Number and numeral of critical importance in mathematics. Zero is known as the additive identity because adding it to any number does not change the number's identity, or value. The product of zero and any number is zero; for most number systems the converse is true—that is, if the product of two numbers is zero, at least one of them must equal zero. The latter property is fundamental to the solution of nearly every problem in mathematics. Division by zero is undefined; efforts to deal with such divisions led to calculus. Various punctuation marks were first used in Mesopotamia beginning about 700 BC to indicate an empty space in positional notation, but never at the end of a number—the difference between, say, 78 and 780 had to be understood from the context. Ptolemy first used 0, or the Greek letter omicron “,” as an empty placeholder, including at the end of a number, to express data in the Babylonian sexagesimal system in his astronomical treatise Almagest (c. 130 AD). The Hindu-Arabic numerals and treatment of zero as a number developed between the 6th and 9th centuries in India. Zero soon followed trade routes to China, the Islamic world, and Europe.

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