Al Khawarizmi Hammad U. Rehman Khawarizmi s Life Ever Wondered where the word Algorithm comes from? Algorithm comes from the Latinized name for a very special man- Al Khawarizmi. He developed the science of mathematics and basically invented Algebra as we see it today and contributed to so many more that his name was used to describe algorithms. Now Al Khawarizmi, born in the area orginally known as the Fertile Cresent and later as Baghdad, was a Muslim mathematical scholar who was active circa 800 CE. Not much is known of his early years, including his birthday, but it can be assumed that he wasa born around 780 CE. His full name was Abu Ja far Muhammad Bin Musa Al Khawarizmi. (The author wants to make the following notification to the reader: Many texts refefer to the scholar as Al Khwrizmi, ommiting the a from his name, I wanted to assure the reader that I am speaking of the same scholar and it is my understanding that with Arabic transliteration, an a must be present in his name to properly pronounce the letter Alif in his name. it is for that reason that throughout tis text, I maintain the a ) From this, we are sure that he was the son of a man named Musa, the Qur anic name for Moses, but due to Arab culture, his mother s name is not known. Tabari and others link his origins to
the area of Qurtubbul as well as Majus (from the less known name of his of Al Majusi) which prompts the question of his Zoroastrian past, if any. Khawarizmi actually makes a not of this issue regarding his origin in his groundbreaking work, due to cultural reasons, and professes his deep love with the religion of Islam and professes his faith of Islam within that text. He lived through the Islamic empire during the Abbasid Period and was under the rule of two kings, Haroon and Ma mun Rasheed. Druring this time, science and religion were coexistant as Muslim scientists ant scholars worked effortlesslyto piece together an understanding of the world. It was here, at this time that AL-Khawarizmi was surrounded by a young brillance in the ages of Post-Greco-Roman science. Though much of his relevant work was recorded and translated through the ages, still we do not know what was the personal ife of the scholar. But predictiong from the area where he was residing (i.e. the Capital Baghdad) and the high level of mastery that he had accomplished and from the sophistication of life in Baghdad compared to other areas of the Muslim World and European Dark Ages, Khawarizmi must have enjoyed a comfortable life and did not suffer from poverty. Not much is know of his marrige if he had married and of his children, if he ad any of those as well. It can be said, though, that the extent of knowledge that he was given and the vastness of research that he had contributed, AL Khawarizmi would have had little time for family. Her had, however, instead of giving lineage to the world, given Algebra, as well as contributed to Cartography, Asronomy, and Trignometry, but most importantly Algebra and the Indian Decimal place system where he had created the Arabic Numerals and popularized them. Khawarizmi s Mathematical Works Al Khawarizmi had, as mentioned before, two major contibutions to mathematics. His book on Algebra gives the Modern World a useful mathematical toolbox for which people could live easier lives. Khawarizmi s other book on Arithmetic was highly praised as he had shown how symbolic denotations for quantites proved to be easier than the early Roman Numerals. His first book was a treatise of Indian Mathematics which he had written thanks to the the construction of Haroon Al Rashid s Bayt Al-Hikmah (Arabic foor House of Wisdom) and was titled Kitab al Jem wa l tafriq bi Hisab al-hind (English Translation: The Indian Book of Combining (as in addition) and Separation (as in subtraction) With Computation). A Separate title has also been used, Kitab Hisab Al-Adad Al-Hindi (English Translation: The Indian Book of Summative Calcuation). In this book, he provides wha ws previously lost knowledge of indian mathematics with respect to thier decimal system. He takes thier concept and expands it to show how the system is used in addition, sunbtraction and other basic mathematical operations. From here on, mathematics now have become less cumbersomne and more understandable to the masses since now instead of writing two as II in the Roman system, which the Arabs had adopted through trade with the Byzanties, it can be written as 2 Sadly, the original text of this book has been lost and not much of any Latin translation has formalized today But, there is still understanding of his work from various contemprary texts. 2
On the Right Image:The top Row shows the original Bhramin Numerals. The middle shows the Arabic Numerals created by Al Khawarizmi. The Bottom Row shows Today s Numerals. On the Left Image: Al Khawarizmi s Logic behind the shape of each digit His second book, which has helped history to retain his name as the Father of Algebra was Al Kitaab Al Mukhtassar fi Hissab Al-Jabr Wa l Muqabalah Translation: The Compendious Book on Calculation and Completion and Balancing. John K. Baumgart translates the title as the Science of Equations. It was written circa 800 CE with over 800 examples ranging from simple linear functions to more complicated quadratic expressions and equations. According to many historians, it is the only book to have existed first on the subject of what is considered Algebra in a modern sense. it is here where Modern English pays respect to Khawarizmi s work. The Arabic word Al-Jabr meaning completion is the basis of the word algebra. Khawarizmi s book is divided into three separate sections for which he had made the following intention in his Introduction, [That Algebra will accomplish] what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, 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. He wanted Algebra to be useful for almost every natural and created situation known to mankind and so in his book, he presents Algebra in a very complete, worded out way. For example, Khawarizmi teaches his system of the Arabic Numerals with base ten in the following manner, When I consider what people generally want in calculating, I found that it always is a number. I also observed that every number is composed of units, and that any number may be divided into units. Moreover, I found that every number which may be expressed from one to ten, surpasses the preceding by one unit: afterwards the ten is doubled or tripled just as before the units were: thus arise twenty, thirty, etc. until a hundred: then the hundred is doubled and tripled in the same manner as the units and the tens, up to a thousand;... so forth to the utmost limit of numeration. After Allowing this to come to understanding, He begins Part One: Algebra. He begins every problem in this section very broadly, and asks the reader to change this problem into one of six mathematical circumstances 3
via addition and subtraction (In Arabic, he referred the following set of operations as Muqabalah, which means Balancing) and by moving terms across the equal sign as we call it today (this movement operation is in Arabic called Jabr which translates to Completion) Al Khawarizmi s six standard mathematical circumstances are: This systematic method proved to be very useful, albeit difficult to first explain. The following is an example from the Mercantile section of the the book itself. The Problem: A man is hired to work in a vineyard 30 days for 10 dollars. He works six days. How much of the agreed price should he receive? Today, this problem would have been expressed as the following equation: 30x = 60 Khawarizmi made the following answer, It is evident that since days are onefifth of the whole time; and it is also evident that the man should receive pay having the same relation to the agreed price that the time he works bears to the whole time, 30 days. What we have proposed, is explained as follows. The month, i.e., 30 days, represents the measure, and ten represents the price. Six days represents the quantity, and in asking what part of the agreed price is due to the worker you ask the cost. Therefore multiply the price 10 by the quantity 6, which is inversely proportional to it. Divide the product 60 by the measure 30, giving 2 Dollars. This will be the cost, and will represent the amount due to the worker. With this novelty, Khawarizmi had also gone to show a sytematical method to finding the root of the sqare and was actually was the first to introduce the concept of roots which he had called Jidhr, meaning root in Arabic. Khawarizmi says, for the following true statement: 5 2 = 25 The folowing is an example of squares equal to roots: a square is equal to five roots. The root of the square then is five, and 25 forms its square, which of course equals five of its roots. He gives a full example and a geometric proof as well for a separate problem: For the equation x 2 + 10 x = 39 He says,... a square and 10 roots are equal to 39 units. The question therefore in this type of equation is about as follows: what is the square which combined with ten of its roots will give a sum total of 39? The manner of solving this type of equation is to take onehalf of the roots just mentioned. Now the roots in the problem before us are 10. Therefore take 5, which multiplied by itself gives 25, an amount which you add to 39 giving 64. Having taken then the square root of this which is 8, subtract from it half the roots, 5 leaving 4
3. The number three therefore represents one root of this square, which itself, of course is 9. Nine therefore gives the square. The Geometric Proof (With Side Annotations) In Part 2, Al Khawarizmi expands on his facination with mensuration. Here, he spent a great time on the work of the Greek and thier facination with pi, which under his sytem of numbers was nothing more than 3 and one seventh (3.1428...) It was with his sytem of numbers that he had nearly found the exact volume of a cone. He also looked into Pyramids as well as circles. It is here that I will note that with his work on mensuration, did the Arabs of the time had nearly pefected their estimation of the circumference of the Earth. (Apparently, they were off by only 37 kilometers). Finally in Part 3, he discusses mathematical solutions to every day situations, as showm with the first example. His work on inheritance was especially useful in Islamic Law where the Qur an provides difficult guidelines for non-scholars to understand, but with his invention of Algebra, many people found it especially helpful. His Collaboration with other scholars Al Khawarizmi definately did nbot work alone and as a result is shown to have much help. For startetrs, three people, which history referrs them to as the Banu Musa (Family of Mosess) Brothers. From whatever research done thus far, there is no evidence that they were in any relation to his lineage (An issue when connection Bin Musa to the Musa Brothers). They were: Jafar Muhhamd ibn Musa ibn Shakir, Ahmad ibn Musa ibn Shakir, and Al-Hassan ibn Musa ibn Shakir. the three of them were major cartographers as well as scholars of the Greek sciences. Their work on Archimedes research of the Circle had proved vital in aiding Al Khawarizmi with his advances in Cartography. beyod these three, Al Khawarizmi also spent a great deal of his time at the Bayt Al Hikmah where he was in te company of various cholars of many faiths and fields of science. There, a legitimate prediction of his collaboration with a number of them can be made, but we cannot be certain. Eventually, Al Khawarizmi s most vital aid was in fact the Khalifa (Caliph), Ma mun Al Rasheed who was a scholar in his own right and helped advance and fund all of Al Khawarizmi s endevors. It was this finacnial help that Al Khawarizmi dedicates his work on Algebra to the Khalifa. 5
Historical Events of the Islamic Empire During His Lifetime A short time before, During and much after his lifetime did Al Khawarizmi get a chance to be a part of what Islamic Historians coin as the Golden Age. In this time. Others like Alk Khawarizmi worked effortlessly on mathematics, with himself and Omar Khayyam at the forefont. Beyond mathematics, Astronomy, Geopraphy and Cartography were rapidly gaining importance as Muslim traders traveled for new routes to India, China, Africa, and Europe. The Rise of the Abbasid Khalifa, after the demise of the Umayyad Khalifa also had a major profound impact on his life as the Abbasid Khalifas opened isnstiutions of high level Astronomy and eventually, under Haroon Al Rasheed, the House of Wisdom. And from this House of Wisdom did Arabs gain access to Greek logic and as a result, a deviations of Islamic faith began to appear, known as the Mu tazilites, whereby which at a time near the end or a little beyond his lifetime, much commotion had spread throughout the empire. Other Events Around the World In Euope, The Dark Ages were uopn them as literature became scarce for the masses. In Africa, Muslim Empires begin thier first steps of civilization through contact with the Muslim Arabs. In East Asia, Chinese Paper and spices become internationally favored goods and China rises to incredible hieghts. In Cenral Asia, the Islamic Golden Age springs and Al Khawarizmi s work is highly acclaimed. Remarks It was a pleasure to study and present Al Khawarizmi. It was an absolute pleasure to refresh myself with Islamic history and being Muslim myself, I found his use of Arabic throughout his work, instead of inventing a symbol for everything, was quite spectacular as it helped me understand what kind of world Islamic mathematics (Arab Mathematics) was like. Having met with many Arabs who studied mathematics in Arabic, I can now see how beautiful of a concept it was back then and what a micacle it is still today that Algebra is extremely relevent to Our time here. My fasination for Al Khawarizmi is now more of a deep love for my religion s deply rooted culture of mathematics and for this, I cannot thank anyone more than Al Khawarizmi himself. References (Mostly all sources are PDF s or webpages that have been converted to PDF s.) All sources are available upon request. http : //www history.mcs.st andrews.ac.uk/history/biographies/banu M usa.html https : //historyof islam.com/contents/the classical period/al khwarizmi/ https : //www.researchgate.net/profile/v ictor K atz/publication/227131361 S tages i n t he H istory o f A lgebra w ith I mplicatio http : //www history.mcs.st andrews.ac.uk/history/biographies/al Khwarizmi.html http : //www.sciencedirect.com/science/article/pii/s0315086005000352 http : //www.storyofmathematics.com/islamic a lkhwarizmi.html http : //www.muslimheritage.com/article/contribution al khwarizmi mathematics and geography http : //bulldog2.redlands.edu/fac/beery/math115/m115 a ctiv c omplsq.htm 6
http : //kathyreck.weebly.com/al khwarizmi f ather of algebra.html http : //mathematics0.blogspot.com/2013 0 3 0 1 a rchive.html 1. http://www.math.uic.edu/\someone/notes.pdf 2. Bak, Newman: Complex Analysis. Springer 1989 7