The presentation given today by Kristi Wallace, after Dr. Barsky's commentary on the mathematician of last Friday, Dr. Paul Erdos, seemed to generate an overall theme of the association of algebra with geometry that the mathematicians of Islam used. The topics covered consisted of the Bedouins, the capture of Alexandria, Mohammed's successors, the caliphs, Al-mansur, Harun Al-Rashid, Al-ma'mun, Bayt Al Hikma (the House of Wisdom), some of the accomplishments of the mathematicians of Islam, Al-Khwarizmi, the Condensed Book on the Calculation of Al-Jabr and Al-Muquabala, Al-Khwarizmi and three types of quantities, Al-Khwarizmi and six types of equations, Omar Khayyam, and the Treatise on Demonstrations of Problems of Al-Jabr and Al-Muqabala.
Among these topics, the six types of equations of Al-Khwarizmi had interested me the most. Particularly in the fourth type of equation, which consisted of squares and roots being equal to numbers, ax^2+bx=c, did I find exceptional interest. The demonstrative problem given in class stated "What must be the square which when increased by ten of its own root amounts to thirty-nine?". Al-Khwarizmi took one half of the roots, which equaled five, and multiplied it by itself equaling twenty-five. He then added thirty-nine equaling sixty-four, took the root of sixty-four equaling eight, and subtracted one half of the number of roots from the root of sixty-four equaling three. Kristi then showed the algebraic example of his as being (x^2+10x+25=39+25) => (the square root of (x+5)^2=the square root of 64) => ((x+5)=8) => (x=8-5=3). She then showed x^2 could be thought of as equaling the area of a square, 10x as equaling the sum of the area of two rectangles attached to the preceding square, and thirty-nine as a given amount. This displayed an example of how the Islamic mathematicians would link geometry to algebra. In my interpretation of this, I thought of the possibility that they knew geometry before they knew algebra. Mathematicians of Islam also may have learned their geometric background from the ancient Greeks, considering the capture of Alexandria. They also may have learned some of the Greek algebraic techniques as well. They seemed to have expanded greatly in the area of algebra. By using geometry to back up their algebra, they knew they took the right course of action. They also may have taken some knowledge from the Indian mathematicians. I also feel that they may have found their algebra by looking for a more concise way to write down what they wanted to geometrically. In other words, it would be quite difficult to write down an extremely long algebraic equation using geometric shapes to explain it. This would take a tremendous amount of time, when it would be much easier to explain them algebraically.
The mathematician of the day was Dr. Paul Erdos, who died on Friday Sept. 20, 1996 at the ripe age of 83. Once again I am encouraged to read of the humanness of a great mathematician. Delving into the history of Islam, we learned that caliph al Mansur founded his new capital in Baghdad, which became the new center for mathematics -- replacing Alexandria. We then moved on to al Khwarizmi and Omar Khayyam.
I liked the organization of the three types of quantities that al Khwarizmi listed. The example also helped, as I saw how completing the square produces a number whose root in turn makes it possible to solve the equation. I will have to take the cubic example down to the math lab to have it explained to me again.
Today's discussion began with Dr. Barsky paying tribute to one of the great mathematicians of the twentieth century, Paul Erdos, who died this past week. Erdos' love of mathematics was so great that he eschewed most material luxuries in order to devote his life entirely to mathematics. Erdos did a great deal of work in number theory, and was acutely involved in the founding of discrete mathematics, which is the basis of computer science. The awe in Dr. Barsky's voice spoke volumes about Erdos' contributions to mathematics.
Student lecturer Kristi Wallace then assumed center stage to speak about the mathematics of Islam. Ms. Wallace began with a brief discussion of the beginnings of Islam, including a short biographical sketch of Mohammed. She then quickly discussed a number of Islamic mathematicians and their contributions, focusing mostly on al-Ma'mun, al-Khwarizmi and Omar Khayyam.
Al-Ma'mun established a research institute called "Bayt al Hikma," which, in English, means "house of wisdom." Scholars working under the patronage of Al-Ma'mun translated the works of Euclid and other Greek mathematicians into Arabic. Ms. Wallace stated that the Islamic mathematicians were primarily responsible for the development of the decimal place system, including decimal fractions. The Islamic mathematicians also systematized the study of algebra, and made great improvements in trigonometry.
Al-Khwarizmi wrote an algebra text and did a great deal of work with quadratics. He spoke of how a quadratic has three parts: the square, the root, and the absolute number, which is the constant term. Al-Khwarizmi placed quadratics into six categories, including bx=c, in which the coefficient of the squared term is zero. We might think of some of these categories as redundant, but since the Islamic mathematicians didn't deal with negatives, "the square plus the root equals the absolute" was not the same to them as "the square equals the root plus the absolute." Ms. Wallace then demonstrated how al-Khwarizmi solved quadratics by completing the square.
Katz talks about how al-Khwarizmi viewed the root not as the side of a square, but rather as just a number that would satisfy certain conditions. I thought this was interesting because up until now, everything has been viewed in geometric terms. But al-Khwarizmi appears to be doing operations that are purely algebraic.
Ms Wallace went on to talk about Omar Khayyam, centering her attention on his method for solving cubic equations by finding the intersection of conic sections. She demonstrated the method of solving x cubed plus cx equals d by finding the intersection points of a circle and a parabola. The discussion of Omar Khayyam as a mathematician got my attention because I have always thought of Khayyam as a poet. This intersection of the arts and the sciences is rather uncommon. It is my experience that most students who are strong in one tend to be weak in the other. For instance, few students have a major in math and a minor in art or music.
Kristi started her presentation by beginning at the end of the fifth Century. As she approached the ninth Century, Kristi talked of what was happening in the world of mathematics. She concentrated the majority of her presentation on Al-Khwarizmi and his work on Al-Jabr, and on Omar Khayyam and his work involving third degree equations.
The types of equations Al-Khwarizmi came up with were an interesting part of the presentation. Kristi listed Al-Khwarizmi's six types of equations which showed how equations, those with a degree of less than the third degree, could be written. The equations dealt with the combinations of the following three elements; the square (x^2), the root (x), and a number.
I am ecstatic to finally learn where the term "algebra" came from.
In fifth century a new Islamic civilization came into being and captured lots of the Middle East. An early ruler, Harun al-Rashid, began to build a new center of mathematics in Baghdad, which can be compared to Alexandria. He established a library in Baghdad, like the one in Alexandria. His successor, al-Ma'mun, established a research institute called Bayt al Hikma, House of Wisdom. During the ninth century many Greek mathematical texts were translated into Arabic. Using Babylonian and Greek Mathematics, Islamic mathematicians created a new algebra, which used the notion of geometric proofs to justify their algebraic rules.
An important Islamic mathematician was al-Khwarizmi, who wrote and algebraic text called The Condensed Book on the Calculations of al-Jabr and al-Mugabala. In it he classified quantities into three types: squares, roots of the square and absolute numbers. He also classified equations into six types: square=roots, squares=number, roots=numbers, squares and roots=numbers, squares and numbers=roots, roots and numbers=squares. All of his equations were written out, for there was no use of symbols yet.
Omar Khayyam was a well known mathematician and poet during the ninth century. He used geometric solutions to show things like intersecting conics solved all cubic equations with possible roots. An example from his book, Treatise on Demonstrations of Problems of al-Jabr and al-Mugabala, was "a cube and sides are equal to a number." The equation is x^3 +cx = d, where all three quantities can be thought of as being the volumes of solids. From what I was able to get, al-Khayyam created the equation (BC)(AB)^2=d, where AB equals the length square root of c. Is this equation something he just made up, or is there some rule that gives you it? I got confused over this point, because I didn't understand where this equation came from. I could see how he drew the graph and then used equations for a parabola and circle to get the original equation.
Kristi's lecture gave a brief overview of the origin and development of the Islamic empire and its contributions to the development of mathematics, particularly the development of algebra. The Moslem rulers of the period, unlike the earlier Romans, encouraged the study of mathematics and created libraries filled with manuscripts of Greek and Babylonian mathematics. These were in turn translated, studied and improved upon. The Moslem mathematicians took much of the earlier mathematics and translated it into algebraic terminology, but apparently still used the Greek technique of using geometric arguments to prove their theorems. The work of two mathematicians from the period was featured in the presentation, the first being al-Khwarizmi.
Al- Khwarizmi classified three types of quantities and six types of equations, and developed systematic methods of solving each type of equation. An example from his text was presented in which he found a solution to a quadratic equation using the Babylonian method of completing the square. The difference now is that the distinction between number and magnitude in the Greek mathematics has been done away with. The solution is not viewed as a magnitude, or the side of a square, but rather as a numeric value satisfying certain conditions, namely the conditions of the equation.
The second mathematician discussed was Omar Khayyam, who developed a systematic method of finding the positive roots of cubic equations by drawing intersecting conics and solving for the point of intersection. This method was a little hard to follow on the board, but the general idea is clear and should be made clearer by the homework assignment