Zero, large numbers and calculation — The World, with the Evidence

An old fragment, brown and damaged, covered with lines of writing in ink. — Photograph of the Bakhshali manuscript. National Geographic, Public domain, source

There is a digit that counts nothing. And yet, without it, almost nothing works.

That digit is zero. It took a very long time to come into existence, and it set mathematicians problems that they did not all manage to solve.

A hole in the number

Two hundred and five is written in digits as 205. Here the zero does not indicate the presence of something: it indicates that there are no tens.

This is its first function: to mark an empty place. Without it, 205 and 25 would be written the same way, and it would no longer be possible to tell them apart.

The idea seems simple. Yet many counting systems lacked it. In Mesopotamia, a scribe perhaps left a little more space, which was ambiguous. In China, the square was left empty on the rod board: the method worked, but only as long as the rods stayed in place.¹

To write a number on a medium and pass it on, a sign was needed.

The day nothing became a number

Marking an empty place is one thing; treating zero as a number with which one can calculate is another.

In 628, in South Asia, the mathematician Brahmagupta took that step. He wrote rules: what you get when you add zero, when you subtract it or when you use it with negative quantities — such as debts.²

But he stumbled over one question: what happens when you divide by zero?

Brahmagupta (628) states that zero divided by zero is zero.
Mahavira (c. 850) states that a number divided by zero does not change.
Bhaskara II (1150) proposes that the result is infinity.³

Three mathematicians, three answers, and none of them works. In mathematics, division by zero is quite simply impossible. It took more than a thousand years to accept this.

Counting grains of sand

At the other end of the learned world, people were asking the opposite question: how far can one go towards the very large?

In the third century BCE, the Greek scholar Archimedes tried to calculate how many grains of sand it would take to fill the universe. To answer, he invented a system capable of naming gigantic numbers, far beyond the words that existed in his language.⁴

In the same period, in South Asia, texts gave names to immense numbers. No merchant needed them: these numbers served to explore abstract ideas, not to count sacks of rice.

Writing down is not calculating

For centuries, people did not calculate directly with written digits. The digits served to record the result, while the calculation was done with objects.

In the Mediterranean, counters were moved on a board called an abacus.
In China, rods were used, then the beads of an abacus.
In the Andes, Inca accountants handled counters on a board with compartments called a yupana.⁵

It was the system that came from South Asia that changed everything: with place value and zero, one can set out a sum and calculate directly on paper.

Around 825, in Baghdad, the scholar al-Khwarizmi set out these methods of calculation in detail. His name gave us the word algorithm, and the title of one of his books gave us the word algebra. A century later, in Damascus, the mathematician al-Uqlidisi adapted these techniques to paper and wrote the oldest known text on digits after the decimal point.⁶

A regulation that does not say why

Between 1299 and 1316, in Florence, the statutes of the moneychangers’ guild forbade its members to enter amounts in Indo-Arabic numerals in their account books. They could still use them to calculate.⁷

It was long claimed that this ban was meant to prevent fraud, since a 0 could be altered into a 6 or a 9. But this explanation is only one hypothesis among others: it does not appear in the text of 1299.⁸

The story went far beyond that

An even stronger version circulates widely: zero is supposed to have frightened medieval Europe, to have been associated with the Devil or with black magic, and the Church is supposed to have banned these numerals.⁹

One historian has gone back over the evidence piece by piece. His conclusion is blunt: this narrative is false or unsubstantiated at nearly every level.¹⁰

He did better than refute it: he dated its birth.

Its foundations were laid in 1905, by a German scholar of Romance languages. Other authors then embellished it. The historian distinguishes two sources: some elements come from an unwarranted interpretation of real medieval sources, others from the imagination of certain modern authors.¹¹

This point is worth pausing on, for it reflects the approach of this whole encyclopaedia.

A legend about the Middle Ages, born at the beginning of the twentieth century, copied for more than a century — and one that can be dated, because someone took the trouble to trace the chain of citations back.

The Middle Ages were not afraid of zero. It is we who needed them to be afraid of it.

The adoption of these numerals was held back not by fear but by the force of habit. As early as 976, in the monastery of San Martín de Albelda, in La Rioja, the copyist Vigila and his two companions were already copying out the Indian numerals, writing that the Indians must be acknowledged to have a most subtle genius, and be granted the advantage in arithmetic and geometry.

But something is missing, and it is precisely the subject of this page: the codex shows only nine figures. Zero is not there. The sign for empty places would take centuries more to make the journey.¹²

Calculating with what?

Instrument	Region	Operating principle
Counter abacus	Mediterranean basin	Moving counters in grooves or columns
Counting rods	China	Arranging wooden or bone sticks on a grid-ruled board
Bead abacus	East Asia	Sliding beads threaded on rigid rods
Yupana	Andes	Board with hollows for distributing stones or seeds
Written calculation	South Asia, then the world	Writing out the steps of the operation directly with written numerals

Note 1

A much more serious problem was the fact that there was no zero to put into an empty position. The numbers sexagesimal numbers 1 and 1,0, namely 1 and 60 in decimals, had exactly the same representation and now there was no way that spacing could help. […] Perhaps the scribe left a little more space than usual between the 6 and the 9 than he would have done had he been representing 6,9. Now if the empty space caused a problem with integers then there was an even bigger problem with Babylonian sexagesimal fractions.

O’Connor, John J. and Robertson, Edmund F., “Babylonian Numerals”, MacTutor History of Mathematics, University of St Andrews, December 2000

A counting board consisted of a checker board with rows and columns. Numbers were represented by little rods made from bamboo or ivory. […] Rods moving slightly along the row, or not being placed centrally in the squares, would lead to the incorrect number being represented. […] There was still no need for a zero on the counting board for a square was simply left blank.

O’Connor, John J. and Robertson, Edmund F., “Chinese Numerals”, MacTutor History of Mathematics, University of St Andrews, January 2004

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Note 2

In particular he wrote Brahmasphutasiddhanta (Correctly Established Doctrine of Brahma), in 628. […] He gave some properties as follows:- When zero is added to a number or subtracted from a number, the number remains unchanged; and a number multiplied by zero becomes zero. He also gives arithmetical rules in terms of fortunes (positive numbers) and debts (negative numbers):-

O’Connor, John J. and Robertson, Edmund F., “Brahmagupta”, MacTutor History of Mathematics, University of St Andrews, November 2000

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Note 3

In 628 CE, the Indian mathematician and astronomer Brahmagupta claimed that “zero divided by a zero is zero.” At around 850 CE, another Indian mathematician, Mahavira, more explicitly argued that any number divided by zero leaves that number unchanged, so then, for example, 24 ÷ 0 = 24. Later, around 1150, the mathematician Bhaskara gave yet another result for such operations. He argued that a quantity divided by zero becomes an infinite quantity.

Martinez, Alberto, “Dividing by Nothing”, Not Even Past, Department of History, University of Texas at Austin, 12 April 2011

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Note 4

The Sandreckoner is a remarkable work in which Archimedes proposes a number system capable of expressing numbers up to […] He argues in this work that this number is large enough to count the number of grains of sand which could be fitted into the universe. There are also important historical remarks in this work, for Archimedes has to give the dimensions of the universe to be able to count the number of grains of sand which it could contain.

O’Connor, John J. and Robertson, Edmund F., “Archimedes of Syracuse”, MacTutor History of Mathematics, University of St Andrews, January 1999

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Note 5

Numerical calculations were done on a variety of yupanas in which piles of tokens, seeds, or pebbles were moved between different compartments of the yupana for the calculation to take place.

Kak, Subhash, “Guáman Poma’s Yupana and Inca Astronomy”, Indian Journal of History of Science, vol. 54, no. 4, 2019, p. 398-404

Quipucamayoc with a quipu and an Andean calculation board, called yupana. […] Researchers believe that one of the methods that the Incas had to make accounts was using small stones for more complex mathematical operations by a kind of ‘abacus’, called yupana.

Museo Chileno de Arte Precolombino, “Does the quipu was analogous to a system of calculation?”, exhibition Quipu: Counting with knots in the Inka Empire, 2003

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Note 6

Algoritmi de numero Indorum in English Al-Khwarizmi on the Hindu Art of Reckoning gave rise to the word algorithm deriving from his name in the title

O’Connor, John J. and Robertson, Edmund F., “Abu Ja’far Muhammad ibn Musa Al-Khwarizmi”, MacTutor History of Mathematics, University of St Andrews, July 1999, open access

His algebra treatise Hisab al-jabr w'al-muqabala gives us the word algebra

O’Connor, John J. and Robertson, Edmund F., “Abu Ja’far Muhammad ibn Musa Al-Khwarizmi”, MacTutor History of Mathematics, University of St Andrews, July 1999, open access

The dust board allowed this in the same sort of way that one can use a blackboard, chalk and a blackboard eraser. However, al-Uqlidisi showed how to modify the methods for pen and paper use. […] Al-Uqlidisi's work is historically important as it is the earliest known text offering a direct treatment of decimal fractions.

O’Connor, John J. and Robertson, Edmund F., “Abu’l Hasan Ahmad ibn Ibrahim Al-Uqlidisi”, MacTutor History of Mathematics, University of St Andrews, November 1999

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Note 7

Ainsi, à Florence, entre 1299 et 1316, il est interdit aux changeurs et prêteurs d’argent de la ville d’utiliser des chiffres arabes dans leur comptabilité (Quod nullus de arte scribat in suo libro per abbacum).

Thus, in Florence, between 1299 and 1316, the city’s moneychangers and moneylenders were forbidden to use Arabic numerals in their accounts (Quod nullus de arte scribat in suo libro per abbacum [that no member of the guild shall write in his book by means of the abacus]).

Conti, Marco, “Chiffres arabes et logiques administratives : premières notes sur l’utilisation des chiffres arabes dans la documentation fiscale de la commune de Bologne (milieu XIVᵉ-début XVᵉ siècle)”, Hypothèses, no. 24, 2023, p. 307-316, open access

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Note 8

The reasons behind this prohibition are a matter of speculation. A concern that the Hindu-Arabic numerals, which existed in a variety of forms, might lend themselves to forgery or fraud has been one of the more popular explanations, but other possibilities exist. […] It should be emphasized that no such comment appears in the source text.