Once they had completed their conquests, the Arabs settled down to build a civilization and a culture. They became interested in the arts and

Transcription

1 The Islamic World

2 We know intellectual activity in the Mediterranean declined in response to chaos brought about by the rise of the Roman Empire. We ve also seen how the influence of Christianity diminished Greek scholarship. If any scholarship remained in Alexandria, it was eventually extinguished by the rise of Islam. Mohammad untied various nomadic Arab tribes in a jihad that continued for a century after his death in 632. Lands from India to Spain, including North Africa and southern Italy, came under Arab control. In 755 the empire split into two independent kingdoms: the eastern kingdom had its capital in Baghdad, and the western kingdom was ruled from Cordoba.

3 Once they had completed their conquests, the Arabs settled down to build a civilization and a culture. They became interested in the arts and sciences, and the capitals in both halves of the empire attracted and supported scholars. In Baghdad the caliph established a House of Wisdom, similar to the former Museum at Alexandria, with an academy, a library, and an astronomical observatory. There Arabs scholars collected and translated as much Greek learning as they could find.

4 Arab scholars improved translations of Greek manuscripts and wrote commentaries on them. In many cases, it is only through these Arabic translations and commentaries that European scholars later gained access to ancient Greek discoveries. An Arabic translation of Euclid s Elements

5 Chapter 9 begins by describing several methods the Arabs used for multiplying, dividing, and extracting roots from numbers using the base ten system developed by the Indians. The grating method is especially nice. The figure below shows that 654 times 321 is 209,934. How and why does this method work?

6 Muhammad al Khwarizmi was one of the more notable scholars at the House of Wisdom. The word algebra comes from the title of one of his works, Al-jabr w al muqabala, which means restoration and completion. In the equation x 2 3x + 6 = 2x, the 3x is restored when we write x = 3x + 2x, and we complete the right hand side when we write x = 5x.

7 In Restoration and Completion, al Khwarizmi distinguishes three types of simple equations and three types of compound equations. (DQ) How would we describe these types using modern notation? Simple: squares equal to roots, squares equal to numbers, roots equal to numbers Compound: squares and roots equal to numbers, squares and numbers equal to roots, squares equal to roots and numbers

8 (DQ) Why did al Khwarizmi work with different types of equations rather than writing all equations using the same form (such as ax 2 + bx + c = 0)?

9 Let s work through al Khwarizmi s solution to the problem one square, and ten roots of the same, amount to thirty-nine dirhems. To what modern equation does this problem correspond? The solution is this: you halve the number of roots, which in the present instance yields five. This you multiply by itself; the product is twenty-five. Add this to thirty-nine; the sum is sixty-four. Now take the root of which, which is eight, and subtract from it half the number of the roots, which is five; the remainder is three. This is the root of the square which you sought for; the square itself is nine. Why does this solution work?

10 (DQ) Use al Khwarizmi s method to solve the problem one square and five roots amount to twenty-four dirhems. To what modern equation does this problem correspond? Draw a picture that shows geometrically why al Khwarizmi s solution works.

11 What equation is implied by Problem 9.2? Problem 9.2: A square and twenty-one in numbers are equal to ten roots of the same square. (DQ) Use modern algebraic notation to show why the solution to Problem 9.2 works.

12 Al Khwarizmi also made important steps in the arithmetization of algebra (doing arithmetic with unknowns). He showed how to multiply two numbers, then extended the process to unknown things by analogy. The process he used for numbers also worked for negative quantities. (DQ) Show geometrically why al Khwarizmi s procedure for computing 10 minus 1 times 10 minus 1 requires the product of two negatives to be positive.

13 (DQ) Do section 9.2 exercises 3 and 4.

14 In the Book of Rare Things in the Art of Calculation, Abu Kamil solved indeterminate equations. He did so before Arab mathematicians knew the methods developed by Diophantus. Al Karaji made important contributions to the advancement of algebra by working with powers of the unknown greater than three. He continued the arithmetization of algebra begun by al Khwarizmi by extending arithmetic to powers of the unknown. In particular he showed how an expression such as 4x 4 +2x 2 could be added, subtracted, or multiplied by an expression such as 3x 3 +x+1.

15 Al-Khayyami s On Completion and Restoration is seen by some as the culmination of medieval algebra in the Middle East. It is the most rigorous of all the algebra texts introduced by Chapter 9, and the proofs require fluency with the geometry of Euclid and Apollonius. Since al-khayyami proved his results geometrically, he became concerned with what it meant to add, for example a cube to a square (this was the problem of homogeneity ). + =?

16 We ve seen how al Kwharizmi solved the equation one square, and ten roots of the same, amount to thirty-nine dirhems (and others like it) by converting the square and ten roots to geometric objects of the same dimension (rectangles in this case). Al-Khayyami did similar things with objects of higher dimensions. 5x x 2 5x

17 To solve cubic equations, al-khayyami needed to find two mean proportionals between two given numbers. Recall that if a/x = x/y = y/b, then x and y are mean proportionals between a and b. Al-Khayyami found mean proportionals by means of intersecting parabolas.

18 Our text demonstrates how al-khayyami solved general cubic equations such as cube and sides equal to a number (x 3 + ax = b). In this case he let a = p 2 and b = p 2 q, and created a geometric construction which led to x being the length of a line segment formed from the intersection of two parabolas. Al-Khayyami s solutions to cubic equations include elegant applications of conic sections, but they are not practical since the required conic sections can be difficult to construct.

19 On the bottom of page 258 in our textbook, we read that al- Khayyami disliked the purely formal nature of ratios as presented in Euclid s Elements. What does this mean? r < r r r r

20 r < r r because r r r > but <

21 Al-Khayyami preferred to compare magnitudes using the Euclidean algorithm. The example on page 258 considers magnitudes of size 28 and 10. The Euclidean algorithm gives 28 = = = 4 2 with sequence 2, 1, 4 of partial quotients. The ratio of any other pair of magnitudes that share the same sequence of partial quotient is equal to the ratio 28/10. This meant that the ratios could be dealt with numerically.

22 So far we ve seen how the arithmetization of algebra began with al Khwarizmi and was extended by al Karaji. The details left out by al Karaji were filled in by al Samaw al in his Shining Book of Calculations. In particular he gave the first examples of polynomial division. Al Samaw al also introduced decimal (as opposed to the more common sexigesimal) notation centuries before its independent invention in Europe.

23 Arab mathematicians also made contributions to geometry, particularly in the accurate construction of conic sections. Why were they so interested in conic sections?

24 Why does the author of our textbook claim that trigonometry as we study it today was largely an invention of the Arab and Persian mathematicians of the Islamic era.

25 Do section 9.4 exercise 5.

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