Category Archives: Mathematics

“Contributions of Al-Khwarizmi to Mathematics and Geography” By Kamal Ayyubi

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Muhammad ibn Musa Al-Khwarizmi is one of the greatest scientific minds of the medieval period and a most important Muslim mathematician who was justly called the ‘father of algebra’. Besides his founding the science of jabr, he made major contributions in astronomy and mathematical geography. In this article, focus is laid on his mathematical work in the field of algebra and his contribution in setting the foundation of the Islamic tradition of mathematical geography and cartography.

Introduction

Islam gave birth to a new civilization that spread from China in the east, India in the south east, Russia in the north, and Anatolia in the west of Asia, to East and North Africa up to the Mediterranean regions of Southern Europe. This civilisation was marked by a deep interest in science. In the heart of the Islamic scientific tradition lays the queen of sciences, mathematics, where the scholars of bilad al-Islam (lands of Islam) excelled in all its branches practiced in pre-modern times.

One of the greatest minds of the early mathematical production in Arabic was Abu Abdullah Muhammad ibn Musa al-Khwarizmi (b. before 800, d. after 847 in Baghdad) who was a mathematician and astronomer as well as a geographer and a historian. It is said that he is the author in Arabic of one of the oldest astronomical tables, of one the oldest works on arithmetic and the oldest work on algebra; some of his scientific contributions were translated into Latin and were used until the 16th century as the principal mathematical textbooks in European universities. Originally he belonged to Khwârazm (modern Khiwa) situated in Turkistan but he carried on his scientific career in Baghdad and all his works are in Arabic. He was summoned to Baghdad by Abbasid Caliph Al-Ma’mun (213-833), who was a patron of knowledge and learning. Al-Ma’mun established the famous Bayt al-Hikma (House of Wisdom) which worked on the model of a library and a research academy. It had a large and rich library (Khizânat Kutub al-Hikma) and distinguished scholars of various faiths were assembled to produce scientific masterpieces as well as to translate faithfully nearly all the great and important ancient works of Greek, Sanskrit, Pahlavi and of other languages into Arabic. Muhammad al-Khwarizmi, according to Ibn al-Nadîm [1] and Ibn al-Qiftî [2] (and as it is quoted by the late Aydin Sayili) [3], was attached to (or devoted himself entirely to) Khizânat al-Hikma. It is also said that he was appointed court astronomer of Caliph Al-Ma’mun who also commissioned him to prepare abstracts from one of the Indian books entitled Surya Siddhanta which was called al-Sindhind [4] in Arabic [5]. Al-Khwarizmi’s name is linked to the translation into Arabic of certain Greek works [6] and he also produced his own scholarly works not only on astronomy and mathematics but also in geography and history. It was for Caliph al-Ma’mun that Al-Khwarizmi composed his astronomical treatise and dedicated his book on Algebra.

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History of Zero

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Zero, zip, zilch – how often has a question been answered by one of these words? Countless, no doubt. Yet behind this seemingly simple answer conveying nothing lays the story of an idea that took many centuries to develop, many countries to cross, and many minds to comprehend. Understanding and working with zero is the basis of our world today; without zero we would lack calculus, financial accounting, the ability to make arithmetic computations quickly, and, especially in today’s connected world, computers. The story of zero is the story of an idea that has aroused the imagination of great minds across the globe.

When anyone thinks of one hundred, two hundred, or seven thousand the image in his or her mind is of a digit followed by a few zeros. The zero functions as a placeholder; that is, three zeroes denotes that there are seven thousands, rather than only seven hundreds. If we were missing one zero, that would drastically change the amount. Just imagine having one zero erased (or added) to your salary! Yet, the number system we use today – Arabic, though it in fact came originally from India – is relatively new. For centuries people marked quantities with a variety of symbols and figures, although it was awkward to perform the simplest arithmetic calculations with these number systems.

The Sumerians were the first to develop a counting system to keep an account of their stock of goods – cattle, horses, and donkeys, for example. The Sumerian system was positional; that is, the placement of a particular symbol relative to others denoted its value. The Sumerian system was handed down to the Akkadians around 2500 BC and then to the Babylonians in 2000 BC. It was the Babylonians who first conceived of a mark to signify that a number was absent from a column; just as 0 in 1025 signifies that there are no hundreds in that number. Although zero’s Babylonian ancestor was a good start, it would still be centuries before the symbol as we know it appeared.

The renowned mathematicians among the Ancient Greeks, who learned the fundamentals of their math from the Egyptians, did not have a name for zero, nor did their system feature a placeholder as did the Babylonian. They may have pondered it, but there is no conclusive evidence to say the symbol even existed in their language. It was the Indians who began to understand zero both as a symbol and as an idea.

Brahmagupta, around 650 AD, was the first to formalize arithmetic operations using zero. He used dots underneath numbers to indicate a zero. These dots were alternately referred to as ‘sunya’, which means empty, or ‘kha’, which means place. Brahmagupta wrote standard rules for reaching zero through addition and subtraction as well as the results of operations with zero. The only error in his rules was division by zero, which would have to wait for Isaac Newton and G.W. Leibniz to tackle.

But it would still be a few centuries before zero reached Europe. First, the great Arabian voyagers would bring the texts of Brahmagupta and his colleagues back from India along with spices and other exotic items. Zero reached Baghdad by 773 AD and would be developed in the Middle East by Arabian mathematicians who would base their numbers on the Indian system. In the ninth century, Mohammed ibn-Musa al-Khowarizmi was the first to work on equations that equaled zero, or algebra as it has come to be known. He also developed quick methods for multiplying and dividing numbers known as algorithms (a corruption of his name). Al-Khowarizmi called zero ‘sifr’, from which our cipher is derived. By 879 AD, zero was written almost as we now know it, an oval – but in this case smaller than the other numbers. And thanks to the conquest of Spain by the Moors, zero finally reached Europe; by the middle of the twelfth century, translations of Al-Khowarizmi’s work had weaved their way to England.

The Italian mathematician, Fibonacci, built on Al-Khowarizmi’s work with algorithms in his book Liber Abaci, or “Abacus book,” in 1202. Until that time, the abacus had been the most prevalent tool to perform arithmetic operations. Fibonacci’s developments quickly gained notice by Italian merchants and German bankers, especially the use of zero. Accountants knew their books were balanced when the positive and negative amounts of their assets and liabilities equaled zero. But governments were still suspicious of Arabic numerals because of the ease in which it was possible to change one symbol into another. Though outlawed, merchants continued to use zero in encrypted messages, thus the derivation of the word cipher, meaning code, from the Arabic sifr.

The next great mathematician to use zero was Rene Descartes, the founder of the Cartesian coordinate system. As anyone who has had to graph a triangle or a parabola knows, Descartes’ origin is (0,0). Although zero was now becoming more common, the developers of calculus, Newton and Lebiniz, would make the final step in understanding zero.

Adding, subtracting, and multiplying by zero are relatively simple operations. But division by zero has confused even great minds. How many times does zero go into ten? Or, how many non-existent apples go into two apples? The answer is indeterminate, but working with this concept is the key to calculus. For example, when one drives to the store, the speed of the car is never constant – stoplights, traffic jams, and different speed limits all cause the car to speed up or slow down. But how would one find the speed of the car at one particular instant? This is where zero and calculus enter the picture.

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