0 (number)
From Wikipedia, the free encyclopedia
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zeroth, noughth
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all other numbers
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02
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03
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04
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05
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06
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08
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012
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016
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020
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036
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٠,0
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০
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० (shunya)
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零, 〇
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零, 〇
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០
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๐
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0 (zero; BrE: /ˈzɪərəʊ/ or AmE: /ˈziːroʊ/) is both a number[1] and
the numerical digit used
to represent that number in numerals.
It fulfills a central role in mathematics as
the additive identity of
the integers, real
numbers, and many other algebraic structures.
As a digit, 0 is used as a placeholder in place value systems. Names for the
number 0 in English include zero, nought or
(US) naught(/ˈnɔːt/), nil,
or — in contexts where at least one adjacent digit distinguishes it from
the letter "O" — oh or o (/ˈoʊ/).
Informal or slang terms for zero include zilch and zip.[2] Ought or aught (/ˈɔːt/)
has also been used historically.[3]
Contents
Etymology
The
word zero came via French zéro from Venetian zero,
via Italian zefiro from ṣafira or ṣifr (Arabic
صفر).[4] The
word ṣifr, even in pre-Islamic time, had the meaning empty.[5] It
got its meaning of word zero when it was used as a translation to mean śūnya (Sanskrit:
शून्य) from India.[5] The
first known English use was in 1598.[6]
Italian zefiro already meant "west
wind" from Latin and Greek zephyrus; this may have influenced the spelling when
transcribing Arabic ṣifr.[7] The
Italian mathematician Fibonacci(c.1170–1250),
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, which was contracted to zero in Venetian.
As
the decimal zero and its new mathematics spread from the Arabic world to Europe
in the Middle Ages,
words derived from ṣifr and zephyrus came to
refer to calculation, as well as to privileged knowledge and secret codes.
According to Ifrah, "in thirteenth-century Paris, a 'worthless fellow' was
called a '... cifre en algorisme', i.e., an 'arithmetical nothing'."[7] From ṣifr also
came French chiffre =
"digit", "figure", "number", chiffrer =
"to calculate or compute", chiffré =
"encrypted". Today, the word in Arabic is still ṣifr, and
cognates of ṣifr are common in the languages of Europe and
southwest Asia.
Modern usage
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[8] are
used. Several sports have specific words for zero, such as nil in football, love in tennis and a
duck in cricket.
It is often called oh in the context of telephone numbers.
Slang words for zero include zip, zilch, nada, scratch and
even duck egg or goose egg.[9]
History
Egypt
|
nfr
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heart with trachea
beautiful, pleasant, good |
|
Ancient Egyptian
numerals were base 10. They used hieroglyphs for
the digits and were not positional. By 1740 BCE, 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.[10]
Mesopotamia
By
the middle of the 2nd millennium BC, the Babylonian mathematics had
a sophisticated sexagesimal 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 placeholder in
the same Babylonian system.
In a tablet unearthed at Kish (dating
from about 700 BC), the scribe Bêl-bân-aplu wrote his zeros with three
hooks, rather than two slanted wedges.[11]
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.
India
Statue
of Aryabhata
The
concept of zero as a number and not merely a symbol or an empty space for
separation is attributed to India, where, by the 9th century AD, practical
calculations were carried out using zero, which was treated like any other
number, even in case of division.[12][13]
The
Indian scholar Pingala, of 2nd century BC
or earlier, used binary numbers in
the form of short and long syllables (the latter equal in length to two short
syllables), making it similar to Morse
code.[14] In
his Chandah-sutras (prosody sutras), dated to 3rd or 2nd century BCE, Pingala
used theSanskrit word śūnya explicitly to refer to zero.
This is so far the oldest known use of śūnya to mean zero in India.[15] The
fourth Pingala sutra offers a
way to accurately calculate large metric exponentiation,
of the type (2)n,
efficiently with less number of steps.[15]
The
earliest text to use a decimal place-value system,
including a zero, is the Jain text from India entitled the Lokavibhāga, dated 458 AD, whereśūnya ("void"
or "empty") was employed for this purpose.[16] The
first known use of special glyphs 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 Chaturbhuja
Temple at Gwalior in
India, dated 876 AD.[17][18] There
are many documents on copper plates, with the same small o in
them, dated back as far as the sixth century AD, but their authenticity may be
doubted.[11]
In
498 AD, Indian mathematician and astronomer Aryabhata stated that "sthānāt
sthānaṁ daśaguṇaṁ syāt;"[19] i.e.,
"from place to place each is ten times the preceding,"[19][20] which
is the origin of the modern decimal-based place value notation.[21][22]
Rules of Brahmagupta
The
rules governing the use of zero appeared for the first time in Brahmagupta's book Brahmasputha Siddhanta (The
Opening of the Universe),[23] written
in 628 AD. Here Brahmagupta 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. Here are
the rules of Brahmagupta:[23]
·
The sum of zero
and a negative number is negative.
·
The sum of zero
and a positive number is positive.
·
The sum of zero
and zero is zero.
·
The sum of a
positive and a negative is their difference; or, if their absolute values are
equal, zero.
·
Zero divided by
a negative or positive number is either zero or is expressed as a fraction with
zero as numerator and the finite quantity as denominator.
·
Zero divided by
zero is zero.
In
saying zero divided by zero is zero, Brahmagupta differs from the modern
position. Mathematicians normally do not assign a value to this, whereas
computers and calculators sometimes assign NaN,
which means "not a number." Moreover, non-zero positive or negative
numbers when divided by zero are either assigned no value, or a value of
unsigned infinity, positive infinity, or negative infinity.
China
This
is a depiction of zero expressed in Chinese counting
rods, based on the example provided by A
History of Mathematics. An empty space is used to represent zero.[24]
The Sunzi
Suanjing, of unknown date but estimated to be dated from the 1st
to 5th centuries, and Japanese records dated from the eighteenth century,
describe how counting rods were
used for calculations. According to A History of Mathematics, the
rods "gave the decimal representation of a number, with an empty space
denoting zero."[24] The
counting rod system is considered a positional notationsystem.[25]
Zero
was not treated as a number at that time, but as a "vacant position",
unlike the Indian mathematicians who developed the numerical zero.[26] Ch'in
Chu-shao's 1247 Mathematical
Treatise in Nine Sections is the
oldest surviving Chinese mathematical text using a round symbol for zero.[27] Chinese
authors had been familiar with the idea of negative numbers by the Han
Dynasty (2nd century CE), as seen in the The
Nine Chapters on the Mathematical Art,[28] much
earlier than the fifteenth century when they became well established in Europe.[27]
Islamic world
The Arabic-language
inheritance of science was largely Greek,[29] followed
by Hindu influences.[30] In
773, at Al-Mansur's behest,
translations were made of many ancient treatises including Greek, Latin,
Indian, and others.
In
813 AD astronomical tables were prepared by Persian al-Khwarizmi
using Hindu numerals,[30] and
about 825 AD, 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.[31] This
book was later translated into Latin 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.[30]
Muhammad
ibn Ahmad al-Khwarizmi, in 976 AD, 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 ṣifr.[32]
Greeks and Romans
Records
show that the ancient Greeks seemed
unsure about the status of zero as a number. They asked themselves, "How
can nothing be something?", leading to philosophical and,
by the Medieval period, religious arguments about the nature and existence of
zero and the vacuum. The paradoxes of Zeno
of Elea 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, Ptolemy, influenced
by Hipparchus and
the Babylonians, was using a symbol for zero (a small circle with a long
overbar) within a sexagesimal numeral system otherwise using alphabetic Greek
numerals. Because it was used alone, not just as a
placeholder, this Hellenistic zero was
perhaps the first documented use of a number 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. In later Byzantine manuscripts
of Ptolemy's Syntaxis Mathematica (also known as the Almagest),
the Hellenistic zero had morphed into the Greek letter omicron (otherwise
meaning 70).
Another
zero was used in tables alongside Roman
numerals by 525 (first known use by Dionysius
Exiguus), 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 computists (calculators of Easter). The
initial "N" was used as a zero symbol in a table of Roman numerals
by Bede or
his colleague around 725.
Medieval Europe
Positional
notation without the use of zero (using an empty space in tabular arrangements,
or the word kha "emptiness") is known to have been
in use in India from the 6th century. The earliest certain use of zero as
a decimal positional digit dates to the 5th century mention in
the text Lokavibhaga. The glyph
for the zero digit was written in the shape of a dot, and consequently
called bindu ("dot"). The dot had been used in
Greece during earlier ciphered numeral periods.
The Hindu-Arabic
numeral system (base 10) reached Europe
in the 11th century, via the Iberian
Peninsula through Spanish Muslims,
the Moors,
together with knowledge ofastronomy and
instruments like the astrolabe,
first imported by Gerbert of Aurillac.
For this reason, the numerals came to be known in Europe as "Arabic
numerals". The Italian mathematician Fibonacci 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, he took charge; 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 their varying methods; 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 Hindus (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.[33][34]
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 algorismus after the Persian
mathematician al-Khwārizmī. The most popular was written byJohannes de
Sacrobosco, 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 Roman
numerals. In the 16th century, they became commonly used in
Europe.
Americas
The
back of Olmec stela C fromTres
Zapotes, the second oldest Long Count 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 Epi-Olmec script.
The Mesoamerican
Long Count calendar developed in
south-central Mexico and Central America required the use of zero as a
place-holder within its vigesimal (base-20)
positional numeral system. Many different glyphs, including this 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.[35]
—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.[35]
Since
the eight earliest Long Count dates appear outside the Maya homeland,[36] it
is assumed 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 4th century BC, 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 did not influence 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.
Mathematics
0 is
the integer immediately
preceding 1. Zero
is an even number,[37] because
it is divisible by 2. 0 is neither
positive nor negative. By most definitions[38] 0
is a natural number,
and then the only natural number not to be positive. Zero is a number which
quantifies a count or an amount of null size.
In most cultures,
0 was identified before the idea of negative things (quantities) that go lower
than zero was accepted.
The
value, or number, zero is not the same as the digit zero,
used in numeral systems using positional notation.
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 leading zero may
be used to distinguish a number.
Elementary algebra
The
number 0 is the smallest non-negative 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 a whole number and hence a rational
number and a real
number (as well as an algebraic
number and a complex
number).
The
number 0 is neither positive nor negative and appears in the middle of a number
line. It is neither a prime
number nor a composite
number. It cannot be prime because it has an infinite number
of factors and
cannot be composite because it cannot be expressed by multiplying prime numbers
(0 must always be one of the factors).[39] Zero
is, however,even.
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 identity
element (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 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 may be left 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 simply 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.
Other branches of mathematics
·
In set
theory, 0 is the cardinality 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 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. 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 ordinal
number, corresponding to the empty set viewed as a well-ordered
set.
·
In abstract
algebra, 0 is commonly used to denote a zero element,
which is a neutral element for
addition (if defined on the structure under consideration) and an absorbing
element for multiplication (if defined).
·
In recursion
theory, 0 can be used to denote the Turing
degree of the partial computable functions.
Related mathematical terms
·
A zero of a function 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 tozeros of
a holomorphic function.
·
The zero
function (or zero map) on a domain D is the constant
function with 0 as its only possible output value,
i.e., the function f defined by f(x) =
0 for all x in D. A particular zero function
is a zero morphism in
category theory; e.g., a zero map is the identity in the additive group of
functions. The determinant on
non-invertible square matrices is
a zero map.
·
Several branches
of mathematics have zero elements,
which generalise either the property 0 + x = x, or
the property 0 × x = 0, or both.
Physics
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 absolute
temperature (as measured in Kelvin) zero is
the lowest possible value (negative temperatures 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 freezing
point of water. Measuring sound intensity in decibels or phons,
the zero level is arbitrarily set at a reference value—for example, at a value
for the threshold of hearing. In physics,
thezero-point energy is
the lowest possible energy that a quantum
mechanical physical
system may possess and is the energy of the ground
state of the system.
Chemistry
Zero
has been proposed as the atomic
number of the theoretical element tetraneutron. It has been shown that a cluster
of four neutrons may be
stable enough to be considered an atom in
its own right. This would create an element with
no protons and
no charge on its nucleus.
As
early as 1926, Professor Andreas von Antropoff coined the term neutronium for a conjectured form
of matter 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 periodic
table. It was subsequently placed as a noble gas in the
middle of several spiral representations of the periodic system for classifying
the chemical elements.
Computer science
The
most common practice throughout human history has been to start counting at
one, and this is the practice in early classic computer
science programming languages such as Fortran and COBOL.
However, in the late 1950s LISP introduced zero-based numbering for
arrays while Algol 58 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 array are
numbered starting from 0 in C,
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 JDBC indexes
parameters from 1 although Java 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 null value.
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 three-valued
logic. No longer is a condition either true or false,
but it can be undetermined. Any computation including a null value
delivers a null result. Asking for all records with value 0 or value not equal
0 will not yield all records, since the records with value null are excluded.
A null
pointer 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 compile
time 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 "negative zero" distinct from
zero. In some signed number representations (but
not the two's complement representation
used to represent integers in most computers today) and most floating
point number representations, zero has two distinct
representations, one grouping it with the positive numbers and one with the
negatives; this latter representation is known as negative
zero.
Other fields
·
In telephony,
pressing 0 is often used for dialling out of a company network or
to a different city or region,
and 00 is used for dialling abroad.
In some countries, dialling 0 places a call for operator assistance.
·
Roulette 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 Formula
One, if the reigning World
Champion 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 Damon Hill driving
car 0, due to the reigning World Champion (Nigel
Mansell andAlain
Prost respectively) not competing in the
championship.
Symbols and
representations
The
modern numerical digit 0 is usually written as a circle or ellipse.
Traditionally, many print typefaces made the capital letter O more
rounded than the narrower, elliptical digit 0.[40] Typewriters 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 displays.[40]
A slashed
zero 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 IBM 3270 displays
and has continued with some modern computer typefaces such as Andalé Mono, 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 infalsification-hindering typeface as
used on German car
number plates 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.
Year label
In
the BC calendar
era, the year 1 BC is the first year before AD 1; no
room is reserved for a year
zero. By contrast, in astronomical
year numbering, the year 1 BC is
numbered 0, the year 2 BC is numbered −1, and so on.[41]
See also
Notes
1.
Jump up^ Russell,
Bertrand (1942). Principles
of mathematics (2 ed.). Forgotten
Books. p. 125.ISBN 1-4400-5416-9., Chapter
14, page 125
2.
Jump up^ Soanes,
Catherine; Waite, Maurice; Hawker, Sara, eds. (2001). The Oxford
Dictionary, Thesaurus and Wordpower Guide (Hardback) (2nd ed.). New
York: Oxford
University Press. ISBN 978-0-19-860373-3.
·
Douglas Harper
(2011), Zero,
Etymology Dictionary, Quote="figure which stands for naught in the Arabic
notation," also "the absence of all quantity considered as
quantity," c.1600, from French zéro or directly from Italian zero, from
Medieval Latin zephirum, from Arabic sifr "cipher," translation of
Sanskrit sunya-m "empty place, desert, naught";
·
Menninger, Karl
(1992). Number
words and number symbols: a cultural history of numbers.
Courier Dover Publications. pp. 399–404. ISBN 0-486-27096-3.;
·
""zero,
n.". OED Online. December 2011. Oxford University Press. (accessed 4 March
2012).". Archived from
the original on 6 March 2012. Retrieved 2012-03-04.French zéro (1515 in
Hatzfeld & Darmesteter) or its source Italian zero , for *zefiro , <
Arabic çifr
·
Smithsonian
Institution, Oriental
Elements of Culture in the Occident,
p. 518, atGoogle Books,
Annual Report of the Board of Regents of the Smithsonian Institution; Harvard
University Archives, Quote="Sifr occurs in the meaning of “empty” even in
the pre-Islamic time. (...) Arabic sifr in the meaning of zero is a translation
of the corresponding India sunya.”;
·
Jan Gullberg
(1997), Mathematics: From the Birth of Numbers, W.W. Norton & Co.,ISBN
978-0393040029, page 26, Quote = ‘‘Zero
derives from Hindu sunya - meaning void, emptiness - via Arabic sifr, Latin cephirum,
Italian zevero.’’;
·
Robert Logan
(2010), The Poetry of Physics and the Physics of Poetry, World
Scientific, ISBN
978-9814295925, page 38, Quote = “The idea
of sunya and place numbers was transmitted to the Arabs who translated sunya or
“leave a space” into their language as sifr.”
7.
^ Jump up to:a b Ifrah,
Georges (2000). The Universal History of Numbers: From Prehistory to
the Invention of the Computer. Wiley. ISBN 0-471-39340-1.
10.
Jump up^ George
Gheverghese Joseph (2011). The Crest of the Peacock: Non-European Roots
of Mathematics (Third Edition). Princeton. p. 86. ISBN 978-0-691-13526-7.
11.
^ Jump up to:a b Kaplan,
Robert. (2000). The Nothing That Is: A Natural History of Zero.
Oxford: Oxford University Press.
12.
Jump up^ Bourbaki,
Nicolas (1998). Elements of the History of Mathematics. Berlin,
Heidelberg, and New York: Springer-Verlag. 46. ISBN
3-540-64767-8.
15.
^ Jump up to:a b Kim
Plofker (2009), Mathematics in India, Princeton University Press, ISBN
978-0691120676, page 55-56. Quote - “In the
Chandah-sutra of Pingala, dating perhaps the third or second century BCE, there
are five questions concerning the possible meters for any value “n”. (...) The
answer is (2)7 = 128, as
expected, but instead of seven doublings, the process (explained by the sutra)
required only three doublings and two squarings - a handy time saver where “n”
is large. Pingala’s use of a zero symbol as a marker seems to be the first
known explicit reference to zero.”
17.
Jump up^ Bill
Casselman (University of British Columbia), American Mathematical Society,
"All
for Nought"
20.
Jump up^ Agarwal,
M.K. (23 May 2012). From
Bharata to India: Chrysee the Golden.
iUniverse. p. 206. ISBN 9781475907650.
21.
Jump up^ O'Connor,
Robertson, J.J., E. F. "Aryabhata
the Elder". School of Mathematics and
Statistics University of St Andrews, Scotland. Retrieved 26 May 2013.
22.
Jump up^ William
L. Hosch, ed. (15 August 2010). The
Britannica Guide to Numbers and Measurement (Math Explained).
The Rosen Publishing Group. pp. 97–98.ISBN 9781615301089.
23.
^ Jump up to:a b Algebra
with Arithmetic of Brahmagupta and Bhaskara,
translated to English by Henry Thomas Colebrooke, London1817
24.
^ Jump up to:a b Luke
Hodgkin (2 June 2005). A
History of Mathematics : From Mesopotamia to Modernity: From Mesopotamia
to Modernity. Oxford University Press.
p. 85. ISBN 978-0-19-152383-0.
26.
Jump up^ Kang-Shen
Shen; John N. Crossley; Anthony W. C. Lun; Hui Liu (1999). The
Nine Chapters on the Mathematical Art: Companion and Commentary.
Oxford University Press. p. 35. ISBN 978-0-19-853936-0. zero
was regarded as a number in India... whereas the Chinese employed a vacant
position
28.
Jump up^ Struik,
Dirk J. (1987). A Concise History of Mathematics. New York: Dover
Publications. pp. 32–33. "In these matrices we find negative numbers,
which appear here for the first time in history."
30.
^ Jump up to:a b c Will
Durant (1950), The Story of Civilization, Volume 4, The Age of
Faith: Constantine to Dante - A.D. 325-1300, Simon & Schuster, ISBN
978-0965000758, p. 241,Quote =
"The Arabic inheritance of science was overwhelmingly Greek, but Hindu
influences ranked next. In 773, at Mansur's behest, translations were made of
theSiddhantas - Indian astronomical treatises dating as far back as
425 BC; these versions may have the vehicle through which the
"Arabic" numerals and the zero were brought from
India into Islam. In 813, al-Khwarizmi used the Hindu numerals in his
astronomical tables."
31.
Jump up^ Brezina,
Corona (2006). Al-Khwarizmi:
The Inventor Of Algebra. The Rosen
Publishing Group. ISBN 978-1-4042-0513-0.
32.
Jump up^ Will
Durant (1950), The Story of Civilization, Volume 4, The Age
of Faith, Simon & Schuster, ISBN
978-0965000758, p. 241, Quote =
"In 976, Muhammad ibn Ahmad, in hisKeys of the Sciences, remarked
that if, in a calculation, no number appears in the place of tens, a little
circle should be used "to keep the rows". This circle the Mosloems
called ṣifr, "empty" whence our cipher."
34.
Jump up^ Grimm,
R.E., "The Autobiography of Leonardo Pisano", Fibonacci
Quarterly 11/1 (February 1973), pp. 99–104.
35.
Jump up^ 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.
37.
Jump up^ Lemma B.2.2, The
integer 0 is even and is not odd, in Penner, Robert C. (1999).Discrete
Mathematics: Proof Techniques and Mathematical Structures. World
Scientific. p. 34. ISBN 981-02-4088-0.
38.
Jump up^ Bunt,
Lucas Nicolaas Hendrik; Jones, Phillip S.; Bedient, Jack D. (1976). The
historical roots of elementary mathematics.
Courier Dover Publications. pp. 254–255. ISBN 0-486-13968-9., Extract
of pages 254–255
39.
Jump up^ Reid,
Constance (1992). From
zero to infinity: what makes numbers interesting (4th
ed.). Mathematical
Association of America.
p. 23. ISBN 978-0-88385-505-8.
40.
^ Jump up to:a b Bemer,
R. W. (1967). "Towards standards for handwritten zero and oh: much ado
about nothing (and a letter), or a partial dossier on distinguishing between
handwritten zero and oh". Communications of the ACM 10 (8):
513–518.doi:10.1145/363534.363563.
41.
Jump up^ Steel,
Duncan (2000). Marking time: the epic quest to invent the perfect
calendar. John Wiley & Sons. p. 113. ISBN 0-471-29827-1. In
the B.C./A.D. scheme there is no year zero. After 31 December 1 BC came AD
1 January 1. ... If you object to that no-year-zero scheme, then don't use it:
use the astronomer's counting scheme, with negative year numbers.
References
·
Diehl, Richard
A. (2004) The Olmecs: America's First Civilization, Thames &
Hudson, London.
·
Ifrah, Georges (2000) The
Universal History of Numbers: From Prehistory to the Invention of the Computer,
Wiley. ISBN
0-471-39340-1.
·
Kaplan, Robert
(2000) The Nothing That Is: A Natural History of Zero, Oxford:
Oxford University Press.
·
Seife,
Charles (2000) Zero: The Biography of a
Dangerous Idea, Penguin USA (Paper). ISBN
0-14-029647-6.
·
Bourbaki,
Nicolas (1998). Elements of the History of
Mathematics. Berlin, Heidelberg, and New York: Springer-Verlag. ISBN
3-540-64767-8.
·
This article is
based on material taken from the Free On-line
Dictionary of Computing prior to 1
November 2008 and incorporated under the "relicensing" terms of
the GFDL,
version 1.3 or later.
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