It is impossible today to imagine a world without what are known as Hindu-Arabic numerals. These nine symbols or digits (1 to 9) and zero (0) – crucially, together with the system of arranging them in place value – are at the very heart of so much of modern life. Without them there would be no computers and no space travel. Indeed, there would be precious little science, technology or medicine of any kind, to say nothing of mathematics itself.
If mathematics is the universal language of science, then Hindu-Arabic numerals and place-value notation are what make that language fluent and indispensable. Each of us uses numbers all of the time, without even thinking about where they came from or what we would do without them. So it may come as a revelation to discover that these numerals were relatively unknown in Europe until the 12th century, and then not widely adopted until well into the 15th century.
Before that era, when Europeans needed to solve a mathematical problem they usually used Roman numerals, which were complicated and unwieldy. Imagine trying to add LIX (59) to VII (7), and you’ll quickly understand why maths would not have evolved far as a discipline with that ancient numeric system. The genius of the Hindu-Arabic system lies in its use of positional notation, also known as place-value notation. Each digit can be placed into columns of units, while the decimal point (introduced in the 16th century) can be moved left or right across columns to increase or decrease the value of a number by the power of 10 – a system that made it possible to express fractions with far greater accuracy.
This simple but brilliant idea means that any number can be written using the same nine symbols and zero, allowing for limitless calculations and permutations. Yet, despite this evident power, it was only centuries after these numerals and the system of positional notation were developed that they came into widespread usage.
The joy of six
Early civilisations developed complex mathematical theories using their own systems. The Babylonians, who emerged from the 19th century BC in the Euphrates valley, developed a sexagesimal system (based around the number 60), the legacy of which survives today in the division of the clock into 60 minutes and seconds. The ancient Greeks used aspects of that sexagesimal system but wrote numbers using letters rather than special characters and, also unlike the Babylonians, used a character to represent zero. Ancient Greek mathematicians were mainly concerned with geometry – lengths, shapes and angles, all labelled using letters rather than numbers – and their ideas had a huge impact on the development of that discipline.
In the early centuries AD, however, a different tradition of mathematics began to flourish in India, probably based on ideas borrowed from Chinese civilisation. Scholars began to use nine special digits to represent the first nine numbers and, in around 600 AD, they began writing these characters in order, according to their value.
The final piece of the puzzle was the zero, which is vital to a system of positional notation based on the number 10. It was originally written as a dot, to denote an empty value in a sequence of numbers. Crucially, this system was described around AD 625 by an Indian mathematician named Brahmagupta in an elaborate astronomical treatise, written in Sanskrit poetry, called the Siddhanta.
There are divergent accounts of how and when this manuscript arrived in Baghdad, the city founded in 762 by the Abbasid Caliph al-Mansur on a bend in the river Tigris as the capital of his burgeoning Muslim empire. One suggests that it was brought directly from India in 773 by a visiting scholar, but it is possible that the Hindu-Arabic numerals were already known in Baghdad by that point. Certainly, they had reached this part of the world some time earlier: in 662, a Syrian priest called Severus Sebokht wrote admiringly of the ‘nine signs’ of the Indians.
By the early ninth century AD, Baghdad was a major centre of scientific learning presided over by the irrepressibly curious and intelligent Caliph al-Ma’mun. It was also the largest and most important city on Earth, capital of the vast Muslim empire that stretched from the Atlantic coast of Africa to the river Indus, spanning an astonishing five million square miles. People came from across the empire to seek their fortune, and the city became a vibrant centre of learning and culture.
Following the lead of several enlightened caliphs, the elite poured their considerable wealth into creating libraries and funding learning. Scholars flocked to be part of the intellectual endeavour, and manuscripts were brought from across the Middle East and beyond to be translated and the knowledge they held put to use.
Astronomy and mathematics were two of the subjects most urgently pursued, and the achievements of the scholars who studied them were truly astonishing. They worked out the Earth’s circumference correctly to within a few hundred miles. They built the first observatory in the Muslim world, where they produced data that transformed human understanding of the universe. They translated, corrected and improved ancient Greek scientific theories, combining them with those from India and with their own ideas, propelling knowledge forward.
Numbers have power
During this intellectual heyday under al-Ma’mun, there were several impressive mathematicians in Baghdad, but the most talented was Muhammad ibn Musa al-Khwarizmi. His name suggests that his origins lay in the province of Khwarazm, far to the north-east on the shores of the Aral Sea. If that’s the case, then he – as with so many of the stars of the Baghdadi intellectual scene – was not Arab but Persian, though he always wrote in Arabic.
Al-Khwarizmi read Brahmagupta’s Siddhanta and realised that the Hindu-Arabic numerals and place-value system had much more potential than the systems currently used in the Muslim empire, which used finger-reckoning and aspects of the sexagesimal system, and expressed fractions in words instead of numbers. Al-Khwarizmi’s Book on Addition and Subtraction after the Method of the Indians was the first book in Arabic explaining the Hindu-Arabic system, with a chapter on each of the nine numerals and demonstrations of how to write numbers using the place-value system.
Al-Khwarizmi was a visionary mathematician. Indeed, we would now describe him as an outlier. With the exception of a few scholarly colleagues, no one seemed very interested in the new system of numbers he extolled, and the profound paradigm shift required to adopt it wasn’t achieved for several centuries. Yet, when this shift finally came, his book was to play a key role: it was translated into Latin in the 12th century, and became an important part of the European intellectual tradition.
Al-Khwarizmi’s name is little known in Europe today, but it has survived in the word algorithm, derived from the Latinised version of his name, ‘Algorismus’. He also gave us the word algebra, derived from al-jabr, part of the Arabic title of his Compendious Book on Calculation by Completion and Balancing, a practical guide to calculating written at the request of the Caliph al-Ma’mun. In it, al-Khwarizmi defined this discipline for the first time by describing different kinds of quadratic equations. Interestingly, he did this in words rather than with the system of notation used in algebra today, which developed during the Renaissance.
Copies of both of al-Khwarizmi’s books were taken from Baghdad to other places where they were studied and translated. By the tenth century, they had reached Spain, most of which was under Muslim rule at that time. During the 11th and 12th centuries, Christian forces in the north of the Iberian peninsula began conquering the great cities of al-Andalus. Toledo fell in 1085, and over the following decades European scholars came to the city in search of Arabic books, including texts by al-Khwarizmi, which they translated into Latin.
These scholars may have already been acquainted with the forms of the numerals themselves, which were present on a certain type of abacus (counting board) thought to have been introduced by a tenth-century monk named Gerbert (later Pope Sylvester II) whose talent and passion for mathematics took him to Spain in search of knowledge. Thus the Hindu-Arabic numerals and system of place-value were gradually introduced to Europe. It was a slow process, in part because of resistance from Christians who regarded the numerals as evil and dangerous – simply because they came from the Muslim world.
Fibonacci and the calculation revolution
The most important figure in the transmission of the Hindu-Arabic system to Europe was not Spanish but Italian, and learned the numerals not in Spain but in Africa: Leonardo of Pisa, known today as Fibonacci (though that name was applied to him only from the 19th century). Born around 1170, he was the son of a successful Pisan merchant posted to the chamber of commerce in Bougie (now called Béjaïa, in Algeria) during an era when Pisa was one of the four major Italian mercantile powers – the others being Amalfi, Venice and Genoa – with trade links and settlements across the Mediterranean world.
In Bougie, Arab mathematicians taught the young Pisan the Hindu-Arabic numerals, probably using al-Khwarizmi’s works, and showed him the brilliance of the place-value system, along with the other wonders of their mathematical tradition.
As a teenager, Fibonacci travelled around the eastern Mediterranean with his father, thereby enjoying opportunities to compare several systems of calculation in use at that time. He quickly recognised the enormous potential of the Hindu-Arabic system to transform learning in the west. In 1202 he wrote a book titled Liber abbaci (Book of Calculation). In this book, the first original work in Latin on the subject, he explained the workings of each of the numerals and the method of writing numbers in order according to their value.
By the 1220s, Fibonacci was back in Pisa, and spent time at the court of Frederick II, Holy Roman Emperor, a man apparently so clever and magnificent that he was nicknamed Stupor Mundi – ‘Wonder of the World’. Fibonacci flourished under Frederick’s patronage and in 1228 he produced a revised edition of the Liber abbaci with a greater focus on practical application in commerce. It detailed how to work out transactions in different currencies, and how to use different systems of weights and measures – methods that became increasingly important as Europe grew in prosperity and the mercantile world expanded and developed. Merchants needed to be able to carry out complex calculations and record their accounts effectively – something that was made possible by the Hindu-Arabic system of numerals as expounded by Fibonacci.
Fibonacci’s initial influence was the introduction of the Hindu-Arabic numerals and the place-value system to people who needed mathematics for practical purposes: merchants, surveyors and architects who used them to carry out the calculations needed in their working lives. The more complex aspects of his writings were taken up later and helped to bring about advances in theoretical mathematics. In the 15th and 16th centuries, scholars began to use the numerals in their exploration of algebra, especially after 1585 when the Flemish mathematician Simon Stevin published an innovative pamphlet on decimal fractions.
The introduction of the decimal point set people thinking in new directions about mathematics, paving the way for alternative concepts such as logarithms, and negative and complex numbers. As usual, there were many practical applications, too: measurement in engineering and surveying, calculation in astronomy and commerce. By this stage, algebraic problems and equations were being written down using digits, symbols and letters, rather than words, enabling scholars to express and compute extremely complex problems.
From the mid-1450s, the printing press – a machine using moveable type that was invented by the German craftsman Johannes Gutenberg around 1440 – transformed the world of knowledge. As the number of books available grew exponentially, their price fell, making them accessible to many more people. It has been estimated that by 1500, just half a century after printing began in Europe, some 20 million books had been produced. Printing also made texts more standardised, helping to fix the form of the Hindu-Arabic numerals and making them widely recognised and known. At the same time, scholars began translating mathematical texts into vernacular languages (Italian, German, English), bringing the mathematical concepts they contained into the everyday lives of Europeans.
By 1550, very few people in Europe were still using the old Roman numeral system to keep their accounts. The simplicity and elegant brilliance of the Hindu-Arabic system had finally won through, opening up astonishing new avenues in the fields of mathematics and the sciences, forging the world we inhabit today – a world that runs on infinite sequences of numbers.
Codes, algorithms, data: these are the lifeblood of our shiny 21st-century universe and the technological changes seen over the past three decades. It is no coincidence that our name for this phenomenon is the ‘digital revolution’.
Violet Moller is a historian and writer. Her latest book is The Map of Knowledge: How Classical Ideas were Lost and Found – A History in Seven Cities (Picador, 2019)