On Algebra

Prof. Ibrahim B. Syed

President

Islamic Research Foundation International, Inc.

Louisville, Kentucky

E-Mail: President@irfi.org

Website: WWW.IRFI.ORG

Every student who goes to high school in every country in the world learns Algebra.

Algebra is a  branch of mathematics in which symbols, usually letters of the alphabet, represent numbers or members of a specified set and are used to represent quantities and to express general relationships that hold for all members of the set.

 A set together with a pair of binary operations defined on the set. Usually, the set and the operations include an identity element, and the operations are commutative or associative.

The question is why learn Algebra?  Careers today demand skills like problem solving, reasoning, decision-making, and applying solid strategies etc. Algebra provides one with a wonderful grounding in those skills - not to mention that it can prepare one for a wide range of careers. Colleges require it and some employers demand it. A required tool for many technical fields, Algebra is a way to increase mathematical skills. Algebra is a great mental workout and it's the only path to moving on to more advanced mathematics. The Basics of Algebra consist of taking the real situation and turn it into an equation. Secondly it makes us find out what the unknown is. By applying Algebraic reasoning one can solve real-life situations. Hence recently the SAT (Scholastic Aptitude Test)  has included  questions on Algebra-II  for high school students aspiring to go to reputable and Ivy League Colleges.

Historically, algebra is the study of solutions of one or several algebraic equations, involving the polynomial (an expression of two or more terms) functions of one or several variables. The case where all the polynomials have degree one (systems of linear equations) leads to linear algebra. The case of a single equation, in which one studies the roots of one polynomial, leads to field theory and to the so-called Galois theory. The general case of several equations of high degree leads to algebraic geometry, so named because the sets of solutions of such systems are often studied by geometric methods.

Modern algebraists have increasingly abstracted and axiomatized the structures and patterns of argument encountered not only in the theory of equations, but in mathematics generally. Examples of these structures include groups (first witnessed in relation to symmetry properties of the roots of a polynomial and now ubiquitous throughout mathematics), rings (of which the integers, or whole numbers, constitute a basic example), and fields (of which the rational, real, and complex numbers are examples). Some of the concepts of modern algebra have found their way into elementary mathematics education in the so-called new mathematics.

Some important abstractions recently introduced in algebra are the notions of category and functor, which grew out of so-called homological algebra. Arithmetic and number theory, which are concerned with special properties of the integers e.g., unique factorization, primes, equations with integer coefficients (Diophantine equations), and congruences are also a part of algebra. Analytic number theory, however, also applies the non algebraic methods of analysis to such problems.

PRINCIPLES OF MODERN ALGEBRA

Modern algebra is yet a further generalization of arithmetic than is classical algebra. It deals with operations that are not necessarily those of arithmetic and that apply to elements that are not necessarily numbers. The elements are members of a set and are classed as a group, a ring, or a field according to the axioms that are satisfied under the particular operations defined for the elements. Among the important concepts of modern algebra are those of a matrix and of a vector space.

21^st Century is the age of Information Technology (IT) and modern computers that  have become ubiquitous and indispensable in everyday life, are the foundations of IT.

Abu Ja'far Muhammad ibn Musa al-Khwarizmi, (780 – 850 CE), is the grandfather of computer science and the father of Algebra. He was the popularizer of Arabic numerals, adopter of zero (the symbol, that is) and the decimal system, astronomer, cartographer, in briefs an encyclopedic scholar.

The modern word algorithm is derived from his name, al-Khwarizmi, the best mathematician of his age and thanks to his book, al-Kitab al-mukhtasar fi Hisab al-jabr w'al-muqabala, (a book showing how to solve equations and problems derived from ordinary life) which means "The Compendious Book on Calculation by Completion and Balancing", was later mangled into algebra, was the first written text on the subject. In al-Khwarizmi's time, algebra was a practical system for solving all kinds of problems "in cases of inheritance, contracts, surveying, tax collection, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computations, and other objects of various sorts and kinds are concerned."   Al-jabr was about removing the negative terms from an equation, while al-muqabala meant "balancing" the values of an equation across an equals sign.

Al-Khwarizmi systematized and corrected Ptolemy’s research in geography and astronomy/astrology, using his own original findings. He supervised the work of 70 geographers to create a map of the then "known world". Amazingly this map  of the "known world" shows the pacific coast of South America-about 700 years before Columbus discovered America.   In 1140 Robert of Chester (who read mathematics in Spain) translated the Arabic title into Latin as Liber algebrae et almucabala, then ultimately gave its name to the discipline of algebra. The Spanish Jew, John of Seville, produced another Latin version.

When his work became known in Europe through Latin translations, his influence made an indelible mark on the development of science in the West: his algebra book introduced that discipline to Europe "unknown till then” and became the standard mathematical text at European universities until the 16^th century. In the 16th century it is found in English as algiebar and almachabel, and in various other forms but was finally shortened to algebra. He is one of the Muslim scholars who laid the foundations for Europe’s Renaissance and the Scientific Revolution. He also wrote on mechanical devices like the clock, astrolabe, and sundial. His other contributions include tables of trigonometric functions, refinements in the geometric representation of conic sections, and aspects of the calculus of two errors.

In addition to an important treatise on Astronomy, Al-Khwarizmi wrote a book on astronomical tables, which were also translated into European languages and, later, into Chinese.

Several of his books were translated into Latin in the early l2th century by Adelard of Bath and Gerard of Cremona. The treatises on Arithmetic, Kitab al-Jam'a wal-Tafreeq bil Hisab al-Hindi, and the one on Algebra, Al-Maqala fi Hisab-al Jabr wa-al-Muqabilah, are known only from Latin translations. Introduction of Arabic numerals provided a pivotal advance over the cumbersome Roman numerals. This development of a more convenient number system assisted progress in science, accounting and bookkeeping. Key to this was the use of the number zero, a concept unknown to the West. The use of this number system (Arabic numerals) spread throughout the Muslim world over the next two centuries, assisting the development of science. The Arabic numeral system was first mentioned in Europe around 1200 CE, but Christian adherence to the Roman system hindered its use and introduction. It was only fully accepted in Europe after it was adopted by the Italian traders in the Renaissance of the 16th century, who followed the practice of their Arab trading partners.

The idea of the college was a concept which was borrowed from Muslims. The first colleges appeared in the Muslim world in the late 600's and early 700's. In Europe, some of the earliest colleges are those under the University of Paris and Oxford they were founded around the thirteenth century. These early European colleges were also funded by trusts similar to the Islamic ones and legal historians have traced them back to the Islamic system. The internal organization of these European colleges was strikingly similar to the Islamic ones, for example the idea of Graduate (Sahib) and undergraduate (mutafaqqih) is derived directly from Islamic terms.

MUSLIM IMPACT ON EUROPE                                                                                                 

During the Middle Ages the Islamic World had a very significant impact upon Europe, which in turn cleared the way for the Renaissance and the Scientific Revolution. In the Medieval age, Islam and Muslims influenced Europe in a number of different ways. One of the most important of these subjects was Science.

Ever since Islam was born, Muslims had made immense leaps forward in the area of Science. Cities like Baghdad, Damascus, Cairo and Cordoba were the centers of civilization. These cities were flourishing and Muslim scientists made tremendous progress in applied as well as theoretical Science and Technology. In Europe, however, the situation was much different. Europe was in the Dark Ages. It had no infrastructure or central government. To the Muslims, Europe was backward, unorganized, carried no strategic importance and was essentially irrelevant. This considering the time period was in fact true. Nevertheless the Catholic Church (which at the time was the strongest institution in Europe) successfully convinced Christian Europe that the Muslims were infidels. This caused Europeans to think that Muslims were culturally inferior to Europe and thus Europe was unable to benefit from the new scientific discoveries being made in the Islamic lands before the 1100’s. By doing this Europe kept itself in the Dark Ages while from China to Spain Islamic Civilization prospered. During the Crusades there was limited contact between Muslims and Christians and not much was transferred. As A. Lewis explains, "The Crusaders were men of action, not men of learning". The real exchange of ideas which led to the Scientific revolution and to the renaissance occurred in Muslim Spain.

Cordoba was the capital of Muslim Spain. It soon became the center for intellectual enlightenment  and learning for the entire Europe. Scholars and students from various parts of the world and Europe came to Cordoba to study. The contrast in intellectual activity is demonstrated best by one example: ‘In the ninth century, the library of the monastery of St. Gall was the largest in Europe. It boasted 36 volumes. At the same time, that of Cordoba contained over 500,000’.            …………..          ( To be continued )