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. Additionally, any number raised to the power of 0 is equal to 1, with exception of 0 to the power of 0 which is sometimes undefined.
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 from the Italian , a contraction of the Venetian form of Italian , itself borrowed from Arabic or .
In pre-Islamic time the word (Arabic ) had the meaning "empty". evolved to mean zero when it was used to translate () from India. The earliest known use of zero as a loanword in English literature was 1598.
The Italian mathematician Fibonacci (), who grew up in North Africa and is credited with introducing the decimal system to Europe, used the term . This became in Italian, and was then contracted to via the Venetian form . The Italian word was already in existence (meaning "west wind" from Latin and Greek ) and may have influenced the spelling when transcribing Arabic ..
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 , "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,, Extract of pp. 254–255 but it is an integer, and hence a rational number and a real number. Zero is evenLemma B.2.2, The integer 0 is even and is not odd, in (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, for nonzero x. But is undefined, because 0 has no multiplicative inverse (no real number multiplied by 0 produces 1), a consequence of the previous rule. Exponentiation: x0 = = 1, except that the case x = 0 is considered undefined in some contexts. For all positive real x, .
The expression , which may be obtained in an attempt to determine the limit of an expression of the form 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 , 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.
Other uses in mathematics The role of 0 as the smallest counting number can be generalized or extended in various ways. In set theory, 0 is the cardinality of the empty set (notated as "{ }", "", or "∅"): if one does not have any apples, then one has 0 apples. In fact, in certain axiomatic developments of mathematics from set theory, 0 is defined to be the empty set. When this is done, the empty set is the von Neumann cardinal assignment for a set with no elements, which is the empty set.
Also in set theory, 0 is the lowest ordinal number, corresponding to the empty set viewed as a well-ordered set. In order theory (and especially its subfield lattice theory), 0 may denote the least element of a lattice or other partially ordered set.
The role of 0 as additive identity generalizes beyond elementary algebra. In abstract algebra, 0 is commonly used to denote a zero element, which is the identity element for addition (if defined on the structure under consideration) and an absorbing element for multiplication (if defined). Examples include identity elements of additive groups and vector spaces. Another example is the zero function (or zero map) on a domain . This is the constant function with 0 as its only possible output value, that is, it is the function defined by for all in . As a function from the real numbers to the real numbers, the zero function is the only function that is both even and odd.
The number 0 is also used in several other ways within various branches of mathematics: A zero of a function f is a point x in the domain of the function such that . In propositional logic, 0 may be used to denote the truth value false. In probability theory, 0 is the smallest allowed value for the probability of any event. Category theory introduces the idea of a zero object, often denoted 0, and the related concept of zero morphisms, which generalize the zero function.
History
Ancient Near East nfr heart with tracheabeautiful, pleasant, goodF35 Ancient Egyptian numerals were of base 10. They used hieroglyphs for the digits and were not positional. In one papyrus written around , a scribe recorded daily incomes and expenditures for the pharaoh's court, using the nfr hieroglyph to indicate cases where the amount of a foodstuff received was exactly equal to the amount disbursed. Egyptologist Alan Gardiner suggested that the nfr hieroglyph was being used as a symbol for zero.eg. Alan Gardiner. “§357. Other Modes of Negation” Egyptian Grammar (1927). p. 266 etc. et al. Cites: Cairo 20003. Turin 1447. British Museum MS 152. The same symbol was also used to indicate the base level in drawings of tombs and pyramids, and distances were measured relative to the base line as being above or below this line.
By the middle of the 2nd millennium BC, Babylonian mathematics had a sophisticated base 60 positional numeral system, but a positional value of zero was indicated by a space between numerals. In a tablet unearthed at Kish (dating to as early as ), the scribe Bêl-bân-aplu used three hooks as a placeholder. By , a punctuation symbol (two slanted wedges) was repurposed as a placeholder. Significantly, however, these placeholder signs were not considered a numerical value, they were never used alone “Therefore, they cannot be interpreted as representations of the concept or the number zero.”Nieder, Andreas (19 November 2019). A Brain for Numbers: The Biology of the Number Instinct. MIT Press. p. 286. They were also not written at the end of a number (so 1 and 60 and 60×60 were all written as ).
Pre-Columbian Americas
The Mesoamerican Long Count calendar developed in south-central Mexico and Central America required the use of zero as a placeholder within its vigesimal (base-20) positional numeral system. Many different glyphs, including the partial quatrefoil were used as a zero symbol for these Long Count dates, the earliest of which (on Stela 2 at Chiapa de Corzo, Chiapas) has a date of 36 BC.
Since the eight earliest Long Count dates appear outside the Maya homeland, it is generally believed that the use of zero in the Americas predated the Maya and was possibly the invention of the Olmecs. Many of the earliest Long Count dates were found within the Olmec heartland, although the Olmec civilization ended by the , several centuries before the earliest known Long Count dates.
Although zero became an integral part of Maya numerals, with a different, empty tortoise-like "shell shape" used for many depictions of the "zero" numeral, it is assumed not to have influenced Old World numeral systems.
Quipu, a knotted cord device, used in the Inca Empire and its predecessor societies in the Andean region to record accounting and other digital data, is encoded in a base ten positional system. Zero is represented by the absence of a knot in the appropriate position.
Classical antiquity The earliest confidently cited exemplar of the Greek use of the Hellenistic zero appears in Hipparchus in 140 CE.