Zero (symbol 0) is the number that stands for an empty quantity. It is also the digit that marks an empty place in positional notation. In modern mathematics, zero is the integer between −1 and 1. It is the additive identity, which means that adding it to any number leaves that number unchanged. Several cultures used a sign to mark an empty position long before anyone treated zero as a number. Indian mathematicians of the first millennium CE were the first to give it full arithmetic status, and it reached the rest of the world through the Hindu–Arabic numeral system.
Mathematical properties
Zero has a set of defining properties. For any number a, a + 0 = a and a × 0 = 0. Zero is neither positive nor negative. It is an even number because it is divisible by 2. Its factorial is defined as 0! = 1. In algebra, a "zero" or "root" of a function or polynomial is any input at which the output is 0. In abstract algebra, the symbol 0 names the identity element of an additive group and the additive identity of a ring or field.
Division by zero is undefined in ordinary arithmetic. No number b satisfies b × 0 = a when a is not zero, and every number satisfies b × 0 = 0, so neither case gives a single answer. In calculus, expressions such as 0/0 or 0⁰ are called indeterminate forms. Their values in a given problem are worked out using limits.
Mathematicians disagree on whether zero is a natural number. Many logicians and computer scientists count from 0, while other traditions start at 1. Some formulations of the Peano axioms take 0 as the first natural number. In set theory, the standard von Neumann construction defines 0 as the empty set and builds every later natural number from it.
Placeholder zeros in early numeral systems
A positional system needs some way to show that a place is empty. In Mesopotamia, the Babylonians at first left an empty space in their cuneiform numbers. When that became confusing, they added a symbol of double angled wedges to mark the empty column. This sign is usually dated to around 300 BCE. Babylonian scribes used it only between digits, not at the end of a number, and it was never used as a number in calculation.
Zero was developed independently in Mesoamerica. The Maya used a base-20 system, and their zero took several forms depending on context: a seed shape for arithmetic, a flower for calendar dates, and sometimes a conch shell or a human head in profile. Long Count calendar dates depend on zero. The earliest known Long Count date, on Stela 2 at Chiapa de Corzo in Chiapas, is from 36 BC. Because the earliest dates come from outside the Maya homeland, scholars still disagree about which culture invented it.
In China, counting-rod arithmetic marked an empty place with a blank space for centuries. A written circle for zero appears in Chinese mathematical texts later. Greek astronomers, working in the sexagesimal tradition inherited from Babylon, sometimes used a small circular sign in astronomical tables. In ancient Greece more broadly, the idea of a void was a contested philosophical question, as in Aristotle's arguments against empty space.
Zero as a number in India
The decisive step came in India, where the decimal place-value system grew up alongside the Sanskrit word śūnya, meaning "empty" or "void." The Brāhmasphuṭasiddhānta of Brahmagupta, written in 628 CE, is the first text that treats zero as a number in its own right and sets out how to calculate with it, including dividing by zero. Brahmagupta gave rules for adding, subtracting and multiplying with zero. His treatment of division differs from modern conventions, and later Indian mathematicians such as Bhāskara II revisited it.
Dating the earliest written Indian zeros has been difficult. The Bakhshali manuscript is a birch-bark arithmetic text that uses a dot as a placeholder. In 2017, Oxford researchers announced that some of its pages dated from between 224 and 383 CE, a claim that was widely reported. However, in October 2024 the University of Oxford revised its earlier dating to 799–1102 CE. The radiocarbon report notes that a second test on Folio 16 showed that the first result had been inaccurate.
The earliest dated zeros in stone come from inscriptions. In Cambodia, a Khmer inscription records the Khmer year 604 (682–683 CE) with a zero digit. In India, an inscription at the Chaturbhuj Temple in Gwalior, dated to 876 CE, records a garden of 187 by 270 hastas and 50 garlands offered every day. The final digits of 270 and 50 are written as small circles, the familiar O-shaped zero.
Transmission and etymology
Indian numerals and zero reached the Islamic world in the period of the Abbasid Caliphate. They were explained in the ninth-century arithmetic of al-Khwarizmi. The Arabic word ṣifr (صفر), which meant "empty" before Islam, came to mean zero when it was used to translate the Sanskrit śūnya.
Zero entered Europe in the 12th century, when al-Khwarizmi's book was translated into Latin. Fibonacci then promoted Indian numerals in his Liber Abaci (1202). In that book he called zero zephirum, from the Arabic ṣifr, meaning "empty" or "nothing." The word became zefiro in Italian, which was then contracted to zero in Venetian. The Medieval Latin form cifra, from the same Arabic root, gave English the word "cipher." Hindu–Arabic numerals made written arithmetic faster than the abacus or Roman numerals, but they were adopted slowly in European commerce.
Applications
Zero is basic to the real and complex number systems, to coordinate geometry (where the origin is the point (0, 0)), and to linear algebra (the zero vector and zero matrix). In binary, 0 and 1 are the only digits, so zero is built into digital computers at the most basic level. Many programming languages number array positions from zero, and the IEEE 754 floating-point standard distinguishes +0 from −0.
In the sciences, zero often marks a reference point rather than "nothing." Temperature scales are an example: the Celsius zero is the freezing point of water, while zero kelvin is absolute zero, the lowest temperature possible in thermodynamics. Calendars show the same gap between conventions. The traditional BC/AD system has no year zero, whereas astronomical year numbering includes one.
References
- The Story of India's Oldest 'ZERO' // '0' Inscription at Gwaliorasiwalkintothesettingsun.substack.com
- Zero in the pre-Columbian Americas Ximena Catepillán1 Alonso Méndez2mayaexploration.org
- Who Came Up With the Concept of Zero and Why It Matters - ScienceInsightsscienceinsights.org
- Maya numeralsen.wikipedia.org
- Ancient Text Reveals New Clues to the Origin of Zeronationalgeographic.com
- Carbon Dating Reveals the History of Zero Is Older Than Previously Thoughtsmithsonianmag.com
- Bakhshali manuscripten.wikipedia.org
- Radiocarbon dating of the Bakhshālī manuscriptora.ox.ac.uk
- Mathematical Treasure: The Cambodian Zeroold.maa.org
- 0en.wikipedia.org
- Much ado about Zeroarxiv.org
- Zerofreemathhelp.com
- Elementary Arithmetic - ID:5c1177751950adocumento.mx