir, where he refers to al-karaji s treatment of these subjects. Conquered by Christians Arab civilization actively reached its peak about the year Between 1100 and 1300, the Christian attacks in the Crusades weakened the eastern Arabs. Subsequently, their territory was overrun and conquered by Mongols. Further destruction by Tartarus under Tamerlane just about wiped out this Arab civilization. In Spain, the Arabs were constantly attacked and finally conquered in 1492 by the Christians. Remark: J. J. O Conner and E. F. Robertson wrote in the MacTutor History of Mathematics archive: Recent research paints a new picture of the debt that we owe to Islamic mathematics. Certainly many of the ideas which were previously thought to have been brilliant new conceptions due to European mathematicians of the 16th, 17th, and 18th centuries are now known to have been developed by Arabic/Islamic mathematicians around four centuries earlier. In many respects, the mathematics studied today is far closer in style to that of Islamic mathematics than to that of Greek mathematics. 109