What Is Zero? The Number That Took Centuries to Accept
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The Romans built aqueducts and ran an empire without a symbol for nothing, and the Greeks, who invented geometry and proof, argued about whether nothing could be something and mostly decided it could not. The zero that sits on every keyboard came from India by way of the Islamic world and reached Europe in the thirteenth century, where it was regarded with suspicion for another three hundred years. It is two ideas in one symbol, and the second is much stranger than the first.
The placeholder
The first zero was a gap. The Babylonians, writing numbers in a system based on sixty, needed a way to tell 61 from 3601, and by around 300 BC they had a symbol of two small wedges to mark an empty column. It was never used at the end of a number and never treated as a quantity; it was punctuation. The Maya, independently, developed a shell-shaped symbol for an empty place in their calendar counts, and the Chinese left a blank space on the counting board. None of these went further, because none was combined with a full system of positional notation in which the value of a digit depends on its place.
That system came together in India. By the fifth century AD Indian mathematicians were writing numbers with nine digits and a dot or small circle for an empty place, so that the same ten symbols could write any number, and the arithmetic of adding and multiplying in columns that every school teaches follows directly. The word itself carries the route: the Sanskrit sunya, empty, became the Arabic sifr, which became the Latin zephirum, the Italian zero, and, by another branch, the English cipher.
The number
The second step was to treat the empty place as a quantity that could be calculated with. The astronomer Brahmagupta, writing in 628 AD, set out the rules: a number plus zero is the number, a number times zero is zero, zero minus a number is a negative number, which he also introduced, and a debt minus zero is a debt. He stumbled on one point, saying that zero divided by zero was zero, and division by zero has remained undefined ever since, for a reason that is easy to see: if six divided by zero were some number, that number times zero would have to be six, and nothing times zero is six.
Zero as a number allowed the whole number line, running through negatives, and it gave algebra its central move: to solve an equation, get everything on one side and set it equal to zero, so that the question becomes where an expression vanishes. Brahmagupta's rules in brief:
- •Adding or subtracting zero leaves a number unchanged
- •Multiplying by zero gives zero
- •Zero is the boundary between positive and negative numbers
- •Dividing by zero is not defined, since no number times zero gives anything but zero
- •Zero to a positive power is zero, and zero to the power zero is left undefined or set to one, depending on the context
The long road west
The Indian system reached Baghdad in the eighth century, and the mathematician al-Khwarizmi, whose name became the word algorithm, wrote a treatise on it around 825 that was translated into Latin three hundred years later. Fibonacci, who had learned the method from Arab merchants in North Africa, introduced it to Europe in his Liber Abaci of 1202, and Italian merchants took it up for bookkeeping because it made multiplication and division practicable without an abacus. The authorities were less keen. Florence banned the new numerals in 1299, on the ground that a zero could too easily be altered into a six or a nine, and Roman numerals remained standard in Europe for accounts until the sixteenth century. The Church had its own unease: nothing, in the theology of the time, was what God had created the world out of, and a symbol for it sat uncomfortably close to the void.
Why the Greeks did without
The Greeks, who could have invented zero, mostly chose not to. Their mathematics was geometric: a number was a length or an area, and a length of nothing was not a length. Aristotle argued that a void could not exist and that the ratio of a number to nothing was meaningless, and the philosophy held. Greek astronomers did use a symbol for an empty place in their tables, borrowed from Babylon, but it never entered their arithmetic. The Romans inherited their numerals from the Etruscans and their mathematics, such as it was, from the Greeks, and had no use for either kind of zero. It is one of the clearest cases in the history of ideas of a discovery blocked not by difficulty but by a conviction that it must be wrong.
Zero now
Zero is the origin of every coordinate system, the baseline of every measurement, the freezing point on one scale and absolute rest on another, and the number every computer is built on, since binary is nothing but zeros and ones. Calculus was invented to handle quantities that shrink towards zero without reaching it, and much of the trouble in physics comes from equations that produce zeros in denominators. A symbol for nothing turned out to be the most productive thing in mathematics, and the argument about whether nothing could be something was settled by finding out what happened when it was.
The takeaway
Zero is two ideas: a placeholder that marks an empty column and makes positional notation work, developed in Babylon and perfected in India, and a number that can be added, multiplied and set equal to expressions, whose rules Brahmagupta laid down in 628 AD. It reached Europe through Arabic mathematics and Fibonacci, was resisted for centuries as a merchant's trick and a philosophical void, and became the foundation of algebra, coordinates, calculus and computing.