How did zero go from a placeholder to a number?

Zero began as a punctuation mark, a way to show an empty place in a written number. It became a number in its own right when Indian mathematicians, most clearly Brahmagupta in the seventh century, treated it as something that could be added, subtracted and multiplied. MacTutor stresses that there was no single moment of invention, and that the placeholder and the number are two different uses of zero. [1] [2]

In a number like 2106, the zero keeps the 2 and the 1 in the right positions, so that it differs from 216. Babylonian tablets from around 1700 BC used a base-60 system and did not distinguish between the two. According to MacTutor, it was around 400 BC that the Babylonians began putting two wedge symbols into the empty place. MacTutor describes this as a kind of punctuation, not a number. Ptolemy, writing around 130 AD, used a similar placeholder in the Almagest and did not regard it as a number either. [1]

Numbers had long been words for collections of objects, and MacTutor notes that zero is not a natural candidate for a number. The difficulty was working out how zero behaves in arithmetic. In 628 AD Brahmagupta gave rules for it. A number minus itself is zero, any number multiplied by zero is zero, and adding zero leaves a number unchanged. He also set out rules for subtracting from zero, which brought in negative numbers. The mathematician Bhargava, speaking to CBC, says Brahmagupta's book is the first place negative numbers appear. [2] [1]

His rules for division were weaker. MacTutor says he was wrong to claim that zero divided by zero is zero, and that his statement about dividing by zero says very little. It still calls his work a brilliant attempt, from the first person known to have tried to extend arithmetic to zero and negatives. [1]

Historians do not agree on a single origin or date. Charles Seife places the zero that reached us in the Fertile Crescent, first appearing in Babylon between 400 and 300 BC, then developing in India. Robert Kaplan argues that zero began to take shape as a number in India in the fifth century AD, though even then it was not fully accepted as one. Kaplan also points to a Sumerian placeholder 4,000 to 5,000 years ago, a dating Seife says he is not entirely convinced by. Both say no single origin can be claimed. Kaplan adds that wherever placeholder notation arises, some way to mark absence is needed. [3]

MacTutor reports that many historians think the Indian use of zero evolved from its use by Greek astronomers. It also notes evidence of an empty-place marker in India as early as 200 AD, which some historians dismiss as later forgery. The earliest dated Indian zero that all accept as genuine is an inscription from 876 at Gwalior. MacTutor says that by around 650 AD zero was in use as a number in Indian mathematics. [1] [4]

Zero also arose independently among the Maya, likely in the first few centuries AD, according to Kaplan. Scientific American says the zero known in the West reached Europe around 1200, brought by Fibonacci along with the other Arabic numerals after his travels in north Africa. The English word comes ultimately from the Arabic sifr, which also gives us cipher. [3] [1]