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The concept of zero as both a placeholder and a number in its own right developed independently in multiple ancient civilisations, including Mesoamerica and India, before spreading via the Islamic world to Europe.

The Indian mathematician Brahmagupta gave the first known formal rules for arithmetic with zero, including its role in negative numbers, in his 628 CE text the Brāhmasphuṭasiddhānta. These rules were later transmitted through the Islamic world (notably via al-Khwārizmī) and reached Europe by the late Middle Ages.

Leonardo of Pisa ("Fibonacci") introduced Hindu-Arabic numerals, including zero, to a wider European audience in his 1202 book Liber Abaci. Adoption was gradual: several Italian cities and merchant guilds in the 13th–14th centuries restricted or banned the use of Hindu-Arabic numerals (including zero) in official ledgers for a time, preferring Roman numerals, which were seen as harder to alter fraudulently.

Zero is the additive identity (n + 0 = n for every number n) and is excluded from the prime numbers because primality is only defined for whole numbers greater than 1. Division by zero is left undefined in standard arithmetic because no number, multiplied by zero, can produce a nonzero result — the operation has no consistent answer.

By mathematical convention, any nonzero number raised to the power of 0 equals 1 (n⁰ = 1), chosen specifically to keep the standard laws of exponents consistent. Similarly, 0! (zero factorial) is defined as 1 — a convention, not a count of arrangements, that keeps formulas like combinations and permutations working correctly when zero items are involved.

Many everyday systems use zero as a fixed reference point rather than "nothing": 0 metres for sea level, 0°C for the freezing point of water on the Celsius scale, the final moment of a countdown, and the origin of a coordinate grid. These uses treat zero as a precise, meaningful value, not an absence.

Several everyday systems deliberately avoid starting at zero: a standard six-sided die is numbered 1–6, not 0–5; ordinary playing cards have no zero card; and most rankings, floor numbering (outside continental European "ground floor" conventions) and jersey numbering conventionally begin at 1. This mirrors a broader pattern — everyday, human-facing counting usually starts at 1, while technical and computing systems more often start at 0.

In binary, the base-2 number system underlying digital computing, 0 and 1 are the only two digits, with 0 conventionally corresponding to "off" or "false" in digital logic. Zero-based indexing — starting a count at 0 rather than 1 — is a widespread convention in programming languages including C, Python and JavaScript, originating from how array memory offsets are calculated. "Null" (the deliberate absence of a value) and "zero" (a specific numeric value) are related but distinct concepts, often confused in casual usage.