Zero is the additive identity: the unique number 0 such that a + 0 = a for every number a. In a place-value numeral system, it is also the placeholder that distinguishes 102 from 12. The full arithmetical treatment of zero — including rules for addition, subtraction, and multiplication — was first explicitly stated by Brahmagupta in 628 CE. The concept has a rich and multi-civilisational history stretching back to Babylonian placeholder notation of c. 300 BCE.
A number that means nothing — and changes everything
Zero is such a basic part of mathematics today that it's easy to forget it had to be invented. The ancient Greeks, who contributed so much to mathematics, had no zero. Roman numerals have no zero. The idea of a number representing nothing — not just an absence, but a quantity that can be added, subtracted, and calculated with — was a conceptual leap that took humanity thousands of years.
Two roles of zero
Zero plays two distinct roles, and these were recognised at different times in history:
Placeholder: In the number 304, zero tells you the tens column is empty. Without a placeholder, positional notation breaks down — you can't distinguish 304 from 34. Babylonian scribes used a placeholder symbol by c. 300 BCE.
Number: Zero as a genuine number you can calculate with — add it, subtract it, multiply by it. This is conceptually harder. What does it mean to add "nothing"? What happens when you multiply by zero? The first explicit, systematic rules for arithmetic with zero appear in Indian mathematics.
Brahmagupta's rules (628 CE)
The Indian mathematician Brahmagupta, in his Brāhmasphuṭasiddhānta (628 CE), gave the first known systematic rules for calculating with zero:
a + 0 = a (zero added to any number gives that number)
a − 0 = a
a × 0 = 0 (any number multiplied by zero is zero)
0 − a = −a (zero minus a number gives its negative)
0 ÷ 0 = 0 (wrong, but an admirable attempt)
His rule for 0 ÷ 0 was incorrect — we now know division by zero is undefined. Bhāskara II (1150 CE) explored the concept of a/0 as infinity, which was also not quite right. The correct modern treatment came much later. But Brahmagupta's rules for zero in addition, subtraction, and multiplication were correct and written nearly 1,400 years ago.
The word zero itself traces back to Sanskrit śūnya (void, empty) → Arabic ṣifr (cipher) → Medieval Latin zephirum → Italian zero. Even the word's journey reflects Indian mathematics travelling west through Islamic scholarship.
History of zero across traditions
Babylonian placeholder (c. 300 BCE)
Babylonian mathematicians worked in base 60. By c. 300 BCE (late Babylonian period), they used a special symbol — two wedge marks — to denote an empty sexagesimal position. This is placeholder zero. Crucially, it was not used at the end of a number (so "360" and "21600" could be ambiguous), and it was not treated as a number in calculations.
Mayan zero (c. 350 CE)
The Maya independently developed a placeholder zero symbol (a shell glyph) in their base-20 calendar and arithmetic system by c. 350 CE, with possible earlier use. The Mayan zero was used at the end of numbers — correcting the limitation of the Babylonian version — but there is no evidence of Mayan arithmetic treating zero as a number.
Indian zero — śūnya
The Indian tradition shows a progression from placeholder to number:
Āryabhaṭa (499 CE): Uses a decimal place-value system in encoding; zero implicitly required as placeholder.
Brahmagupta (628 CE): First systematic rules for zero as a number — arithmetic with 0. Introduces negative numbers in the same text.
Mahāvīra (850 CE): Notes that division by zero leaves the number unchanged — an early but incorrect attempt at 1/0.
Bhāskara II (1150 CE): Proposes a/0 = ∞ (khahara), a precursor to the modern limit concept.
The oldest unambiguous written zero as a number is in the Bakhshali manuscript, which contains a dot symbol for zero. The manuscript was carbon-dated (2017) to folia from different periods; the oldest pages date to 224–383 CE, which would make this the oldest known zero symbol.
Chinese zero
Chinese rod numerals used an empty space for zero by at least the 2nd century BCE. The word 零 (líng, zero) appears in Chinese mathematical texts from the 13th century CE onwards. The abstract number zero was likely influenced by Indian mathematics via Buddhist scholarly exchange.
Transmission to Europe
Zero reached European mathematics via Arabic scholarship. Al-Khwārizmī's works (c. 820 CE) used Indian numerals including zero. The word "cipher" (empty, zero) entered European languages via Arabic ṣifr. Leonardo of Pisa (Fibonacci) introduced Indian-Arabic numerals including zero to Europe in his Liber Abaci (1202 CE).
Modern mathematical role
Zero is the additive identity in any group, ring, or field. In set theory, 0 = ∅ (the empty set) in the von Neumann construction. The natural numbers ℕ are defined with or without zero depending on convention; ISO 80000-2 includes 0 in ℕ. Zero is neither positive nor negative.
Division by zero, limits, and formal construction
Why division by zero is undefined
In a field, every non-zero element a has a multiplicative inverse a⁻¹. Zero cannot have an inverse because: if 0 · x = 1 for some x, then 0 = 0 · x = 1, contradicting 0 ≠ 1 (required in a non-trivial ring). Equivalently, in any ring, 0 · a = 0 for all a (provable from the axioms), so no element maps to 1 under multiplication by 0.
The expressions 0/0 and a/0 (a ≠ 0) are undefined in standard arithmetic. In extended real number systems (projective line ℝ∪{∞}), one can define 1/0 = ∞ consistently for some purposes, but this is not a field.
Zero in limits and calculus
The indeterminate forms (0/0, 0·∞, 0⁰ etc.) that arise in limits are indeterminate because different functions approaching 0 can produce different limit values. L'Hôpital's rule, Taylor series, and other tools resolve specific cases. The epsilon-delta definition of a limit (Cauchy-Weierstrass) makes precise what it means for a quantity to "approach zero" without dividing by it.
The Bakhshali manuscript dating
The Bakhshali manuscript (discovered 1881, now Bodleian Library, Oxford) contains a hollow dot symbol used as a placeholder zero. Carbon dating by the Bodleian (2017) gave dates of 224–383 CE, 680–779 CE, and 885–993 CE for different folia — indicating the manuscript is a compilation from multiple periods. The date of the oldest folia (224–383 CE) would make it the earliest written zero symbol, predating the Mayan shell glyph. Historians debate whether all folia share a common mathematical tradition; the dating is not universally accepted as settling the question of zero's origin.
Chapter 18 (Kuṭṭākadhyāya) of the Brāhmasphuṭasiddhānta contains the rules for arithmetic with zero and negative numbers. The Sanskrit text is available in the critical edition by Dvivedin (1902) and in English translation by Plofker (2009). The relevant verses (18.29–18.35) give: addition, subtraction, and multiplication of positive, negative, and zero quantities. Division by zero is given as "zero divided by zero is zero" — the known error. Brahmagupta uses the term kha (sky, void) for zero in this context.
Sources
Plofker, K. (2009). Mathematics in India. Princeton University Press. pp. 150–156. — Brahmagupta's rules for zero in context. Tier 1
Colebrooke, H.T. (1817). Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara. John Murray, London. — Translation of Brāhmasphuṭasiddhānta (1817 edition; primary text accessible). Tier 1 (primary source)
Kaplan, R. (1999). The Nothing That Is: A Natural History of Zero. Oxford University Press. — Accessible scholarly history. Tier 2
Seife, C. (2000). Zero: The Biography of a Dangerous Idea. Viking. — Popular account; use with care for precise historical claims. Tier 2
Neugebauer, O. (1969). The Exact Sciences in Antiquity. 2nd ed. Dover. — Babylonian placeholder context. Tier 2
Joseph, G.G. (2011). The Crest of the Peacock. 3rd ed. Princeton University Press. pp. 214–223. Tier 3
Relationships
Formalised by Brahmagupta, Brāhmasphuṭasiddhānta 628 CE — first systematic arithmetic rules
Placeholder origin Babylon, c. 300 BCE
Independent invention Maya, c. 350 CE
Earliest written symbol Bakhshali manuscript, 224–383 CE (oldest folia)
Algebraic role Additive identity in any Group, Ring, or Field