The Invention of Zero
We often say that India gave the concept of Zero to the world. How did it happen? Here’s the story.
There has always been a need for counting and manipulating quantities. This need became more important after man invented agriculture. The move to a settled lifestyle, invention of agriculture, trading of crops - all required handling quantities and numbers. Construction of houses, villages and towns again required measurements.
Also property needed to be protected. This led to rulers who promised protection, armies, and bureaucracies. The ruler’s need for legitimacy with the public led to their assuming kinship with gods, or even claiming the mantle of gods. Priestly classes arose. The Heavens became involved in worldly affairs. This meant astronomical calculations were required to work out auspicious times for events. Mathematics was again needed.
There are Egyptian papyri dating to 1800-1500 BCE which describe mathematical techniques for calculating area, volume, worker output, and even the strength of date beer!
Dealing with numbers requires a notation to represent numerals. The Egyptians had hieroglyphic symbols for one, ten, hundred, thousand, ten thousand, hundred thousand, and even for a million (the hieroglyph for one million is also the hieroglyph for “many”!). The number 413 for example, would need the “hundreds” symbol four times, “tens” symbol once and “ones” symbol three times. The principle behind this system is very similar to what the Romans used 2000 years after the Egyptians (such as MCMXVI, and the like.).
The first use of zero was as a “place holder”. For instance in the number 403 there are 4 “hundreds” and 3 “ones”, but zero “tens”. The Babylonians used something like the zero as a place holder for this around 1500 BC. However the use of zero as a placeholder was not consistent. The Greeks, a thousand years after the Babylonians, did not use zero as a place holder. They had philosophical objections about using something to represent nothing!
The need for familiarity with numbers must also have been there in India. As early as 2500 BC, the Harappans were building cities, and trading widely with areas across the Arabian Sea. They would surely have needed calculations. But as the Harappan script has not yet been conclusively deciphered we don’t know if they had a notation for numerals, what that notation was, and what kind of calculations they were doing.
Till the middle of the first millennium BCE, knowledge was passed orally from generation to generation. This is true of Vedic, Buddhist and Jain teachings, all of which were passed on orally. It is only in Emperor Ashoka’s time (3rd century BCE) that the written word became widespread. The Brahmi script used then spread across much of today’s India. Ashokan Brahmi had symbols for the numerals 1-9, 10, 20, 30..90, 100, and 1000. However there was no symbol for zero, even as a place holder.
After Ashoka converted to Buddhism, he sent several missions to many lands to propagate the Buddhist faith. The missions to the north west, for example, crossed the Pamirs and reached China and beyond. Opening of these routes to China also encouraged commerce. The expansion in the propagation of Buddhism and increase in commerce continued under the Kushans, who controlled much of what is now Tajikistan, Uzbekistan, Afghanistan, Pakistan, Eastern Iran and Northern India till about 375 AD.
But for 500 years starting from about 100 BCE, India’s largest trading partner was not any Asian country. It was the Roman Empire. This is clear from several archaeological finds in northern Africa – the gateway of the Romans to the Arabian Sea. There is evidence that during this period perhaps as much as 30% of Imperial Rome’s revenues came from the customs duties collected at ports in northern Africa on imports from India – specially from south India. The growth in trade and commerce continued during the “Golden Age” of the Guptas, who followed the Kushans and ruled for almost 300 years. India was one of the richest regions of the world.
The Roman Empire started declining in the 5th century CE, but India's prosperity continued because trade from India then turned eastward towards south east Asia. For several centuries thereafter there was large scale infusion of India’s cultures, scripts, languages, religions, philosophy and iconography in these lands.
These prosperous periods saw tremendous progress in subjects as diverse as architecture, literature, medicine, etc. It is around this time that the concept of zero saw extraordinary intellectual development in India – development which would lead to a revolution in the way the world counts, and calculates. It lay the foundation for modern science and technology.
Early references to a decimal system are there even in Patanjali’s Yog Sutra (c 150 CE). The use of zero, represented by a dot, appears as a place holder around this time. Mathematical techniques required for calculations became increasingly sophisticated in the early centuries CE. Many unknown mathematicians may have contributed to this development. But it is in the Aryabhatiya of Aryabhata (476 – 550 CE) that we find a comprehensive record of formulas, techniques and concepts which cover topics such as squares, cubes, square roots, cube roots, triangles, circles, fractions, quadratic equations and sines. Aryabhata uses the decimal system, and also uses zero as a place holder. Applying this mathematics Aryabhata could calculate the value of Pi to a close approximation, and had even worked out the exact length of the solar year.
But the real revolution which demonstrated the true power of zero, occurred when Brahmagupta (598-668 CE), a hundred years after Aryabhata, changed zero from being a mere place holder to a number in its own right like the other nine digits. Brahmagupta’s astounding intellectual leap was built upon the work of Aryabhata and also Varahmihira (who had brought in Greek ideas from a text known as Yavana Sutra). In his work Brāhma-sphuṭa-siddhānta Brahmagupta describes zero as the number you get when you subtract a number from itself. He described the positional number system and set out rules for arithmetic operations with zero. Suddenly it became possible to represent any arbitrarily large number in a simple manner and perform calculations with it. Brahmagupta also gave the concept of negative numbers and how to operate with them.
In his second work, the Khaṇḍakhādyaka, Brahmagupta used these techniques in calculations dealing with rotation of planets, eclipses, and many more practical problems in astronomy and other subjects.
Brahmagupta had just laid the foundation of modern mathematics and astronomy in India.
In line with the tradition of the day, Aryabhata and Brahmagupta had written their works in Sanskrit verse. Their books were taught in several institutions of learning such as universities and Buddhist viharas, and the concepts and techniques explained in the books became widely used.
It is around this time that a chain of events occurs outside India which ultimately leads to this knowledge of mathematics spreading from India to the rest of the world.
At the time that Brahmagupta was completing his work on astronomy, a massive churn was taking place in Arabia. Prophet Mohammed had died in 632 CE. He was followed by the first four “Rashidun” Caliphs. The struggle for power after them led to the establishment of the Second Caliphate – the Umayyad Caliphate - in 650 CE. Military operations by Islamic armies initiated at the time of Prophet Muhammad were continued by the Umayyads. These military operations lead to dramatic and lasting political changes. The Arab armies swept into North Africa, Libya, Tunisia and Spain to the west, Central Asia to the north, and to Sind in the east.
Steady flow of commerce and propagation of Buddhism to China and central Asia had led to several Buddhist centres coming up in modern-day Afghanistan and northern Iran. One of these was a monastery called “Nav Vihara” (New Vihara), which had come up near Balkh in Persia (the place is now called “Nowbahar”). Nav Vihara played a key role in the passage of zero and Indian mathematics to Arabia and from there to the rest of the world.
The Umayyad armies reached Nav Vihara too. The monastery was unguarded and unfortified. The head priest (Pramukh) surrendered. He and his family were taken to Damascus, the Umayyad capital. There they converted to Islam.
The Umayyad capital was a melting pot of various nationalities. The “Pramukh”, being highly educated, was given respect and honour. The family name became “Barmak”, an Arabic corruption of “Pramukh”.
The Pramukh’s young son had been studying in Kashmir at the time the rest of the family moved to Damascus. The son was especially interested in mathematics, astronomy and medicine. After his return from his studies in Kashmir he was brought to Damascus. There he took the name Khalid, and became Khalid ibn Barmak.
Khalid is a crucial character in the next step in our account.
In 750 CE the Umayyads were overthrown by the Abbasids. The first Abbasid Caliph was Al Mansur, and it so happens that Khalid had attended college in Damascus along with Al Mansur. Both were close friends.
Khalid was a whiz kid and was put in charge of Abbasid finances. His proficiency led to his becoming the Grand Vazir. He personally planned and oversaw the design of the new capital – Baghdad. He designed Baghdad as a series of concentric circles, much like a Buddhist mandala. He set up a paper factory. He became well known as a man of intellect and curiosity.
(Stories about the Barmakis’ generosity later appeared in the Arabian Nights - “The Barmecide’s Feast”, for example. People of a certain vintage in India, who went to middle school in 1970s, may remember reading this story in their English Reader)
Khalid’s interest in Indian astronomy and mathematics, which he had studied in Kashmir, led to his asking for the latest works on these subjects from India. The combined works of Brahmagupta were brought to him by a delegation from Sind, in the form of a manuscript - “Sindhind”.
Khalid got Al-Mansur to commission a translation of the Sindhind into Arabic. This translation, done by al-Fazārī, was titled “Zīj as‐Sindhind al‐kabīr” or "Great astronomical tables of the Sindhind". It rendered Brahmagupta’s somewhat obscure verse format into staid Arabic prose. The translation was housed in the Bayt al-hikma (the House of Wisdom) which was created by Al-Mansur for housing not just translations of the Indian works, but also works from Greece and other places on subjects such as medicine, literature and others.
So the mathematics of India was translated into Arabic. But it required our next principal character, to take it further. This character is Al Khwarizmi, who made his appearance in Baghdad around 815 CE. He was a polymath from an area near Samrkand which was deeply influenced by Buddhism and Indian thought. He was later appointed Head Astronomer at the Abbasid Court.
While studying translations in the Bayt al-hikma, al Khwarizmi was dazzled by Brahmagupta’s work. Finding the earlier Arabic translation somewhat obscure, he translated it again – this time into easily understandable Arabic Prose. His translation was called “Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr waʾl-muqābala” (“The Compendious Book on Calculation by Completion and Balancing”), or simply “Kitab al-Jabr”. It was from this title that our word “algebra” was born. Al Khwarizmi’s own name later gave rise to the word for another Indian concept - “algorithm”.
Al Khwarizmi’s writing was clear. He combined Euclid with Indian mathematics. His work encompassed Geometry and Trigonometry too. In simple language he explained the Indian positional number system and the importance of zero. He showed how this simplifies calculations. His work spread fast across the Arab world, and even to al-Andalus, or Spain, then under the Arabs.
In 1085 the Qazi of Toledo in Spain, which had become the repository of knowledge in Europe, writes appreciatively about the Indian contribution to mathematics (and also of another Indian contribution – the game of chess).
Soon thereafter, Toledo, and Spain, were wrested back by Christian rulers. But the knowledge repositories of Toledo were not ravaged by the new rulers. The knowledge from Toledo – including the mathematical knowledge of Sindhind - slowly diffused into European courts.
But widespread induction of this mathematical knowledge into Europe occurred only a hundred years later. The character responsible this time was Leonardo Bonacci (1170-1240), also called Leonardo of Pisa. Being the son of Guglielmo Bonacci, an Italian merchant and customs official, Leonardo later came to be known as Leonardo filius Bonacci (son of Bonacci), or Leonardo Fibonacci.
As a young boy Fibonacci stayed with his father in the port of Bugia in Algeria, where the father was posted as a customs official. Fibonacci went to school there and learned about the Hindu – Arabic number system. He was fascinated by this system, which was much simpler than the Roman system still in vogue in much of Europe. On his return to Pisa he wrote a book in 1205 CE, Liber Abaci (The Book of Calculations). Among other things, Fibonacci explained the theory and basis of the Hindu number system. The book was read only by a few scholars.
At this time the ruler was King (later Emperor) Frederick II, a king who was inquisitive, interested in knowledge and one who encouraged discovery. On a visit to Pisa, Fibonacci had the fortune of being introduced to Frederick. He told Frederick about the Hindu number system and mathematics. To test this system Frederick posed some tough mathematical problems after dinner that night. Fibonacci solved all the problems quickly. Frederick was impressed, and asked Fibonacci to write down these solutions for him. Fibonacci also became friendly with Michael Scot, Frederick’s astronomer. Scot encouraged Fibonacci to explain the Indian mathematical concepts not just theoretically, but also give practical examples. Fibonacci did so, and gave several application examples of using the Hindu number system for commercial calculations such as computing interest.
The knowledge spread like wildfire. Very soon, zero and the place-value system of Aryabhata and Brahmagupta became the only way that the world represented numbers and carried out mathematical calculations. The Renaissance, and the scientific discoveries of Kepler, Copernicus, Newton, all used and built upon this system.
This, then, is how zero and the Indian number system travelled from India to Arabia and then to Europe, and transformed the world.
(Partly based on William Dalrymple’s book “The Golden Road”)
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