What Is Zero? The Number That Took Centuries to Accept

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 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, for which he also gave rules, 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. The rules as now understood:

• •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 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.

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 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.

Zero causes arguments even among people comfortable with numbers. Is it even or odd? It is even, because an even number is one that divides by two with no remainder, and zero divided by two is zero. Is it positive or negative? Neither: it is the boundary between them. Is it a natural number? That depends on the convention, since some textbooks start the counting numbers at one and others at zero. And in the rule that the factorial of zero equals one, zero is not being awkward; that value keeps formulas for counting arrangements working consistently.

The calendar is the most famous place where zero is missing. The system of numbering years from the birth of Christ was devised in the sixth century, before zero had reached Europe, so the year before AD 1 is 1 BC, with no year zero in between. That is why purists insisted that the third millennium began on 1 January 2001, not 2000, since a first millennium starting in AD 1 ran through the end of 1000. Astronomers, who need to do arithmetic across the gap, avoid the problem by using a numbering that includes a year zero.

Programmers meet zero in ways that surprise newcomers. Most programming languages number the items in a list starting from zero, so the first item is item 0 and the tenth is item 9. The habit comes from how memory works: a list is stored as a block, and an item's number is really its distance from the start of the block, so the first item is zero steps away. The computer scientist Edsger Dijkstra wrote a short note in 1982 arguing that counting from zero is also the tidier convention for describing ranges of numbers, and most modern languages follow it.

Division by zero is handled in two different ways. When a program divides one whole number by zero, it usually stops with an error, since there is no answer to give. The standard for fractional, or floating-point, arithmetic used by almost every modern processor, first agreed in 1985, takes another route: dividing a positive number by zero gives a special value representing infinity, and dividing zero by zero gives a marker called NaN, short for not a number. The same standard even contains two zeros, positive and negative, which compare as equal but record which side of zero a calculation approached from, a detail that matters in some scientific work.

Most temperature scales put zero at a convenient but arbitrary point: the freezing point of water on the Celsius scale, or a cold salt mixture on Fahrenheit's original scale. Negative temperatures are common in winter, so zero on these scales is not the lowest possible value. Physics, however, has a temperature with a true zero.

Absolute zero, minus 273.15 degrees Celsius, is the point at which a system has the least possible thermal energy. The Kelvin scale starts there, so zero kelvin means the bottom, and nothing can be negative on it. Scientists have cooled atoms to within a tiny fraction of a degree of absolute zero, but the laws of thermodynamics say it can never quite be reached, which makes it a zero that is both real and unreachable.

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.