Zero and the decimal system: how humanity learnt to count nothing

Whether it is a mobile number, an electricity bill or a shop receipt, we get by every day with just ten digits: one to nine, and zero. With these ten symbols we can write the smallest and the largest sums, because the value of each digit is fixed by its place. In 205, the two means two hundred, and the zero shows that the tens place is empty. Humans learnt this simple idea over centuries.

A mark for an empty place

The MacTutor archive at the University of St Andrews has a detailed article on the history of zero. According to it, the people of ancient Babylon used a place-value system for more than a thousand years, but it had no zero. Around 400 BC they began to put two slanted wedges to show an empty place. The Greek astronomer Ptolemy used a round symbol around 130 AD, but he saw it not as a number but as a kind of mark for a gap.

According to MacTutor, steps in this direction were taken in other parts of the world too. The Maya civilisation of Central America was using a base-20 place-value system with a symbol for zero by 665 AD. In India, the mathematician Aryabhata used a place-value system around 500 AD, although it did not yet have a symbol for zero. According to MacTutor, by about 650 AD zero was being used as a number in Indian mathematics.

Brahmagupta’s rules

According to MacTutor, Brahmagupta was born in 598 AD, probably in Ujjain. At about the age of 30, in 628 AD, he wrote the "Brahmasphutasiddhanta" at Bhillamala (today’s Bhinmal). It is a book on astronomy and mathematics in 25 chapters. He later became head of the Ujjain observatory, which was then the greatest centre of mathematics in India.

What sets Brahmagupta apart is that he treated zero not merely as a mark for an empty place but as a proper number, and wrote rules for calculating with it. He called positive and negative numbers "fortunes" and "debts". MacTutor quotes some of his rules as follows: a debt taken from zero gives a fortune; the product or quotient of two debts is a fortune; the sum of zero and a negative number is negative. These are the same rules every schoolchild learns today, for example that a negative times a negative is a positive.

Not every rule was correct. Brahmagupta wrote that zero divided by zero is zero, which MacTutor calls wrong. He gave no clear answer for dividing a number by zero. According to MacTutor, two hundred years later, in 830 AD, the mathematician Mahavira also gave a wrong answer to this question.

The Bakhshali manuscript

One chapter of this story is linked to present-day Pakistan. According to MacTutor, in the summer of 1881 a farmer near the village of Bakhshali in Mardan district found a mathematical manuscript written on birch bark while digging in an old stone enclosure. About 70 leaves survive. They contain rules and examples of arithmetic, algebra and some geometry, and a thick dot is used for zero. It has been kept in the Bodleian Library, Oxford, since 1902. Experts do not agree on its date. In 2017, radiocarbon testing of three leaves gave different dates ranging from the third to the tenth century AD, and the interpretation of these results is also disputed among scholars.

According to MacTutor, the first record of zero in India whose date is known and whose authenticity everyone accepts is an inscription at Gwalior from 876 AD.

From Baghdad to Europe

How did the system spread further? According to MacTutor, around 770 AD an astronomical text from India reached the court at Baghdad. It later became the basis of the astronomical tables of Muhammad ibn Musa al-Khwarizmi. Al-Khwarizmi (about 780 to 850) was a scholar of the House of Wisdom in Baghdad. He wrote a book on Indian numerals. Its Arabic text is lost, but a Latin translation survives under the title "Algoritmi de numero Indorum". The computer science term "algorithm" comes from al-Khwarizmi's name through this Latin title. According to MacTutor, his book described the whole system, including zero.

The system reached Europe on a large scale through Leonardo Pisano of Italy, known as Fibonacci. According to MacTutor, he was born in Pisa around 1170. His father was the representative of Pisan merchants at the customs office in Bugia in North Africa (today’s Bejaia in Algeria), and it was there that Fibonacci learnt these numerals. In 1202 he wrote "Liber abaci", which described this method of calculation with the nine digits and the symbol for zero.

Zero today

Today bank accounts, school results, shop receipts and computer calculations are all written in these same ten digits and this place-value system. The story of zero passed through many hands, from Babylon to Bhillamala, Baghdad and Pisa, and one precious leaf of it also came out of the soil of Mardan. The lesson is that if even "nothing" is given a name and a rule, it can become the most powerful digit.