When I was a child at school we learned about two types of numerals: Roman and Arabic. The Arabic ones are what we use today, where a sequence of digits such as 513 represent five hundreds, one ten, and three units. The Roman ones were letters of the alphabet such as I (meaning one), V (meaning five) and X (meaning 10), where for example XXVI meant 26. The reason the Romans chose those particular letters is that they started recording numbers by using special symbols for 1, 5, 10, 50, 100, 500, 1000, etc. In time these symbols were replaced by similar-looking letters of the Roman alphabet I, V, X, L, C, D, M, etc., familiar to us from dates on monuments, such as MDCLXVI, meaning 1666, the date of the Fire of London.

When alphabets came into widespread use, in the first millennium BC, their letters were also used for numbers, starting with A for one, B for two, and so on. The first nine were used for the numbers one to nine, the next nine for the tens from 10 to 90, then the hundreds for 100, 200, etc. For instance, the Greek alphabet, supplemented with three archaic letters no longer in use, supplied 27 symbols, providing for the nine units, the tens from 10 to 90, and the hundreds from 100 to 900. Beyond that, other symbols were used. It was a similar thing with other alphabets in the Ancient Near East.

Systems of counting date back to pre-history and when writing was invented various methods were used to represent numbers. Among Britons, for example, writing was done in runes, and there was also a runic system of numbers. This changed when the country was incorporated into the Roman Empire, and Roman numerals became standard.

Arabic numerals have a rather different history. Forget about the symbols themselves, which in Arabic countries usually have a different form from those we are used to. The main point is that they use what we call a place-value system, where, for instance, 513 is a quite different number from 135. This base-10 system originated in India by the eighth century AD, and by the first half of the ninth century had penetrated the highly literate Arabic world. It was described in Arabic texts, notably one by the Persian scholar Al-Khwarizmi, On the Calculation with Hindu Numerals. Translated into Latin this became Algoritmi de numero Indorum, which led to our modern term ‘algorithm’. Al-Khwarizmi’s later book, Calculation by Completion and Balancing (Al-jabr), which presented the first systematic solution of linear and quadratic equations, led to the modern term ‘algebra’. His slightly younger contemporary Al-Kindi (Latin: Alkindus), from southern Iraq wrote, a four-volume work, On the Use of the Indian Numerals, which helped spread the system.

They knew very well that the place-value system came from India, and we in the West now refer to Indo-Arabic numerals in recognition of this fact. The system had found its way to Europe via North Africa. A young man called Leonardo had learned it as a boy in what is now Algeria, where his father was the director of a trading post in the port city of Béjaïa. Later called Leonardo Bigollo Pisano (‘Leonardo the Traveller from Pisa’), he travelled widely around the Mediterranean coast, meeting various merchants and learning about their systems of arithmetic. In his book Liber Abaci (Book of Calculation), written in Latin and published in 1202, he popularised the Indo–Arabic numeral system.

Leonardo of Pisa was the foremost mathematical mind of his day, known to modern mathematicians as Fibonacci (Filius Bonacci — son of Bonacci). His name is associated with a sequence of numbers that he used in his book, where each number is the sum of the previous two: 0, 1, 1, 2, 3, 5, 8, 13, 21, etc. It is important to natural growth, and the number of petals on a flower tend to be one of those in the sequence.

This history all seems fairly simple. The Indians invented the place-value system taken up by the Arabic world, and its advantages for arithmetic led to a replacement of more cumbersome systems such as Roman numerals. But then there is the question of fractional quantities. How were these treated? Fractions such as two fifths were already part of ancient Indian and Greek mathematics, but decimal fractions seem to have evolved in the Arabic world. The Persian mathematician Jamshīd al-Kāshī claimed to have invented them in the 15th century, but an Arabic mathematician named al-Uqlidisi, active in Damascus and Baghdad, got there well before him. In the 10th century he wrote the earliest surviving book on the place-value system, The Arithmetics of India, which appeared in around 952 and is notable for its treatment of decimal fractions.

This story has a very long prologue that is sometimes forgotten, and unknown to many. In the Ancient Near East a place-value system already existed, along with its use for fractional quantities, in southern Iraq. This was the location of the Sumerians and Akkadians, who built the world’s first cities in the Tigris and Euphrates delta near the coast of the Gulf. They wrote in cuneiform script and, before 2000 BC, invented a place-value system that worked to base-60 rather than base-10. For European historians, cuneiform was unknown until the 19th century, and its modern decipherment only began in the second half of that century. Little wonder then that 19^th-century historians of mathematics knew nothing about the vital contributions from this part of the world.

Such a great civilisation had trading links by sea via the Gulf, southwards along the coast of Arabia to Oman, and eastwards around the coast of Iran to the Indus Valley. Sea trade along both routes flourished during the third millennium BC, so it is entirely possible that the place-value system was known to scribes in India long before the Indo-Europeans arrived around 1500 BC. The problem is always a lack of evidence, particularly with ancient remains subject to decomposition.

The reason we know so much about the Sumerians and can date their invention of place- value numbers is that they wrote on clay, which preserved their work – better than the more modern use of parchment, or indeed paper, which can be destroyed in a fire. Clay tablets caught in a fire are preserved; for records of particular value they would fire them in a kiln.

In summary the civilisations of the ancient Near East used the place-value system for writing whole numbers and fractions well before the Arabs or the Indians, and it was all done to base-60, which made calculations easier than base-10. When we learned of base-60 through their astronomy, where fractional quantities were essential, it was via the Greeks – we had no idea where it came from.