0 

Cardinal 
0, zero, "oh" ( ), nought, naught, nil, null, zilch, nada 
Ordinal 
0th, zeroth, noughth 
Factorization 
0 
Divisors 
all numbers 
Roman numeral 
N/A 
Arabic 
٠ 
Bengali 
০ 
Devanāgarī 
० 
Chinese 
〇，零 
Japanese numeral 
〇，零 
Khmer 
០ 
Thai 
๐ 
Binary 
0 
Octal 
0 
Duodecimal 
0 
Hexadecimal 
0 
0 (
zero) is both a
number and the
numerical
digit used to represent that number in
numerals. It plays a central role in
mathematics as the
additive identity of the
integers,
real numbers,
and many other
algebraic structures. As a
digit, zero is used as a placeholder in
place value systems. In the
English language, zero may also be called
oh,
null,
nil,
or
nought.
As number
0 is the
integer preceding
1. In most systems, 0 was identified
before the idea of negative things that go lower than zero was
accepted.
Zero is an even number. 0
is neither positive nor negative. By some definitions 0 is also 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. Almost all
historians omit the
year
zero from the
proleptic
Gregorian and
Julian
calendar, but
astronomers include it
in these same calendars. However, the phrase
Year Zero may be used to
describe any event considered so significant that it serves as a
new base point in time.
As digit
The modern numerical digit 0 is usually
written as a circle, an ellipse, or a rounded rectangle. In most
modern
typefaces, the height of the 0
character is the same as the other digits. However, in typefaces
with
text figures, the character is
often shorter (
xheight).
On the
sevensegment displays
of calculators, watches, and household appliances, 0 is usually
written with six line segments, though on some historical
calculator models it was written with four line segments.
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 rare instances, a leading 0 may distinguish a number. This
appears in
roulette in the United States,
where '00' is distinct from '0' (a wager on '0' will not win if the
ball lands in '00', and vice versa). Sports where competitors are
numbered follow this as well; a
stock car
numbered '07' would be considered distinct from one numbered '7'.
This is most common with singledigit numbers.
Distinguishing the digit 0 from the letter O
Traditionally, many print typefaces made the capital letter
O more rounded than narrower, elliptical digit 0.
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.
The digit 0 with a dot in the centre seems to have originated as an
option on
IBM 3270 displays. Its appearance
has continued with the
Microsoft
Windows typeface
Andalé Mono.
One variation used a short vertical bar instead of the dot. This
could be confused with the
Greek
letter Theta on a badly focused display,
but in practice there was no confusion because theta was not (then)
a displayable character and very little used anyway.
An alternative, the
slashed zero
(looking similar to the letter O except for the slash), was
primarily used in handwritten coding sheets before transcription
to punched cards or tape, and is also used in oldstyle
ASCII graphic sets descended from the default
typewheel on the
ASR33 Teletype.
This form is similar to the symbol \emptyset, or "∅" (
Unicode character U+2205), representing the
empty set, as well as to the letter
Ø used in several
Scandinavian languages. Some
Burroughs/
Unisys equipment displays a digit 0 with a
reversed slash.
The opposing convention that has the letter O
with a slash
and the digit 0
without was advocated by SHARE, a
prominent
IBM user group, and recommended by IBM
for writing
FORTRAN programs, and by a few
other early mainframe makers; this is even more problematic for
Scandinavians because it means two of
their letters collide. Others advocated the opposite convention,
including IBM for writing
Algol programs.
Another convention used on some early
line
printers left digit 0 unornamented but added a tail or hook to
the capital O so that it resembled an inverted
Q
or cursive capital letterO (\mathcal O).
Some fonts designed for use with computers made one of the
capitalO–digit0 pair more rounded and the other more angular
(closer to a rectangle). The
Texas Instruments TI99/4A
computer featured a more angular capital O and a more rounded digit
0, whereas others made the choice the other way around.
German license plate with slit
zeros
The typeface used on most
European vehicle registration plates
distinguishes the two symbols partially in this manner (having a
more rectangular or wider shape for the capital O than the digit
0), but in several countries a further distinction is made by
slitting open the digit 0 on the upper right side (as in
German plates using
the
fälschungserschwerende
Schrift, "hardertofalsify script").
Sometimes the digit 0 is used either exclusively, or not at all, to
avoid confusion altogether.
For example, confirmation numbers used by Southwest
Airlines use only the capital letters O and I instead of the
digits 0 and 1, while Canadian
postal codes use only the digits 1 and 0 and never the capital
letters O and I, although letters and numbers always
alternate.
Names
In 976
Muhammad ibn
Ahmad alKhwarizmi, in his Keys 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
Arabs called sifr.
Will Durant, 'The Story of Civilization', Volume 4,
'The Age of Faith', pp. 241.
The word "
zero" came via
French zéro from
Venetian zero, which (together
with
cipher) came via
Italian zefiro from Arabic صفر,
ṣafira = "it was empty",
ṣifr = "zero", "
nothing".
Italian
zefiro already meant "west wind" from Latin and
Greek
zephyrus; this may have
influenced the spelling when transcribing Arabic
ṣifr. The
Italian mathematician
Fibonacci
(c.11701250), who grew up in Arab 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 Arab
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 thirteenthcentury Paris, a
'worthless fellow' was called a "... cifre en algorisme", i.e., an
"arithmetical nothing"." 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.
History
Early history
By the middle of the
2nd millennium
BCE, 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 BCE, a
punctuation symbol (two slanted wedges) was coopted as a
placeholder in the same
Babylonian system.
In a tablet unearthed
at Kish (dating from
about 700 BCE), the scribe Bêlbânaplu 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.
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.
The
concept of zero as a number and not merely a symbol for separation
is attributed to Indiawhere
by the
9th century CE practical
calculations were carried out using zero, which was treated like
any other number, even in case of division. The Indian scholar
Pingala (circa
5th
2nd century
BCE) 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. He and his contemporary Indian
scholars used the Sanskrit word
śūnya to refer to zero or
void.
History of zero
The
Mesoamerican Long Count
calendar developed in southcentral Mexico and Central America required the use of zero as
a placeholder within its vigesimal
(base20) 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 BCE. Since the eight earliest Long Count dates appear outside
the Maya homeland, 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 BCE, several centuries before
the earliest known Long Count dates.
Although zero became an integral part of
Maya numerals, 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.
The use of a blank on a counting board to represent 0 dated back in
India to 4th century BCE.
In
China, counting rods were
used for calculation since the 4th
century BCE. Chinese mathematicians understood negative
numbers and zero, though they had no symbol for the latter, until
the work of
Song Dynasty mathematician
Qin Jiushao in 1247 established a symbol
for zero in China.
The Nine Chapters on
the Mathematical Art, which was mainly composed in the
1st century CE, stated "[when
subtracting] subtract same signed numbers, add differently signed
numbers, subtract a positive number from zero to make a negative
number, and subtract a negative number from zero to make a positive
number."
By
130 CE,
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). An isolated use of the initial, N, was used
in a table of Roman numerals by
Bede or a
colleague about
725, a zero symbol.
In 498 CE, Indian mathematician and astronomer
Aryabhata stated that "Sthanam sthanam dasa gunam"
or place to place in ten times in value, which may be the origin of
the modern decimalbased place value notation.
The oldest known text to use a decimal
placevalue system, including a zero, is
the Jain text from India entitled the
Lokavibhâga,
dated 458 CE. This text uses Sanskrit numeral words for the digits,
with words such as the Sanskrit word for
void for zero.
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 CE. There are many
documents on copper plates, with the same small
o in them,
dated back as far as the sixth century CE, but their authenticity
may be doubted.
The HinduArabic numerals and the positional number system were
introduced to the
Islamic
civilization by
AlKhwarizmi.
Will Durant, 'The Story of Civilization',
Volume 4, 'The Age of Faith', pp. 241. AlKhwarizmi's
book on arithmetic synthesized Greek and Hindu knowledge and also
contained his own fundamental contribution to mathematics and
science including an explanation of the use of zero.
It was only centuries later, in the 12th century, that the Arabic
numeral system was introduced to the
Western world through
Latin translations of his
Arithmetic.
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), written in
628. 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:
 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.
 A positive or negative number when
divided by zero is a fraction with the zero as
denominator.
 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,
nonzero positive or negative numbers when divided by zero are
either assigned no value, or a value of unsigned infinity, positive
infinity, or negative infinity. Once again, these assignments are
not numbers, and are associated more with computer science than
pure mathematics, where in most contexts no assignment is
done.
Zero as a decimal digit
 See also: History of the
HinduArabic numeral system.
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
HinduArabic numeral
system (base 10) reached Europe in the 11th century, via the
Iberian
Peninsula through
Spanish Muslims, the Moors, together with knowledge of astronomy 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 giveandtake
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 preeminent 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
algorimus after the Persian
mathematician alKhwarizmi. The most popular was written by
Johannes de Sacrobosco, about
1235 and was one of the earliest scientific books to be
printed in
1488. Until the late 15th
century, HinduArabic 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.
In mathematics
Elementary algebra
The number 0 is the least
nonnegative 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, neither a
prime number nor a
composite number, nor is it a
unit. It is, however,
even (see
evenness of zero).
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, for nonzero x. But is undefined, because 0 has no
multiplicative inverse, a consequence of the previous rule; see
division by zero.
 Exponentiation: x^{0} =
^{x}/_{x} = 1, except that the
case x = 0 may be left undefined in some contexts; see
Zero to the zero
power. For all positive real x, 0^{x}
= 0.
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 socalled "
indeterminate form". That does not simply
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 is 0, and
the product of 0 numbers is 1. The
factorial 0! evaluates to 1.
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 least ordinal number, corresponding to the empty
set viewed as a wellordered set.
 In propositional logic, 0
may be used to denote the truth value
false.
 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 lattice theory, 0 may denote
the bottom element of a bounded lattice.
 In category theory, 0 is
sometimes used to denote an initial object of a category.
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. See also
zero for zeros 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
noninvertible square matrices
is a zero map.
In science
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, on the
Kelvin
temperature scale, zero is the coldest possible temperature
(
negative temperatures exist
but are not actually colder), whereas on the
Celsius scale, 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, the
zeropoint 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. It is at the
centre of the
Chemical Galaxy
(2005).
Medical science
In computer science
Numbering from 1 or 0
The most common practice throughout human history has been to start
counting at one. Nevertheless, in
computer science zero is often used as the
starting point. For example, in the vast majority of
programming languages, the elements of
an
array are numbered starting from
0 by
default. The
popularity of the
C
programming language in the 1980s has made this approach
common.
One advantage of this convention is in the use of
modular arithmetic. Every integer is
congruent modulo
N to
one of the numbers 0, 1, 2, ..., , where . Because of this, many
arithmetic concepts (such as hash tables) are more elegantly
expressed in code when the array starts at zero.
A second advantage of zerobased array indexes is that this can
improve efficiency under certain circumstances. To illustrate,
suppose
a is the
memory
address of the first element of an array, and
i is the
index of the desired element. In this fairly typical scenario, it
is quite common to want the address of the desired element. If the
index numbers count from 1, the desired address is computed by this
expression:
 a + s \times (i1) \,\!
where
s is the size of each element. In contrast, if the
index numbers count from 0, the expression becomes this:
 a + s \times i \,\!
This simpler expression can be more efficient to compute in certain
situations.
Note, however, that a language wishing to index arrays from 1 could
simply adopt the convention that every "array address" is
represented by
a′ =
a –
s; that is,
rather than using the address of the first array element, such a
language would use the address of an imaginary element located
immediately before the first actual element. The indexing
expression for a 1based index would be the following:
 a' + s \times i \,\!
Hence, the efficiency benefit of zerobased indexing is not
inherent, but is an artifact of the decision to represent an array
by the address of its first element.
A third advantage is that ranges are more elegantly expressed as
the halfopen
interval,
[0,
n), as opposed to the closed interval, [1,
n],
because empty ranges often occur as input to algorithms (which
would be tricky to express with the closed interval without
resorting to obtuse conventions like [1,0]). On the other hand,
closed intervals occur in mathematics because it is often necessary
to calculate the terminating condition (which would be impossible
in some cases because the halfopen interval isn't always a
closed set) which would have a
subtraction by 1 everywhere.
This situation can lead to some confusion in terminology. In a
zerobased indexing scheme, the first element is "element number
zero"; likewise, the twelfth element is "element number eleven".
Therefore, an analogy from the ordinal numbers to the quantity of
objects numbered appears; the highest index of
n objects
will be and referred to the
nth element. For this reason,
the first element is often referred to as the
zeroth element to avoid confusion.
Null value
In databases a field can have a
null value. This is equivalent
to the field not having a 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
threevalued 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.
Null pointer
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).
(Note that on most common architectures, the null pointer is
represented internally the same way an integer of the same byte
width having a value of zero is represented, so C compilers on such
systems perform no actual conversion.)
Negative zero
In mathematics 0 = 0 = +0, both
−0 and +0 represent the exact 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.
In other fields
 In some countries and some company phone networks, dialing 0 on
a telephone places a call for operator assistance.
 In Braille, the numeral 0 has the same
dot configuration as the letter J.
 DVDs that can be played in any region are
sometimes referred to as being "region
0"
 In classical music, 0 is very rarely used as a number for a
composition: Anton Bruckner wrote a
Symphony No. 0 in D minor and a Symphony No. 00; Alfred
Schnittke also wrote a Symphony No. 0.
 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).
 A chronological prequel of a series may be numbered as 0.
 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 and Alain
Prost respectively) not competing in the championship.
 In the educational series Schoolhouse Rock!, the song My Hero,
Zero is about the use of zero as a placeholder. The song
explains that by appending zeros to a number, it is multiplied by
10 for each one added. This enables mathematicians to create
numbers as large as needed.
Quotations
See also
Notes
 Lemma B.2.2, The integer 0 is even
and is not odd, in
 R. W. Bemer. "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, Volume 10, Issue 8 (August 1967),
pp. 513–518.
 Bo Einarsson and Yurij Shokin. Fortran 90 for the Fortran
77 Programmer. Appendix 7: "The historical development of
Fortran"
 Merriam Webster online Dictionary
 Georges Ifrah. The Universal History of Numbers: From
Prehistory to the Invention of the Computer. Wiley (2000).
ISBN 0471393401.
 Kaplan, Robert. (2000). The Nothing That Is: A Natural
History of Zero. Oxford: Oxford University Press.
 Bourbaki, Nicolas (1998). Elements of the History of
Mathematics. Berlin, Heidelberg, and New York:
SpringerVerlag. 46. ISBN 3540647678.
 Britannica Concise Encyclopedia (2007), entry
algebra
 Binary Numbers in Ancient India
 Math for Poets and Drummers (pdf, 145KB)
 No long count date actually using the number 0 has been found
before the 3rd century CE, 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.
 Diehl, p. 186
 Robert Temple, The Genius of China, A place for zero;
ISBN 1853752924
 Needham, Joseph (1986). Science and Civilization in China:
Volume 3, Mathematics and the Sciences of the Heavens and the
Earth. Taipei: Caves Books, Ltd. Page 43.
 The statement in Chinese, found in Chapter 8 of The Nine
Chapters on the Mathematical Art is 正負術曰:
同名相除，異名相益，正無入負之，負無入正之。其異名相除，同名相益，正無入正之，負無入負之。The word 無入 used here,
for which zero is the standard translation by mathematical
historians, literally means: no entry. The full Chinese
text can be found at :wikisource:zh:九章算術.
 Aryabhatiya of Aryabhata, translated by Walter Eugene
Clark.
 Ifrah, Georges (2000), p. 416.
 Feature Column from the AMS
 Ifrah, Georges (2000), p. 400.
 Algebra with Arithmetic of Brahmagupta and
Bhaskara, translated to English by Henry Thomas
Colebrooke, London1817
 Sigler, L., Fibonacci’s Liber Abaci. English
translation, Springer, 2003.
 Grimm, R.E., "The Autobiography of Leonardo Pisano",
Fibonacci Quarterly
11/1 (February 1973), pp. 99104.
References
 Barrow, John D. (2001) The
Book of Nothing, Vintage. ISBN 0099288451.
 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
0471393401.
 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
0140296476.
 Bourbaki, Nicolas (1998).
Elements of the History of Mathematics. Berlin,
Heidelberg, and New York: SpringerVerlag. ISBN 3540647678.
 Isaac Asimov article "nothing counts" in "Asimov on Numbers"
Pocket Books, 1978
External links