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Science · Mathematics

Zero

c. 3rd century BC – 628 AD

Mathematics — number systems

The additive identity, the placeholder that makes positional notation possible, and the last of the everyday numbers to be accepted as a number in its own right. Babylonian scribes used a placeholder symbol, and the Maya had a true zero in their calendar, but the decisive step was Indian: the Bakhshali manuscript carries dots used as placeholders, and in 628 AD Brahmagupta's Brahmasphutasiddhanta gave the first surviving rules for arithmetic with it, including that a number minus itself gives it. It reached Europe through Arabic mathematics and Fibonacci's Liber Abaci of 1202.

How to spot them

  • ▸The additive identity: adding it changes nothing
  • ▸Brahmagupta's Brahmasphutasiddhanta of 628 AD gave the first rules for calculating with it
  • ▸Neither positive nor negative, and even; division by it is undefined
  • ▸The word came to English via Italian from the Arabic sifr, 'empty', which also gave 'cipher'
  • ▸There is no year of this number in the Gregorian calendar — 1 BC is followed by AD 1

Key facts

Brahmagupta's 628 AD treatise giving the first surviving arithmetical rules for it (628)

The Brahmasphutasiddhanta, which also handled negative numbers as debts

Dots used as placeholders on birch bark found in a Peshawar village in 1881 (1881)

The Bakhshali manuscript, now in Oxford; Bodleian radiocarbon dating of 2017 claimed the 3rd or 4th century, but Oxford's own 2024 report revised it to roughly 799–1102 AD

A shell-shaped glyph in the Maya Long Count calendar

An independent invention of a true placeholder in Mesoamerica

The number that, added to any other, leaves it unchanged

The additive identity — the defining algebraic property

The Arabic word sifr, meaning 'empty', which also gave English the word 'cipher'

Via Latin zephirum and Italian zefiro to its modern name

The year that does not exist between 1 BC and AD 1

The Gregorian calendar has no such year, which is why centuries begin in years ending 01

The operation that is undefined, however tempting it looks

Division by it — the quotient has no consistent value

Key ideas

  • Two distinct ideas had to merge: an empty-column placeholder, which several cultures invented, and a number you can add, subtract and multiply, which is an Indian achievement.
  • Brahmagupta correctly stated that a number minus itself gives it, and that it added to a negative gives a negative, but stumbled over division by it — a gap that would take a thousand years and the calculus to resolve properly.

Quick-fire facts

  • •Brahmagupta's Brahmasphutasiddhanta (628 AD) contains the first known rules for arithmetic with it, and also rules for negative numbers
  • •The Bakhshali manuscript, held at the Bodleian Library in Oxford, was radiocarbon-dated in 2017 to as early as the 3rd or 4th century AD — a claim historians disputed, and Oxford's own 2024 dating of five folios placed the surviving manuscript at about 799–1102 AD
  • •An inscription at the Chaturbhuj Temple in Gwalior, dated 876 AD, is often cited as the oldest firmly dated Indian inscription using it in the modern way
  • •The Babylonians used a two-wedge placeholder from around the 3rd century BC but never at the end of a number
  • •Maya astronomers used a true zero in the Long Count calendar, independently of the Old World
  • •The name descends from the Arabic sifr ('empty' or 'nothing'), which also gives 'cipher' and 'decipher'
  • •Fibonacci's Liber Abaci of 1202 helped introduce the Hindu-Arabic numerals, including it, to European commerce
  • •It is an even number, is neither positive nor negative, and the factorial of it is defined to be 1
  • •Anything raised to the power of it is conventionally 1, while it raised to the power of it is usually taken as 1 in combinatorics and left undefined in analysis
  • •Absolute zero, −273.15 °C, is the zero of the Kelvin scale; the Gregorian calendar, by contrast, has no year of this number

Drill it — 7 cards

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Sources