0

0 (zero, ) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and complex numbers, as well as other algebraic structures. Multiplying any number by 0 results in 0, and consequently dividing by 0 is generally considered to be undefined in arithmetic. As a numerical digit, 0 plays a crucial role in decimal notation: it indicates that the power of ten corresponding to the place containing a 0 does not contribute to the total. For example, "205" in decimal means two hundreds, no tens, and five ones. The same principle applies in place-value notations that uses a base other than ten, such as binary and hexadecimal. The modern use of 0 in this manner derives from Indian mathematics that was transmitted to Europe via medieval Islamic mathematicians and popularized by Fibonacci. It was independently used by the Maya. Common names for the number 0 in English include zero, nought, naught (), and nil. In contexts where at least one adjacent digit distinguishes it from the letter O, the number is sometimes pronounced as oh or o (). Informal or slang terms for 0 include zilch and zip. Historically, ought, aught (), and cipher have also been used.

Etymology

The word zero came into the English language via French zéro from the Italian zero, a contraction of the Venetian zevero form of Italian zefiro, itself borrowed from Arabic ṣafira or ṣifr. In pre-Islamic time the word ṣifr (Arabic صفر) had the meaning "empty". Sifr evolved to mean zero when it was used to translate śūnya (Sanskrit: शून्य) from India. The earliest known use of zero as a loanword in English literature was 1598. The Italian mathematician Fibonacci (c. 1170 – c. 1250), who grew up in North Africa and is credited with introducing the decimal system to Europe, used the term zephyrum. This became zefiro in Italian, and was then contracted to zero via the Venetian form zevero. The Italian word zefiro was already in existence (meaning "west wind" from Latin and Greek Zephyrus) and may have influenced the spelling when transcribing Arabic ṣifr.

Modern usage Depending on the context, there may be different words used for the number zero, or the concept of zero. For the simple notion of lacking, the words "nothing" (although this is not accurate) and "none" are often used. The English words "nought" or "naught", "nil", and null are also synonymous. It is often called "oh" in the context of reading out a string of digits, such as telephone numbers, street addresses, credit card numbers, military time, or years. For example, the area code 201 may be pronounced "two oh one", and the year 1907 is often pronounced "nineteen oh seven". The presence of other digits, indicating that the string contains only numbers, avoids confusion with the letter O. For this reason, systems that include strings with both letters and numbers (such as postcodes in the UK) may exclude the use of the letter O. Slang words for zero include "zip", "zilch", "nada", and "scratch". In the context of sports, "nil" is sometimes used, especially in British English. Several sports have specific words for a score of zero, such as "love" in tennis – possibly from French l'œuf, "the egg" – and "duck" in cricket, a shortening of "duck's egg". "Goose egg" is another general slang term used for zero.

Mathematics

The concept of zero plays multiple roles in mathematics: as a digit, it is an important part of positional notation for representing numbers, while it also plays an important role as a number in its own right in many algebraic settings.

As a digit

In positional number systems (such as the usual decimal notation for representing numbers), the digit 0 plays the role of a placeholder, indicating that certain powers of the base do not contribute. For example, the decimal number 205 is the sum of two hundreds and five ones, with the 0 digit indicating that no tens are added. The digit plays the same role in decimal fractions and in the decimal representation of other real numbers (indicating whether any tenths, hundredths, thousandths, etc., are present) and in bases other than 10 (for example, in binary, where it indicates which powers of 2 are omitted).

Elementary algebra

The number 0 is the smallest nonnegative integer, and the largest nonpositive integer. The natural number following 0 is 1 and no natural number precedes 0. The number 0 may or may not be considered a natural number, but it is an integer, and hence a rational number and a real number. Zero is even (that is, a multiple of 2), and is also an integer multiple of any other integer. It is neither a prime number nor a composite number. The number 0 is regarded as neither positive nor negative, and is usually displayed as the origin of a number line. When the real numbers are extended to form the complex numbers, 0 becomes the origin of the complex plane.

The following are some basic rules for dealing with the number 0. These rules apply for any real or complex number x, unless otherwise stated.

Addition: x + 0 = 0 + x = x. That is, 0 is an identity element (or neutral element) with respect to addition. Subtraction: x − 0 = x and 0 − x = −x. Multiplication: x · 0 = 0 · x = 0. The converse also holds: If x · y = 0 then x=0 or y=0. Division: ⁠0/x⁠ = 0, for nonzero x. But ⁠x/0⁠ is undefined, because 0 has no multiplicative inverse (no real number multiplied by 0 produces 1), a consequence of the previous rule. Exponentiation: x0 = ⁠x/x⁠ = 1, except that the case x = 0 is considered undefined in some contexts. For all positive real x, 0x = 0. The expression ⁠0/0⁠, which may be obtained in an attempt to determine the limit of an expression of the form ⁠f(x)/g(x)⁠ as a result of applying the lim operator independently to both operands of the fraction, is a so-called "indeterminate form". That does not mean that the limit sought is necessarily undefined; rather, it means that the limit of ⁠f(x)/g(x)⁠, if it exists, must be found by another method, such as l'Hôpital's rule. The sum of 0 numbers (the empty sum) is 0, and the product of 0 numbers (the empty product) is 1. The factorial 0! evaluates to 1, as a special case of the empty product.

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