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History of Mathematics: Contributions of the Arabs to Algebra and Arithmetic

Introduction

            Mathematics has three main departments namely geometry with the subdivisions of conic and trigonometry sections; geometry; and analysis with calculus and algebra sections. Trigonometry can be defined as “the art of computation of figures or the science of numbers” (Puritz, 2007, p10). Algebra, which belongs to the analysis department, can be defined as the “branch of mathematics which treats the relations and properties of quantity by means of letters and other symbols” (ibid, p.11). In the analysis process, arithmetic can be translated to algebra.

            Many scholars have contributed to making algebra and arithmetic successful. However, recent research has shown that we are highly indebted to the Arabic mathematics (Bháscara, 1817). They have contributed so much to the earlier days of arithmetic and algebra. It has been thought that the European mathematicians made earlier successful contributions to arithmetic and algebra. However, it has been found that these contributions originated from the Arabic mathematicians.

            Algebra is divided into two big sections namely classical and abstract/modern algebra. Classical algebra is older than abstract algebra having been developed around 4,000 years ago while the latter was formed around 200 years ago. The main contributors of algebra are known to be Arabs, Egyptians, Babylonians, Greeks, Diophantine’s, Hindus, and Europeans (Smith, 1958).

            The Egyptians mathematicians came up with ways of solving the algebra problems through a method known as “method of false position” (Kennedy, 1983, p. 289) which solved the problems through verbal means. The Babylonians’ methods were more improved but had to generate only positive numbers. In fact, they solved their problems through examples and no explanations were made as to why the formulas have been followed. For this reason, it was very difficult to grasp the concept.

            The Greeks did not at all recognize irrational numbers and for this reason, they only solved simple equations through geometric methods. The Diophantine’s development was also not well developed. He had a different formula for any sum he handled and ignored negative rational roots. It can be argued that no deductive structure was followed in the formulas (Greenes & Rubenstein, 2008).

           The Hindus succeeded the Greeks in the development of Mathematics. Although they included zero and negative numbers in their Algebra, they only gave the steps, which were to be followed for the computation but did not give explanations and proofs as to why they had to follow those specific proofs. However, they recognized that quadratic equations had two roots.

            To start with, the name algebra originated from the Greeks (Smith, 1958). The language used was mainly Arabic but they made noticeable contribution to mathematics up to date. They also contributed to the Hindu’s positional notations with much invention made by their own. They further came up with ways of solving cubic equations by geometric methods. Most of their algebra was rhetorical and that is the reason it is significant to date.

The Europeans contributed to the History of Algebra by just making changes on the formulas followed by the Arabs. They came with ways of solving cubic equations, and gave explanations to them (Kennedy, 1983). They also introduced the involvement of complex equations. The theory of equations was invented, in which the relationship between the roots and the coefficients of an equation had to be understood.

            Algebra grows out of arithmetic and for this reason, arithmetic is very important in the history of algebra. Acknowledgment of new numbers including zero, irrationals, complex numbers and negative numbers is crucial. This implies that it will be impossible to study about the history of Algebra without mentioning arithmetic. The two departments of mathematics are highly related.

The main steps through which Algebra developed are three. These are rhetorical, syncopated and symbolic steps. During the rhetorical stage, most of the work was verbally. Later, in the syncopated stage, abbreviated words were used while in the last stage, the symbolic stage, letters were used. The symbolic stage is the one everybody knows because it the recent one until now (Greenes & Rubenstein, 2008).

Even though there are several people who are thought to have contributed to the development of algebra and arithmetic, Arabic mathematicians are seen to have played the greatest role in the development (Kennedy, 1983). The essay below focuses on the contribution the Arabic mathematicians made in Algebra and Arithmetic.

Contributions of the Arabs to Algebra and Arithmetic

            The type of algebra and arithmetic used in today’s mathematics is highly related to that contributed by the Arabs than that contributed by the Greeks (Rāshid, 1994). Even though the Greeks were doing brilliant in developing the style to be followed in Arithmetic and Algebra, their development was stagnant after the sixteenth century. The Europeans took over the development from the Greeks to make it better. On the contrary, the modern mathematics developed by Arabs was a finished business and was not left to be finished by the Europeans.

            Many scholars who have been involved with the research of the history of mathematics do not acknowledge the contribution made by the Greeks. In fact, they avoid mentioning the name Greeks in their study or make very queer statements. An example of this is the remark made by Duhem in his study when he argues, “Arabic science only reproduced the teachings received from Greek science” (Totah, 2002, p. 13). This proves that they do not acknowledge the contributions they made.

            Arabs have played a big role in the invention of Arithmetic. It is through them that the numerals written from right to left came into being. Further, as mentioned earlier, the first person to introduce number Zero in 967 AD was an Arabian by the name Muhammad Bin Ahmad. Although the modern society does not take this introduction with the necessary weight, it should be noted that through it, great mathematical breakthroughs were made possible (Rāshid, 1994). Zero should be seen as a significant number because it is used in both Algebra and Arithmetic and as a result, this contribution should be recognized.

            In order to be able to solve Algebra by applying Arithmetic formulas, the Arabs came up with the principle of errors and fractions. Al-Khawarzmi, an Arabian mathematician introduced the first treatise. He was able to solve first as well as second degree quadratic equations, which are very useful in both engineering and science (Kennedy, 1983). To make the work easier of solving the first and second-degree quadratic equations, he introduced geometrical method, which is been used to date. This implies that even with the new discoveries, it remains that all the methods applied today have their origin from the Arab mathematicians.

            It was through Al-Khawarzmi inventions that we realize that quadratic equations should be seen to have two roots. Thabet Bin Qura, another Arabian mathematician, developed Al-Khawarzmi’s methods further. Through Thabet Bin Qura, it was realized that Algebra could be applied to geometry (Al-Daffa, 1978). This made it possible for the Arabs to be able to solve quadratic equations of both third and fourth degrees since the formulas used were developed and perfected.                

            Abul Wafa, an Arabian mathematician made great contributions to mathematics. He invented and succeeded in developing a geometry branch that involved higher degree problems in Algebra. His contributions had much influence in polyhedral theory (Bháscara, 1817). His workings are used to date because he had greatly thought of them and as a result, they are of the best applicable ways.

            Al-Karaki is an Arabian of the 11th century who is included in the list of the greatest Arabian mathematicians. He came up with very good books in Arithmetic and Algebra. In fact, his contribution is one among the greatest because it is in writing and can be referred to any time one needs clarification on anything. In his books, he has methods of calculating square roots, and a number of theories. The theories include intermediate quadratic equations, mathematical induction, and indices.      

            Arabs contribution to geometry was great and recognizable. It started with Euclid transition and Apolonios conic section of the Greeks. These contribution shave been preserved for the modern society. Arab mathematicians have made later discoveries in this field. Their contribution to the works of the Greeks is significant in that it makes the whole concept clear.

            Ibn al-Haitham wrote his book in Arabic. Roger Bacon later translated it into English because it was thought to be of great help up to date. Since it is in written form, it is easy to refer from. It has been noted that the book is solving geometric problems, which would have been hard to solve even today with the increased technology. This implies that nobody else has been able to solve similar problems because they have been proved difficult to solve even with the great inventions made to Mathematics (Bháscara, 1817).

            Although the Greeks developed the geometry of conic section, it was greatly developed by the Arabs. It was earlier not understood and in deed had calculations made but no explanations made as to why certain formulas were followed. It was only possible to follow the calculation from the books not able to get the concept. This means that it was very difficult to apply the concept communicated across in similar problems. However, with the development made by the Arabs on this issue, it was possible to understand the whole concept. Each computation was explained.

            The greatest achievement by the Arabs that has left a great impact in mathematics was by Abu Jafar Muhammad Ibn Muhammad Ibn al-Hassan (Al-Daffa, 1978). This contribution was named Nassereddine al-Tusi and its main idea was that of separating trigonometry from astronomy. This invention criticized Euclid's theory of parallels invented by the Greeks. It can be thus named as non-Euclidian geometry and it should be noted that, it had concrete calculations.           

            It is of paramount importance to note that the contribution of the Arabs in mathematics began with the Greek translations. The ruler (Caliph) of the Arabs, Caliph Harun al-Rashid who reigned in 786, influenced these translations. In order to ensure that people were encouraged to do the translations, he provided scholarships to the first translators (Totah, 2002). This motivated many translators who had the initial intentions of doing translations but later changed to inventions and developments.

The next ruler (Caliph) was al-Ma'mun, al-Rashid’s son. He was encouraged to continue with his father’s initiated project. In efforts to make the project more successful, he made the learning environment more conducive. He set up a house in Baghdad knows as the House of Wisdom from where interested mathematicians had to do the translations and research (Al-Daffa, 1978). The first Arab mathematicians to work from this house are Al-Kindi and the three brothers of Banu Musa. The famous translator named Hunayn ibn Ishaq worked from this house as well.

Only Mathematicians rather than language professionals who are poor in Mathematics made the translations from Greek to Arabic (Totah, 2002). The level of research influenced the translations at that time. The translations had to be done to fit the research of the time. This implies that if the translations were below the level of the research, developments were made and this marked the beginning of the contribution of the Arabs to mathematics.

    All the translations made initiated research to come up with improved methods of dealing with Arithmetic and Algebra problems. This means that not all earlier works made in different languages were translated. Only the works that deemed necessary to aid in research. Nevertheless, it is notable that the translations were made from all the other different languages. These include the works done by the Greeks, Diophantus, Egyptians, and Babylonians among others.

These translations initiated the contributions made by al-Khwarizmi, who led to the beginning of Algebra. His moves made mathematics move from the initial concept of the Greeks in which mathematics was essentially geometry. He needed mathematics to be seen as a broader concept rather than one-single topic, geometry.

Algebra needed to be seen as a unifying theory allowing geometrical magnitudes, irrational numbers as well rational numbers to be viewed as algebraic objects. These developments in mathematics should be viewed in a significant way because it is through them that mathematics is seen in a broader concept. In fact, they have created an urge for future developments in mathematics. It is also through algebra inventions that mathematics was applied in a new perspective, a way that had not been seen before (Bháscara, 1817).

Al-Khwarizmi's Successors and Their Contribution in Algebra and Arithmetic

The successors of Al-Khwarizmi were encouraged to see his work developed as he would have wished. They “undertook a systematic application of arithmetic to algebra, algebra to arithmetic, both to trigonometry, algebra to the Euclidean theory of numbers, algebra to geometry, and geometry to algebra” (O'Connor & Robertson, 1999). These applications led to geometric equations construction, new elementary numbers theory, numerical equation solutions, numerical analysis, combinatorial analysis and polynomial algebra.

            The first notable mathematician was al-Mahani, who was born in 820, and came into being forty years after al-Khwarizmi. His main contribution was the idea of geometrical problems reduction for example duplicating cube to algebra problems. The next scholar was Abu Kamil of born in 850, who linked the developments made in algebra of al-Khwarizmi to those of al-Karaji. Although he did not use symbols in his algebra equations, he recognized powers in words. This shows that he had the idea of what should be done but had not researched thoroughly on it or the technology in use then did not allow his to understand the whole concept fully. Symbols were used in algebraic equations of the Arabs later around the 14th century. The first notable Arab mathematicians to use it are al-Qalasadi and Ibn al-Banna in the 15th century (Totah, 2002).     

            Al-Karaji was born in 953 and his main contributions are notable to date. He freed algebra from geometry and replaced them with arithmetical operations. These arithmetic operations are in mathematics to date. He is the first mathematician to define monomials such as x, x1, x2, and x3 etc and 1/x---1/x5 etc (Rāshid, 1994). He also came up with rules for the products generated from any two of the monomials. He started an algebraic school, which has been in use for many years after.

            Al-Samawal was born in 1130 and has been an associate of al-Karaji's school since 200 years after it was formed. He was able to come up with a precise description of algebra. He described algebra as the art of “operating on unknowns using all the arithmetical tools, in the same way as the arithmetician operates on the known” (O'Connor & Robertson, 1999).

            Omar Khayyam was born in 1048 and classified cubic equations by use of geometric equations. He promised to give further classifications later as he wrote, “If the opportunity arises and I can succeed, I shall give all these fourteen forms with all their branches and cases, and how to distinguish whatever is possible or impossible so that a paper, containing elements, which are greatly useful in this art will be prepared” (O'Connor & Robertson, 1999). It shows his commitment in simplifying algebraic situations.

           The next Arab mathematician was Sharaf al-Din al-Tusi who was born in 1135. He initiated the application of algebraic geometry as he introduced the study of curves using algebraic equations. He did not follow the developments from al-Karaji's school but rather developed Khayyam's algebra application.

            The three Banu Musa brothers taught another Arab mathematician notable for his contribution in algebra. He was Thabit ibn Qurra, born in 836. His main contributions were made in the number theory. He came up with a beautiful theorem, which made it possible to find a pair of amicable numbers. The two numbers were supposed to be related in that each was the summation of the proper divisor of the other (Al-Thaqāfīyah, 1977).

Thabit ibn Qurra’s contribution was improved further by Al-Baghdadi of 980 although by making a slight variant of the earlier findings. However, al-Haytham of 965 made the fist attempt in classifying perfect numbers, numbers which equaled to the summation of the proper divisors to be of the form 2k-1(2k - 1) with 2k - 1 being the prime (Rāshid, 1994). Further, al-Haytham came up with Wilson’s theorem that p being a prime number, it follows that 1+ (p-1) should be divisible by p. it is however not clear whether he could prove his argument. The theorem is known as Wilson’s theorem because it was later in 1770 argued by Waring that John Wilson knew how to prove the result. Nonetheless, Lagrange proved it in 1771. Since it was found to be a true argument, the contribution by Al-Haytham remains crucial.

            Al-Farisi who was born in 1260 proved Thabit ibn Qurra's theorem although by introducing new concepts of combinatorial and factorisation methods. He also gave pairs of amicable numbers although they were known earlier before his birth. He invented further on them and made explanations on them to make them understood by different people even the non-mathematicians.

Different Arithmetic Counting Systems

            There were three types of arithmetic counting in early mathematics applied by the Arabs. The first one was finger-reckoning arithmetic. This system involved the counting of fingers but with written numerals. The business community mostly used it as they did their daily operations. Abu'l-Wafa of 940 is also remarkable for his use of this counting method. He has written a number of treatises by use of this system (Totah, 2002). Although he was familiar with the Indian numerals, he did not find them to be of help to the business community. As a result, he preferred the use of the finger-reckoning counting.

            The second system was known as Sexagesimal system. In this method, Arabic alphabet letters denoted the numerals. Although its origin was by the Babylonians, the Arabs developed it to be useful in their systems. The Arabic mathematicians used this method in astronomical work.   

            The third method was Indian numeral system. This method involved the use of numerals, fractions and decimal places. The numerals used by the Arabs were from India and with no standard and as a result, there were different standards used in different parts of Arab. There was the use of dust boards to help in the calculations, which required rubbing as the computations progressed (Rāshid, 1994). In order for this system to fit in the Arab community, developments had to be made. The most noticeable development was made by al-Uqlidisi, born 920, as he introduced the use of pen and paper to replace the use of the duct boards. Al-Baghdad also did some noticeable contributions as he improved the decimal place use to make clear meaning to the users.

            The third system used was very helpful to the Arab Mathematicians. It allowed major advancements to be made on the numerals. It enabled roots extraction by great Arab mathematicians mainly Omar Khayyam born in 1048 and Abu’l- Wafa. The main notable development made on this system was as a result of binomial theorem by al-Karaji. Al-Kashi who was born in 1380 developed the decimal fractions as well as real numbers like π (pie). The impact of his improvement on decimal places was great to the extent that he has been considered the inventor of decimals (Al-Thaqāfīyah, 1977). The fact that he came up with an algorithm applicable to calculating nth roots made him a special contributor in the topic. His developments are in use even to date.

            The Arab Mathematicians did not contribute to number systems, number theory and algebra only but also had major contributions made to mathematical astronomy, trigonometry and geometry. Ibrahim ibn Sinan of 908 made a significant contribution in integration. These were revival figures in geometry. Omar Khayyam crowned it all by combining both approximation theories with trigonometry to solve algebraic equation by use of geometrical methods.

            Trigonometrical and geometrical research conducted by Arab Mathematicians was motivated by geography, time keeping and astronomy. For instance, Ibrahim ibn Sinan with the help of Thabit ibn Qurra, his grandfather, studied curves necessary in sundials construction. Abu Nasr Mansur and Abu'l-Wafa used spherical geometry in astronomy and further used other formulas involving tan and sin. Biruni of 973 applied sin formula in latitudes and longitudes calculation as well as in astronomy. Additionally, the projection of a hemisphere onto a plane by al-Biruni was motivated by geography and astronomy. Most of the Arabic mathematicians applied trigonometric functions in astronomy studies (Totah, 2002).

Conclusion

            Arabic mathematicians have a played a major role in Algebra and Arithmetic. The fact that their contribution is felt up to date is enough evidence that their contribution was based on real situations. It has however been difficult for the Arabs to prove their inventions mostly in formulas but with the current and advanced technology, their inventions have been proved true.

            It has been noted that most of the Algebra and Arithmetic formulas applied in today’s mathematics have their origin in Arabic mathematics. Developments and improvements have been done on their inventions ton match with current demand. It is not possible to have formulas, which have no explanations in today’s world and be accepted in mathematics. As a result, explanations have to be done and this is the main improvement made in Arab mathematics.  


References

Al-Daffa', A. A. (1973). The Muslim Contribution to Mathematics. London: Taylor & Francis publishers.

Al-Thaqāfīyah, M. T. (1977). Islamic and Arab Contribution to the European Renaissance. Egypt: General Egyptian Book Organization publishers.

Bháscara, B. (1817). Algebra, with Arithmetic and Mensuration. U.K. J. Murray publishers.

Greenes, C. E. & Rubenstein, R. N. P. (2008). Algebra and Algebraic Thinking in School Mathematics. New York: National Council of Teachers of Mathematics.

Kennedy, E. S. (1983). Studies in the Islamic Exact Sciences. Beirut. American University of Beirut press.

O'Connor, J. J. & Robertson, E. F. (1999). Arabic Mathematics: Forgotten Brilliance? Retrieved 03 April 2011 from http://www-history.mcs.st-andrews.ac.uk/HistTopics/Arabic_mathematics.html

Puritz, C. (2007). Algebra and Arithmetic. New York: Parkwest Publications.

Rāshid, R. (1994). The Development of Arabic Mathematics: Between Arithmetic and Algebra. New York: Springer publishers.

Smith, D. E. (1958). History of mathematics. London: Courier Dover Publications.

Totah, K. A. (2002). The Contribution of the Arabs to Education. New Jersey: Gorgias Press LLC.

 

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