Why Zero Was Such a Difficult Idea

Nobody needed a mathematician to explain the experience of having no money. Humanity understood absence long before it agreed to give absence a place in arithmetic. An empty storeroom could communicate its condition perfectly well without a special symbol.

The difficult step was making nothing behave like something you could calculate with. Once admitted, zero would sit beside other numbers, change their appearance and occasionally bring the entire operation to a halt. It remains capable of ruining a spreadsheet without occupying any more space than a perfectly cooperative seven.

Zero looks inevitable because we learn it before we have much opportunity to object. Its history suggests that the obvious can take a considerable amount of work.

A vacancy needs an address

Start with the difference between 25 and 205. The zero in the second number tells us there are no tens, while keeping the two in the hundreds position. Remove it and the remaining digits describe something else.

A blank could perform that job, but blanks are ambiguous. Has somebody deliberately left a position empty, spaced the symbols generously or failed to finish the number? These are uncomfortable possibilities when the number concerns how much somebody owes you.

In a positional system, a digit’s location contributes to its value. An empty position therefore needs a dependable way of announcing that it is empty.

Babylonian mathematics developed a placeholder for this purpose within its base-sixty notation. As the University of St Andrews’ history of zero explains, that did not amount to treating zero as a number in its own right.

It was a sign telling the other signs where to stand. Giving it arithmetic responsibilities was another step.

Nothing is not the same as unknown

The distinction is easy to appreciate through a form. A box containing zero says that a quantity has been established and found to be none. An empty box may mean nobody has checked.

A warehouse with zero spare parts has a known problem. A warehouse whose spare-parts column is blank has an additional problem: it does not yet know which problem it has.

We still muddle these states. Software can treat a missing value as though it were zero, turning an absence of information into a confident numerical claim. The resulting calculation may be impeccably executed and entirely misleading.

Zero allows absence to become explicit. It says that the position has been considered, rather than overlooked.

That is a surprisingly substantial intellectual achievement for a mark whose principal visual feature is having nothing in the middle.

A number with unusual manners

Counting begins comfortably with things. One animal, two jars, three people attempting to claim the same inheritance. Zero asks us to include the quantity left when there are no things to count.

Subtraction supplies a route. Three minus three leaves zero. Once that result belongs to the number system, it becomes reasonable to ask what happens when we add it, multiply by it or use it elsewhere.

The Indian mathematician Brahmagupta set out rules involving zero and negative numbers in his Brahmasphutasiddhanta in 628. The St Andrews biography describes his treatment of positive quantities as fortunes and negative quantities as debts.

Zero could occupy the point at which the two cancelled. Nothing now had a formal relationship with something, rather than simply marking its disappearance.

The financial vocabulary is helpful. You can understand the difference between possessing five and owing five without requiring either condition to appear as a pile of objects. Arithmetic was becoming capable of describing relationships as well as inventories.

Division refuses to cooperate

Most of zero’s behaviour is reassuring. Add it and the number stays the same. Multiply by it and the result is zero. Division is where the apparently modest newcomer begins asking difficult questions.

Six divided by two is three because three multiplied by two gives six. To divide six by zero on the same basis, we would need a number that produces six when multiplied by zero. There is no such number in ordinary arithmetic.

Zero divided by zero creates the opposite difficulty. Every number multiplied by zero gives zero, so the operation cannot select a unique answer.

Brahmagupta’s rules did not resolve division by zero as we understand it today. That is part of the history of working out a new mathematical object’s behaviour. Introducing a useful idea does not mean immediately discovering every condition under which it behaves sensibly.

Nor does declaring the answer “infinity” settle the issue. Some expressions grow without bound as a denominator approaches zero, but approaching a value and dividing by it are different operations.

Zero was not merely another symbol to add to the cupboard. It required people to reconsider what the existing rules permitted.

There was more than one route to nothing

The Maya also developed zero notation independently, using it in their numerical and calendrical systems. Archaeologist Diane Davies’ account of the Maya calendar describes the familiar combination of dots, bars and a shell-shaped sign for zero.

This gives us a less cinematic history than a single genius inventing nothing while everybody else continued counting sheep. Different societies encountered related representational problems and developed their own solutions.

The Indian development of zero within decimal arithmetic was nevertheless central to the system that travelled through the Islamic world and eventually became familiar in Europe.

A concept can have several histories without every contribution becoming interchangeable. The placeholder, the symbol and the rules for calculating with it are connected achievements, not necessarily one event.

We prefer inventions to have birthdays. Ideas are often inconsiderate about supplying them.

Merchants needed more than a revelation

Leonardo of Pisa, better known as Fibonacci, helped explain Hindu-Arabic calculation to a European readership in Liber abaci, first written in 1202.

The Museo Galileo’s account of the medieval abacus tradition places the book within the transmission of mathematical knowledge through Mediterranean contacts and the development of schools teaching practical calculation.

The setting is instructive. Better notation had to become usable knowledge. People needed methods they could learn and apply to the transactions confronting them.

The advantages were considerable, but a useful system still had to enter habits, teaching and working practices. Knowing that something is possible and being able to rely on it are different stages.

Fibonacci did not invent zero. His contribution belongs to the equally important business of making existing knowledge available to people who could do something with it.

The difficulty we have forgotten

Looking backwards, it is tempting to interpret the slow development of zero as a failure to grasp a simple truth. But the people involved did not begin with our notation and somehow neglect one of its buttons.

They worked within systems that could already perform useful calculations. A new symbol or rule had to solve a problem without creating unacceptable confusion elsewhere.

We inherit the result after generations of development have made it teachable. The child writing 205 benefits from decisions whose difficulty has disappeared into the convention.

That is one of the quieter forms of progress. An achievement becomes so dependable that it no longer looks like an achievement.

Zero gave us a way to distinguish an empty position from a missing mark, a balanced account from an unfinished calculation, and a known absence from something nobody had bothered to establish.

Somewhere, a spreadsheet is still treating a blank cell as zero. Several centuries of mathematical work have supplied the distinction. Somebody now has to persuade the spreadsheet’s owner to use it.