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Mathematics and astronomy

Al Khwarizmi's lifelong questions about time, place and missing quantities gave rise to algebra and higher mathematics

Mohammed al Khwarizmi was the father of algebra

A distinctive 12th century Syrian sundial


The genius of Al Khwarizmi

Although the Almagest held a special place in the mind of early Arab mathematicians, the greatest numerical thinker of the time, Mohammed Al Khwarizmi, was entranced by documents from farther south and east. While many of his colleagues at the House of Wisdom pored exclusively over the Greek and Byzantine texts gradually coming into Arabic, Al Khwarizmi was drawn to writings from India.

His research into mathematical and astronomical texts from India would enable him to revolutionise the Arab concept of numbers... and lay the foundation for modern higher mathematics and the rise of the West.

Al Khwarizmi was exceptionally gifted with numbers. At some point in the early 9th century, he joined the House of Wisdom in Baghdad during the reign of Caliph Al Mamun.

Even as his colleagues were pursuing their own projects, he had the caliph's librarians trawl their archives for papers brought from India in the time of the caliph's late father, Harun al Rashid. After much searching, a treasure trove of Indian materials was produced, including the Brahma Sphuta Siddhanta, roughly translated as The Opening of the Universe.

Al Khwarizmi's rediscovery of this single paper touched off a process of mathematical exploration and discovery that still resonates in the modern era.

Using bilingual aides, Al Khwarizmi began to render the foreign characters into Arabic. And with each day, he uncovered ideas and symbols that initially mystified him, but gradually became clear in the blinding light of understanding.

First, he found the Indian method of representing numerical quantities with symbols. Until that time, the Arabs and others in the Mediterranean-Mesopotamian region had been using several methods of representing quantities. One was a verbal method, writing out the numbers as words. The other was a complicated finger-counting method, which was perhaps as versatile as the Chinese abacus, but inadequate to the needs of ever more abstract calculations of star positions. There were also the Greek letters such as pi, and finally, the Roman numeral system.

As he opened the pages before him, Al Khwarizmi saw a system of Indian symbols representing the quantities 1 to 10 and then combinations of those symbols to represent ever increasing quantities rising to infinity. This new system of symbols told him many things.

First, he saw that the Indians had settled upon the decimal system as the most efficient way to manage ever more complex calculations. And although he and other Arab and Muslim thinkers knew about the decimal system, it was not universally used.

He also saw that using this system of symbols was infinitely easier to do than the other ways of representing quantities.

He would take this system of symbols and make it his own. It would gradually come to be known as the Hindu-Arabic numerals, and would eventually be adopted by Europe and the West - a revolutionary simplification of mathematic representation into a universal, efficient, and coherent language.

But buried within this Hindi way of writing numbers was another symbol, one that by itself was just as revolutionary. In the Hindi writings of Brahmagupta was a single black dot. When Al Khwarizmi asked his translators for an explanation, they replied that it meant nothing. Thinking they were playing with him, he pressed for an explanation. And they responded by saying the black dot represented the quantity of nothing.

We now know this quantity to be zero - a concept that opened new levels of abstraction. Al Khwarizmi was stunned by its implications. He was baffled and excited by the meaning of zero, its mystery and its evasiveness. He even tried to use the zero in division, and was puzzled by the absence of an answer, unique in division.

But there was more in the papers from India. Al Khwarizmi could see a whole system of numbers on the lesser side of zero, numbers stretching as far into the negative side of infinity as those on the positive side.

Today we know these numbers as negative numbers. Since they, like nothingness, cannot be seen or measured in the physical world, they are pure abstraction. But they enable all sorts of abstract calculations that would not be possible without their existence.

Mohammed Al Khwarizmi was swift to take all these foreign symbols and ideas into his heart and mind and make them his own. Had he not done so, the history of our world could have unfolded in a different way. What we call the modern era might not have come until centuries later.

Al Khwarizmi knew the importance of sharing these concepts as widely as possible. He sensed the tragedy that might have ensued had the papers of Brahmagupta been eaten by moths in the caliph's library or in India and turned his energy into setting down all these old-new concepts in a document that the rising Arabic civilisation could use.

He called his Arabic version the Sindhind, which he completed in about the year 825. As he worked with a particular mathematical form known as polynomial equations, he became especially fascinated with the methods of finding missing quantities. By 830 he had published a book roughly translated as the Compendius Book on Calculation by Completion and Balancing. In the Arabic title, one of the key words was the word al jabr, which means restoration or completion.

Al Khwarizmi was formalising a mathematical process that would one day have more impact on the world than even his rediscovery of the papers of Brahmagupta. Today we know this process as algebra, and it is one of the foundations of modern higher mathematics and all that flows from it in the sciences, engineering, financial management, and all the manifestations of the digital world.

Were this Al Khwarizmi's only contribution to modern higher mathematics, he would merit a place of honour on a level with Euclid, Newton, Einstein and a few others. But there is more.

Among the many derivatives of algebra is a very esoteric and complicated formula for dealing with large numbers of data. It is called the algorithm, and in the modern world it is the basis for the writing of software, of pattern recognition, of financial market modelling, of Google searches, and of human gene sequencing among many other things. The modern digital world could not exist without it.

For some centuries, mathematical historians believed that the word algorithm was of Greek origin, because the 'ithm' ending resembles Greek words like 'rhythm' and 'hymn'.

Recently, researchers realised that the word was really a Latinised version of Al Khwarizmi's name. Mediaeval European scholars had been well aware of the origin of the word, but in the intervening centuries, it had been forgotten.

Al Khwarizmi is considered the father of the algorithm, which 1,200 years later would revolutionise the worlds of computation, software, and web-based communications.

His works were not translated into Latin and absorbed into Europe for another 300 years, long after he was dead. Until then, Dark Age Europe had no intellectual or scholastic community that could have made use of it, outside of a few learned monks locked away in monastic libraries. When the English monk Robert of Chester translated the Compendius Book into Latin in Spain in 1145, Europe was only beginning to stir from the long depression begun with the fall of Rome.

The Compendius Book was not Al Khwarizmi's only work. After its completion, he created a work on arithmetic that was translated into Latin in about 1126 by another English monk, Adelard of Bath. This work was important because it introduced the Indo-Arabic numerals into Europe, laying the foundation for the later explosion of science and thought on that continent.

But Al Khwarizmi, like all good mathematicians of the early Arab caliphate, was also deeply interested in the movements of the skies. Around 820 he completed a visionary compendium on his understanding of the movements of the heavenly bodies. This work, known as the Zij al Sindhind, or 'Star Tables Based on the Indian Calculation Method', was the first Arabic astronomical work to go beyond translating older existing astronomical works from older civilisations.

While it used Indian and other tools, the book applied them in new ways to track the movements of the sun, moon, and five known planets at that time. Al Khwarizmi also developed a sundial that could be used at any latitude to tell the correct time. Many mosques eventually installed his sundial, to help with calculating correct times of prayer.

He also seems to have developed several versions of the quadrant, a competitor of the astrolabe. Both were analogue computers that used sky positions to tell time and location. His sine quadrant could help map the movements of stars, and his horary quadrant could accurately tell the hour from the position of the sun or stars, especially helpful in selecting the time of prayer.

In his Zij al Sindhind, Al Khwarizmi also began a long and proud Arab-Muslim tradition of creating calendars and star information including 117 star tables and 37 chapters on calculating the calendar.

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