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0, zero, "oh" (), nought, naught, nil
Zeroth, noughth
All numbers
0 (zero; ) is both a
used to represent that number in . The number 0 fulfills a central role in
of the , , and many other
structures. As a digit, 0 is used as a placeholder in .
include zero, nought (UK), naught (US) (), nil, or—in contexts where at least one adjacent digit distinguishes it from the letter "O"—oh or o (). Informal or slang terms for zero include zilch and zip. Ought and aught (), as well as cipher, have also been used historically.
The word zero came into the English language via French zéro from
zero, Italian contraction of Venetian zevero form of 'Italian zefiro via ?afira or ?ifr. In pre-Islamic time the word ?ifr (Arabic ???) had the meaning "empty". Sifr evolved to mean zero when it was used to translate ?ūnya (Sanskrit: ?????) from India. The first known English use of zero was in 1598.
The Italian mathematician
(c. ), 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 in Venetian. The Italian word
was already in existence (meaning "west wind" from Latin and Greek ) and may have influenced the spelling when transcribing Arabic ?ifr.
There are different words used for the number or concept of zero depending on the context. For the simple notion of lacking, the words nothing and none are often used. Sometimes the words nought, naught and aught are used. Several sports have specific words for zero, such as nil in
(soccer), love in
and a duck in . It is often called oh in the context of telephone numbers. Slang words for zero include zip, zilch, nada, and scratch. Duck egg and goose egg are also slang for zero.
heart with trachea
beautiful, pleasant, good
were . They used
for the digits and were not . By ;BC, the Egyptians had a symbol for zero in accounting texts. The symbol nfr, meaning beautiful, 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, the
had a sophisticated
positional numeral system. The lack of a positional value (or zero) was indicated by a space between sexagesimal numerals. By 300 BC, a punctuation symbol (two slanted wedges) was co-opted as a
in the same . In a tablet unearthed at
(dating from about 700 BC), the scribe Bêl-b?n-aplu wrote his zeros with three hooks, rather than two slanted wedges.
The Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the end of a number. Thus numbers like 2 and 120 (2×60), 3 and 180 (3×60), 4 and 240 (4×60), looked the same because the larger numbers lacked a final sexagesimal placeholder. Only context could differentiate them.
The back of
stela C from , the second oldest
date discovered. The numerals 7.16.6.16.18 translate to September, 32 BC (Julian). The glyphs surrounding the date are thought to be one of the few surviving examples of .
developed in south-central Mexico and Central America required the use of zero as a place-holder within its
(base-20) positional numeral system. Many different glyphs, including this partial ——were used as a zero symbol for these Long Count dates, the earliest of which (on Stela 2 at Chiapa de Corzo, ) 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 . Many of the earliest Long Count dates were found within the Olmec heartland, although the Olmec civilization ended by the 4th century BC, several centuries before the earliest known Long Count dates.
Although zero became an integral part of , with a different, empty -like "" used for many depictions of the "zero" numeral, it is assumed to have not influenced
numeral systems.
, a knotted cord device, used in the
and its predecessor societies in the
region to record accounting and other digital data, is encoded in a
positional system. Zero is represented by the absence of a knot in the appropriate position.
had no symbol for zero (μηδ?ν), and did not use a digit placeholder for it. They seemed unsure about the status of zero as a number. They asked themselves, "How can nothing be something?", leading to philosophical and, by the
period, religious arguments about the nature and existence of zero and the . The
depend in large part on the uncertain interpretation of zero.
Example of the early Greek symbol for zero (lower right corner) from a 2nd-century papyrus
By 130 AD, , influenced by
and the Babylonians, was using a symbol for zero (a small circle with a long overbar) in his work on
called the Syntaxis Mathematica, also known as the . The way in which it is used can be seen in his
in that book. Ptolemy's zero was used within a
numeral system otherwise using alphabetic . Because it was used alone, not just as a placeholder, this
was perhaps the earliest documented use of a numeral representing zero in the Old World. However, the positions were usually limited to the fractional part of a number (called minutes, seconds, thirds, fourths, etc.)—they were not used for the integral part of a number, indicating a concept perhaps better expressed as "none", rather than "zero" in the modern sense. In later
manuscripts of Ptolemy's Almagest, the Hellenistic zero had morphed into the Greek letter
(otherwise meaning 70).
Another zero was used in tables alongside
by 525 (first known use by ), but as a word, nulla meaning "nothing", not as a symbol. When division produced zero as a remainder, nihil, also meaning "nothing", was used. These medieval zeros were used by all future medieval . The initial "N" was used as a zero symbol in a table of Roman numerals by
or his colleagues around 725.
This is a depiction of zero expressed in Chinese , based on the example provided by A History of Mathematics. An empty space is used to represent zero.
The , of unknown date but estimated to be dated from the 1st to 5th centuries AD, and Japanese records dated from the 18th century, describe how the c. 4th century BC Chinese
system enables one to perform decimal calculations. According to A History of Mathematics, the rods "gave the decimal representation of a number, with an empty space denoting zero." The counting rod system is considered a
promulgated , one of which was "〇". The word is now used as a synonym for the number zero.
Zero was not treated as a number at that time, but as a "vacant position". 's 1247
is the oldest surviving Chinese mathematical text using a round symbol for zero. Chinese authors had been familiar with the idea of negative numbers by the
(2nd century AD), as seen in , much earlier than the 15th century when they became well-established in Europe.
(c. 3rd/2nd century BC), a
scholar, used
in the form of short and long syllables (the latter equal in length to two short syllables), a notation similar to . Pingala used the
explicitly to refer to zero.
It was considered that the earliest text to use a decimal , including a zero, is the , a
surviving in a medieval Sanskrit translation of the
original, which is internally dated to AD 458 ( 380). In this text,
("void, empty") is also used to refer to zero.
A symbol for zero, a large dot likely to be the precursor of the still-current hollow symbol, is used throughout the , a practical manual on arithmetic for merchants, the date of which was uncertain. In 2017 three samples from the manuscript were shown by
to come from three different centuries: from 224-383 AD, 680-779 AD, and 885-993 AD, making it the world's oldest recorded use of the zero symbol. It is not known how the
bark fragments from different centuries that form the manuscript came to be packaged together.
The origin of the modern decimal-based place value notation can be traced to the
(c. 500), which states sthānāt sthāna? da?agu?a? syāt "from place to place each is ten times the preceding." The concept of zero as a digit in the decimal place value notation was developed in , presumably as early as during the
(c. 5th century), with the oldest unambiguous evidence dating to the 7th century.
The rules governing the use of zero appeared for the first time in 's
(7th century). This work considers not only zero, but negative numbers, and the algebraic rules for the elementary operations of arithmetic with such numbers. In some instances, his rules differ from the modern standard, specifically the definition of the value of zero
The number 605 in Khmer numerals, from the Sambor inscription ( 605 corresponds to AD 683). The earliest known material use of zero as a decimal figure.
There are numerous copper plate inscriptions, with the same small o in them, some of them possibly dated to the 6th century, but their date or authenticity may be open to doubt.
A stone tablet found in the ruins of a temple near Sambor on the , , , includes the inscription of "605" in
(a set of numeral glyphs of the
family). The number is the year of the inscription in the , corresponding to a date of AD 683.
The first known use of special
for the decimal digits that includes the indubitable appearance of a symbol for the digit zero, a small circle, appears on a stone inscription found at the
in India, dated 876. Zero is also used as a placeholder in the , portions of which date from AD 224–383.
The -language inheritance of science was largely , followed by Hindu influences. In 773, at 's behest, translations were made of many ancient treatises including Greek, Roman, Indian, and others.
In AD 813, astronomical tables were prepared by a
mathematician, , using H and about 825, he published a book synthesizing Greek and Hindu knowledge and also contained his own contribution to mathematics including an explanation of the use of zero. This book was later translated into
in the 12th century under the title Algoritmi de numero Indorum. This title means "al-Khwarizmi on the Numerals of the Indians". The word "Algoritmi" was the translator's Latinization of Al-Khwarizmi's name, and the word "Algorithm" or "Algorism" started meaning any arithmetic based on decimals.
, in 976, stated that if no number appears in the place of tens in a calculation, a little circle should be used "to keep the rows". This circle was called ?ifr.
(base 10) reached Europe in the 11th century, via the
through Spanish , the , together with knowledge of
and instruments like the , first imported by . For this reason, the numerals came to be known in Europe as "Arabic numerals". The Italian mathematician
or Leonardo of Pisa was instrumental in bringing the system into European mathematics in 1202, stating:
After my father's appointment by his homeland as state official in the customs house of Bugia for the Pisan merchants who thronged to it, and in view of its future usefulness and convenience, had me in my boyhood come to him and there wanted me to devote myself to and be instructed in the study of calculation for some days. There, following my introduction, as a consequence of marvelous instruction in the art, to the nine digits of the Hindus, the knowledge of the art very much appealed to me before all others, and for it I realized that all its aspects were studied in Egypt, Syria, Greece, Sicily, and Provence, with t and at these places thereafter, while on business. I pursued my study in depth and learned the give-and-take of disputation. But all this even, and the algorism, as well as the art of Pythagoras, I considered as almost a mistake in respect to the method of the
(Modus Indorum). Therefore, embracing more stringently that method of the Hindus, and taking stricter pains in its study, while adding certain things from my own understanding and inserting also certain things from the niceties of Euclid's geometric art. I have striven to compose this book in its entirety as understandably as I could, dividing it into fifteen chapters. Almost everything which I have introduced I have displayed with exact proof, in order that those further seeking this knowledge, with its pre-eminent method, might be instructed, and further, in order that the Latin people might not be discovered to be without it, as they have been up to now. If I have perchance omitted anything more or less proper or necessary, I beg indulgence, since there is no one who is blameless and utterly provident in all things. The nine Indian figures are: 9 8 7 6 5 4 3 2 1. With these nine figures, and with the sign 0 ... any number may be written.
Here Leonardo of Pisa uses the phrase "sign 0", indicating it is like a sign to do operations like addition or multiplication. From the 13th century, manuals on calculation (adding, multiplying, extracting roots, etc.) became common in Europe where they were called
after the Persian mathematician al-Khwārizmī. The most popular was written by , about 1235 and was one of the earliest scientific books to be printed in 1488. Until the late 15th century, Hindu–Arabic numerals seem to have predominated among mathematicians, while merchants preferred to use the . In the 16th century, they became commonly used in Europe.
immediately preceding .
because it is divisible by
with no remainder. 0 is neither positive nor negative.[] By most definitions 0 is a , and then the only natural number not to be positive. Zero is a number which quantifies a count or an amount of
size. In most , 0 was identified before the idea of negative things, or quantities less than zero, was accepted.
The value, or number, zero is not the same as the digit zero, used in
using . Successive positions of digits have higher weights, so inside a numeral the digit zero is used to skip a position and give appropriate weights to the preceding and following digits. A zero digit is not always necessary in a positional number system, for example, in the number 02. In some instances, a
may be used to distinguish a number.
The number 0 is the smallest
integer. The
following 0 is 1 and no natural number precedes 0. The number 0 , but it is a whole number and hence a
(as well as an
The number 0 is neither positive nor negative and is usually displayed as the central number in a . It is neither a
nor a . It cannot be prime because it has an
number of , and cannot be composite because it cannot be expressed as a product of prime numbers (0 must always be one of the factors). Zero is, however,
(as well as being a multiple of any other integer, rational, or real number).
The following are some basic (elementary) 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
(or neutral element) with respect to addition.
Subtraction: x - 0 = x and 0 - x = -x.
Multiplication: x · 0 = 0 · x = 0.
Division: 0/x = 0, for nonzero x. But x/0 is , because 0 has no
(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 may be left undefined in some . 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
operator independently to both operands of the fraction, is a so-called "". That does not simply mean that the limit sought is n rather, it means that the limit of f(x)/g(x), if it exists, must be found by another method, such as .
The sum of 0 numbers (the ) is 0, and the product of 0 numbers (the ) is 1. The
0! evaluates to 1, as a special case of the empty product.
In , 0 is the
of the empty set: 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
to be the empty set. When this is done, the empty set is the
for a set with no elements, which is the empty set. The cardinality function, applied to the empty set, returns the empty set as a value, thereby assigning it 0 elements.
Also in set theory, 0 is the lowest , corresponding to the empty set viewed as a .
In , 0 may be used to denote the
In , 0 is commonly used to denote a , which is a
for addition (if defined on the structure under consideration) and an
for multiplication (if defined).
In , 0 may denote the
In , 0 is sometimes used to denote an
In , 0 can be used to denote the
f is a point x in the domain of the function such that f(x) = 0. When there are finitely many zeros these are called the roots of the function. This is related to
The zero function (or zero map) on a domain D is the
with 0 as its only possible output value, i.e., the function f defined by f(x) = 0 for all x in D. The zero function is the only function that is both
and . A particular zero function is a e.g., a zero map is the identity in the additive group of functions. The
on non-invertible
is a zero map.
Several branches of mathematics have , which generalize either the property 0 + x = x, or the property 0 · x = 0, or both.
The value zero plays a special role for many physical quantities. For some quantities, the zero level is naturally distinguished from all other levels, whereas for others it is more or less arbitrarily chosen. For example, for an
(as measured in )
is the lowest possible value ( are defined, but negative-temperature systems are not actually colder). This is in contrast to for example temperatures on the Celsius scale, where zero is arbitrarily defined to be at the
of water. Measuring sound intensity in
or , the zero level is arbitrarily set at a reference value—for example, at a value for the threshold of hearing. In , the
is the lowest possible energy that a
may possess and is the energy of the
of the system.
Zero has been proposed as the
of the theoretical element . It has been shown that a cluster of four
may be stable enough to be considered an
in its own right. This would create an
and no charge on its .
As early as 1926, Andreas von Antropoff coined the term
for a conjectured form of
made up of neutrons with no protons, which he placed as the chemical element of atomic number zero at the head of his new version of the . It was subsequently placed as a noble gas in the middle of several spiral representations of the periodic system for classifying the chemical elements.
The most common practice throughout human history has been to start counting at one, and this is the practice in early classic
programming languages such as
and . However, in the late 1950s
introduced
for arrays while
introduced completely flexible basing for array subscripts (allowing any positive, negative, or zero integer as base for array subscripts), and most subsequent programming languages adopted one or other of these positions. For example, the elements of an
are numbered starting from 0 in , so that for an array of n items the sequence of array indices runs from 0 to n-1. This permits an array element's location to be calculated by adding the index directly to address of the array, whereas 1-based languages precalculate the array's base address to be the position one element before the first.[]
There can be confusion between 0- and 1-based indexing, for example Java's
indexes parameters from 1 although
itself uses 0-based indexing.[]
In databases, it is possible for a field not to have a value. It is then said to have a . For numeric fields it is not the value zero. For text fields this is not blank nor the empty string. The presence of null values leads to . No longer is a condition either true or false, but it can be undetermined. Any computation including a null value delivers a null result.[]
is a pointer in a computer program that does not point to any object or function. In C, the integer constant 0 is converted into the null pointer at
when it appears in a pointer context, and so 0 is a standard way to refer to the null pointer in code. However, the internal representation of the null pointer may be any bit pattern (possibly different values for different data types).[]
In mathematics -0 = +0 = 0; both -0 and +0 represent exactly the same number, i.e., there is no "positive zero" or "negative zero" distinct from zero. However, in some computer hardware , zero has two distinct representations, a positive one grouped with the positive numbers and a negative one groupe this kind of dual representation is known as , with the latter form sometimes called negative zero. These representations include the
binary integer representations (but not the
binary form used in most modern computers), and most
number representations (such as
floating point formats).
In binary, 0 represents the value for "off", which means no electricity flow.
Zero is the value of false in many programming languages.
(the date and time associated with a zero timestamp) begins the midnight before the first of January 1970.
epoch (the date and time associated with a zero timestamp) begins the midnight before the first of January 1904.
that require applications to return an integer value as an
typically use zero to indicate success and non-zero values to indicate specific
or warning conditions.
In telephony, pressing 0 is often used for dialling out of a
or to a different , and 00 is used for dialling . In some countries, dialling 0 places a call for .
DVDs that can be played in any region are sometimes referred to as being ""
wheels usually feature a "0" space (and sometimes also a "00" space), whose presence is ignored when calculating payoffs (thereby allowing the house to win in the long run).
In , if the reigning
no longer competes in Formula One in the year following their victory in the title race, 0 is given to one of the drivers of the team that the reigning champion won the title with. This happened in 1993 and 1994, with
driving car 0, due to the reigning World Champion ( and
respectively) not competing in the championship.
On the U.S. , in most states exits are numbered based on the nearest milepost from the highway's western or southern terminus within that state. Several that are less than half a mile (800 m) from state boundaries in that direction are numbered as Exit 0.
The modern numerical digit 0 is usually written as a circle or ellipse. Traditionally, many print typefaces made the capital letter
more rounded than the narrower, elliptical digit 0.
originally made no distinction in shape between O and 0; some models did not even have a separate key for the digit 0. The distinction came into prominence on modern character .
can be used to distinguish the number from the letter. The digit 0 with a dot in the center seems to have originated as an option on
displays and has continued with some modern computer typefaces such as , and in some airline reservation systems. One variation uses a short vertical bar instead of the dot. Some fonts designed for use with computers made one of the capital-O–digit-0 pair more rounded and the other more angular (closer to a rectangle). A further distinction is made in
as used on
by slitting open the digit 0 on the upper right side. Sometimes the digit 0 is used either exclusively, or not at all, to avoid confusion altogether.
, the year 1 BC is the first year before AD 1; there is not a . By contrast, in , the year 1 BC is numbered 0, the year 2 BC is numbered -1, and so on.
(zero as an )
No long count date actually using the number 0 has been found before the 3rd century AD, but since the long count system would make no sense without some placeholder, and since Mesoamerican glyphs do not typically leave empty spaces, these earlier dates are taken as indirect evidence that the concept of 0 already existed at the time.
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This article is based on material taken from the
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Chris Woodford (2006), , Evans Brothers,  
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