The History of Zero

The Problem of Counting Without Zero

For thousands of years, different civilizations developed systems for writing numbers without a symbol for the absence of quantity. The Egyptians and Romans, for example, used a different symbol for each order of magnitude (units, tens, hundreds), which made representing large numbers cumbersome and made arithmetic operations, adding, subtracting, multiplying, very difficult to carry out in writing. That's why much of everyday calculation in the ancient world was done with abacuses or counting boards, rather than written procedures.

The problem became even more concrete in positional systems, where the value of a symbol depends on the place it occupies. Without a sign to mark an empty position, it was impossible to distinguish in writing, for example, 61 from 601 or from 6,001: ambiguity was inevitable whenever a digit was missing in some order. Solving that gap, both in writing and, later, in calculation, was the challenge that different civilizations faced along separate paths, and it's the starting point of this story.

Babylon: The First Placeholder

The starting point is in Mesopotamia. The Babylonians inherited from the Sumerians a positional numeral system in base 60, the same principle we use today to measure hours and minutes, developed about 4,500 years ago and passed down through the Akkadian Empire around 300 BCE.

In a positional system, the value of a symbol depends on the place it occupies. The problem was that the Babylonians had no way to distinguish, for example, the number 61 from the number 3,601. In both cases, without a symbol marking the absence of a digit, the writing was ambiguous. The solution they found was to leave a blank space between symbols, and later to adopt a mark that served that function. But that marker was not a number, it wasn't used for calculation, only to indicate an empty place within a figure.

The Americas: The Maya Zero

On the other side of the planet, with no contact whatsoever with Mesopotamia, the Maya civilization independently arrived at a similar idea, around 350 CE. Their base-20 numeral system included a glyph shaped like a stylized shell to represent zero. They used it mainly in their calendar calculations, where it was essential for keeping an exact count of time cycles.

The Maya conceived of zero autonomously, with no link to Babylon. However, their numeral system remained confined to Mesoamerica and never connected with the mathematical traditions that would later shape modern arithmetic.

China: The Empty Slot in the Rod-Numeral System

In China, the numeral system using bamboo counting rods, dating back to at least the second millennium BCE, made it possible to represent any quantity by arranging rods in columns corresponding to units, tens, hundreds, and so on. It was a decimal positional system, very advanced for its time, but for centuries there was no symbol for zero. When a digit was missing, a blank space was simply left in the corresponding column.

It wasn't until the 13th century, during the Yuan dynasty, that China officially adopted a circular symbol (〇) to mark those empty positions. It appears documented in the Shushu Jiuzhang ("Mathematical Treatise in Nine Sections," 1247), a work by the mathematician Qin Jiushao. There are records of earlier contact with the Indian numeral system, through an Indian astronomer who worked at the Chinese court in the 8th century, although that system was not adopted at the time.

India: Zero Becomes a Number

The decisive turning point occurred in India, between the 5th and 7th centuries. It was there that zero stopped being merely a placeholder and came to be treated as a real quantity, with its own properties and rules of operation.

The central figure in this shift is Brahmagupta, a mathematician and astronomer born around 598 CE in Bhillamala (present-day Bhinmal, Rajasthan). In 628 CE, in his work Brahmasphutasiddhanta, Brahmagupta formalized zero, defining it as the result of subtracting a number from itself, and for the first time established explicit rules for adding, subtracting, and multiplying with it. He is considered the first mathematician to treat zero as a number in its own right, rather than as a mere gap between digits. For this and other contributions, including foundational work in algebra, he is recognized as one of the central figures of Indian mathematics.

The very name we use today preserves that origin. In Sanskrit, zero was called shunya, meaning "empty" or "void."

The Bridge to Europe: Baghdad and Pisa

Indian arithmetic, with its zero and its nine digits, would not have reached the West without two key links.

The first is Al-Khwarizmi, a 9th-century Persian mathematician who worked at the House of Wisdom in Baghdad. Al-Khwarizmi meticulously studied and spread the Indian numeral system throughout the Arab world, and his Latinized name is the origin, centuries later, of the word "algorithm." Through Arabic, the Sanskrit word shunya was translated as sifr, also meaning "empty," a term from which both "cipher" and, later, "zero" derive.

If you're interested in learning more about the origin of the algorithm, we recommend reading → What Is an Algorithm?

The second link is Leonardo of Pisa, known as Fibonacci. In 1202, he published the Liber Abaci (Book of Calculation), in which he introduced Europe to the Hindu-Arabic numeral system, including zero, as a tool for commercial and accounting calculations. Fibonacci had learned this system during his travels in North Africa, where his father worked as a merchant, and became convinced of its superiority over Roman numerals, which lacked a zero and were far more limited for calculation.

From sifr to the medieval Latin zephirum, and from there to the Italian zefiro and finally to "zero," the word's journey passed through three languages before settling into the modern European languages.

Why This Mattered So Much

Before positional zero, carrying out a long arithmetic operation, an addition, a subtraction, a multiplication of several digits, was a craft-like task, with no general rules that worked for any number. Zero changed that at its root:

• It allowed a single set of simple rules to work for calculating with any quantity, regardless of its size.
• It freed mathematicians from the mechanical repetition of calculations.
• It opened the door to deeper questions about the properties of numbers.
• It laid the foundations of algebra as we know it.