Zero, term applied to the number representing naught, denoted by the symbol 0. The fundamental arithmetic properties of the number 0 are: a + 0 = a,a - 0 = a, and a × 0 = 0, in which a is any number; and 0 ÷ b = 0, in which b is any number other than 0. Division by 0 is not defined and therefore is an inadmissible operation (See also Exponent). In the real-number system, 0 is the only number that is neither negative nor positive, and it represents the boundary between the negative and the positive numbers. This property makes 0 the natural starting point, or origin, on many scales, as on the coordinate axes and on thermometers.

In the development of written notation, a symbol for zero was evolved long after symbols for the other numbers were invented. The Babylonians used written symbols for numbers thousands of years before they invented a symbol for zero. Zero was introduced initially, not as a number to be used in computation, but as a position marker to distinguish between such numbers as 123, 1203, 1230, and 1023. The Maya, about the 1st century ad, used a small oval containing an inner arc to denote zero. About five centuries later the Hindus began to use a circle or a dot as a symbol for zero; the dot later fell into disuse. These Indian mathematicians wrote numbers in columns, and they used the zero to represent a blank column. The Hindu word for zero was śŭnya, meaning empty, or void; this word, translated and transliterated by the Arabs as sifr, is the root of the English words cipher and zero.

See Number Systems; Numerals.