The Genius of Muhammad Bin Musa Al-Khwarizmi

The mathematics of the early Islamic period gave rise to many Arabs who were eager to absorb, improve and transmit the culture and science of all the countries that fell under their sway. They also wanted to learn all they could from ancient, intellectual cultures. The Arabs were able to culminate the various ideas that were coming in from around the world and achieved much success in their academic endeavours. They were committed to excellence and promoted further studies and investigations in math, chemistry, algebra, trigonometry and astronomy. Their goal was to give a rebirth to human knowledge. In Baghdad, the Kahalifah, Al-Mamun, opened an academic institute called "Darul Hikma", the "House of Wisdom". Those who studied and researched there did a great service to the progress of civilization by writing many textbooks on arithmetic, algebra, trigonometry and many other subjects. Between the years 750 C.E. to 1450 C.E., Arabs wrote approximately five hundred treatises on various subjects. Overall, they had great fame for their advancement in math. Atiq Ul Haque describes the situation to be that "... the most outstanding contribution of the Arabs to human knowledge was in the field of mathematics." Out of many Arabs who dominated these fields of study at the time, the Persian, Abu Abdullah Muhammad Ibn Musa Al-Khwarizmi, has the greatest honour and distinction. He produced many works that were translated into Latin and were the basis and foundation of mathematical concepts that were built upon by future mathematicians.

Al-Khwarizmi was born in 770 in a small town south of the river Oxus in present Uzbekistan. His parents migrated south to Baghdad and it was here that he spent most of his childhood. His father was a very educated man who stressed the importance of religious awareness and educational excellence in his son’s life. Despite being an influential Central Asian mathematician and philosopher, little is known about Al-Khwarizmi’s personal life. His contributions to the fields of astronomy, geography, and mathematics flourished under the rule of caliph Al-Mamun and his "House of Wisdom" in Baghdad in which he did his studies and research. This institution reached a major milestone with the work of Al-Khwarizmi. Al-Khwarizmi is described as "…one of the greatest minds of Islam, he influenced mathematical thought to a greater extent than any other medieval writer." His works played a very important and influential role in the history of mathematics because they introduced the Western European world to Indian numerals, Arabic algebra and arithmetic.

His works on arithmetic were translated into Latin, under the name Algorimus, and were continually used until the fourteenth century. The word ‘algorithm’ is derived from his name and also happened to be a medieval European term for decimal numeration. His arithmetic explained the Hindu system of numeration and introduced the Arabic numbers to the West. The number writing system used by Arabs used tens, hundreds, and thousands and this became the basis for modern mathematics. This led to the use of nine Arabic numerals, along with the zero sign, as the most useful and most basic tool in science. His book on arithmetic was one of the resources by which Western Europe became familiar with the decimal point and was the first book ever to explain the operation of the decimal numbers. Although it is lost in its original Arabic text, a Latin translation still exists.

The title of the translation, Algorithmi de numero Indorum¸ added the word algorithm, a Latininzed version of the author’s name, into mathematical terminology. An algorithm is a "step-by-step systematic procedure used to accomplish an operation." The information on algorithms that Al-Khwarizmi gathered were very useful to the Arabs in solving problems and helped others to whom this knowledge was dispersed. McLeish describes his contributions by saying that, "Arabic numerals and algorithms, presented by Al-Khwarizmi, made arithmetic so simple that one could throw away all auxiliary aids, such as the abacus, and work directly with the numbers themselves. This made it easier to understand and comprehend the abstract nature of numbers.

A whole chapter of Al-Khwarizmi’s book was devoted to his work with fractions. He introduced the work "kasra", meaning broken numbers, to represent fractions. The same connection between the words "fractions" and "fracture" in English exists between "kasra" and the Arabic counterpart for fracture. He described how each single unit can be divided in smaller pieces by each number of the decimal system. He went on to give a name to each fraction that further described them.

Al-Khwarizmi also introduced a method of multiplying fractions together. He would set up a table in which he would multiply all the denominators to find the common denominator. Then he would express the whole numbers as fractions to these denominators. Next he would multiply numerator by numerator and denominator by denominator. In another section of his book he deals with multiplying mixed numbers. McLeish illustrates an example of how he solved the problem of multiplying 3 ½ by 8 ¾. This example shows how hard these new methods must have been for people trained in classical mathematics. Only through their hard efforts could these techniques be mastered and used to further investigate math.

His book on algebra, Hisb al Jabr wal Muqabula, was translated into Latin by Robert Chester in the twelfth century. Commonly referred to as, The Book of Summary in the Process of Calculation for Completion and Equation, it is said to be the place where the science of algebra originated. This book served as the principal mathematical textbook in the European universities. In it, Al-Khwarizmi dealt with equations, algebraic multiplication and division, measurement of surfaces and other questions. The reason that Al-Khwarizmi wrote this book on algebra was to help scholars solve complex yet practical problems. For example, his algebra made it simpler to calculate the division of an estate among the legitimate heirs in accordance with Islamic law. But Al-Khwarizmi, being a true mathematician, wasn’t simply concerned with such mundane matters. He also was interested in the theoretical aspects of algebra as the science of equations.

The concept of jabr, completion, is illustrated when an equation such as x – 4 = 10 becomes x = 14. The left side of the equation, where the x is reduced by 4, is "restored" or "completed" back to x. The operation of muqabalahI, "cancelling" or "balancing", can be seen in an equation like, x2 + x = x2 + 4 that becomes x = 4. The two sides of an equation are being "cancelled" by subtracting x2 from both sides.

Having written his book on algebra, he was given the distinction of being one of the founders of algebra who developed this branch of science to an exceptional level. His book contained and illustrated calculations of integration and equation presented in over 800 examples. He also introduced the negative signs to the Arabs to which they had been completely oblivious before.

Al-Khwarizmi also dealt with equations in his book, The Book of Algebra and Almucabola Containing Demonstrations of the Rules of the Equations of Algebra. He discussed the numbers of restoration and opposition, which are roots, squares, and numbers. Al-Khwarizmi’s major contributions in this area was to the theory of equations. Five of his equations are linear, meaning that the equation is of the first power of the unknown. He was able to systemize the Babylonian treatment of quadratics by reducing all such problems to the six basic types.

Also in The Book of Algebra and Almucabola Containing Demonstrations of the Rules of the Equations of Algebra, Al-Khwarizmi deals with geometric figures. He says that it is not enough to just speak about these concepts in terms of numbers but that these concepts must also be proven by geometrical demonstrations and proofs. He commonly used squares with sides of unknown values to help prove that the manipulation of numbers that were performed were correct.

Al-Khwarizmi was a genius who contributed a tremendous amount of knowledge to the early developments of mathematics. As can be seen from his work with arithmetic, fractions, algebra, and quadratic equations, Al-Khwarizmi made many ground-breaking discoveries that changed the course of mathematical history. It is hard to believe that procedures that seem so obvious to us in the twenty-first century had to actually be discovered and invented by someone. It is interesting to think that if this man had not worked with mathematics to the extent that he did, we may not have had the mathematical concepts that we do today. Al-Khwarizmi has left a lasting mark on the mathematical concepts and procedures that are still taught and used today.