zero

Zero is a number that represents the absence of quantity and serves as the additive identity in arithmetic, meaning that adding zero to a number leaves it unchanged. In numeral systems, zero also functions as a placeholder that allows place-value notation to distinguish values like 10 from 1.

The idea of nothingness appears in many cultures, but zero as a number with its own value originated in ancient India. Indian mathematicians, notably Brahmagupta in the 7th century, developed rules for arithmetic that included operations involving zero. The concept spread to the Islamic world and was later introduced to Europe through the Hindu–Arabic numeral system, gaining widespread use after translations and continued adoption in the Middle Ages and Renaissance.

In mathematics, zero has several key properties: it is the additive identity (a + 0 = a), multiplication by zero yields zero (a × 0 = 0), and zero divided by a nonzero number is zero (0 ÷ a = 0). Division by zero is undefined, and exponent rules involve zero in nuanced ways (for example, 0^n = 0 for positive n). The value 0^0 is considered indeterminate in many contexts, though some areas define it for convenience.

Beyond arithmetic, zero appears in other disciplines. In set theory, the empty set has cardinality zero. In geometry and physics, zero often marks an origin or reference point, such as the coordinate system’s origin. In computing, zero is a fundamental binary digit, representing the false value in many logical systems and data encoding schemes. Zero thus plays a central role as both a concept and a numerical symbol across mathematics and science.