0 (Number)

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. Historically, it
was the most recent digit to come into use. In the English language, zero may also be called null or
nil when a number, "oh" (IPA: [oʊ]) or cipher (archaic) when a numeral, and nought or naught[1] in
either context.

0 As A Number

0 is the integer between 1 and −1. In most systems, 0 was identified before the idea of 'negative
integers' was accepted. Zero is an even number.[2]

Zero is a number which quantifies a count or an amount of null size; that is, if the number of your
brothers is zero, that means the same thing as having no brothers, and if something has a weight of
zero, it has no weight. If the difference between the number of pieces in two piles is zero, it means
the two piles have an equal number of pieces. Before counting starts, the result can be assumed to be
zero; that is the number of items counted before you count the first item and counting the first item
brings the result to one. And if there are no items to be counted, zero remains the final result.

While mathematicians accept zero as a number, some non-mathematicians would say that zero is not
a number, arguing that one cannot have zero of something (for example, 'zero oranges'). Others hold
that if one has a bank balance of zero, one has a specific quantity of money in that account, namely
none.

Almost all historians omit the year zero from the proleptic Gregorian and Julian calendars, 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.

0 As A Digit

The modern numerical digit 0 is usually written as a circle, an ellipse, or a rounded rectangle. While
the height of the 0 character is the same as the other digits in most modern typefaces, in typefaces
with text figures the character is often less tall (x-height).

On the seven-segment displays of calculators, watches, etc., 0 is usually

written with six line segments, though on some historical calculator
models it was written with four line segments. The latter is less common
than the former.

The value, or number, zero (as in the "zero brothers" example above) 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: bijective numeration provides a possible counterexample.
Distinguishing zero from O

The oval-shaped zero and circular letter O together came into use on
modern character displays. The zero with a dot in the centre seems to have
originated as an option on IBM 3270 displays (in theory 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 a displayable
character). An alternative, the slashed zero (looking similar to the letter O
other than the slash), was primarily used in hand-written coding sheets before transcription to
punched cards or tape, and is also used in old-style ASCII graphic sets descended from the default
typewheel on the ASR-33 Teletype. This form is similar to the symbol , representing the empty
set, as well as to the letter Ø used in several Scandinavian languages.

The convention which has the letter O with a slash and the zero without was used at IBM [citation needed]
and a few other early mainframe makers; this is even more problematic for Scandinavians because it
means two of their letters collide. Some Burroughs/Unisys equipment displays a zero with a reversed
slash. Another convention used on some early line printers left zero unornamented but added a tail or
hook to the letter-O so that it resembled an inverted Q or cursive capital letter-O ( )

The typeface used on some European number plates for cars distinguish the two symbols by making
the O rather egg-shaped and the zero more angular, but most of all by slitting open the zero on the
upper right side, so the circle is not closed any more (as in German plates). The typeface chosen is
called fälschungserschwerende Schrift (abbr.: FE Schrift), meaning "script which is harder to
falsify". Note that those used in the United Kingdom do not differentiate between the two as there
can never be any ambiguity if the design is correctly spaced; the same applies to UK Postcodes.

Sometimes the number zero is used exclusively or not at all to avoid confusion altogether. For
example, confirmation numbers used by Southwest Airlines use only the letters O and I instead of
the numbers 0 and 1.

Etymology

The word "zero" came via French zéro from Venetian language zero, which (together with "cipher")
came via Italian zefiro from Arabic ‫صفر‬, şafira = "it was empty", şifr = "zero", "nothing", which was
used to translate Sanskrit śūnya ( शूनय ), meaning void or empty.

Italian zefiro already meant "west wind" from Latin and Greek zephyrus; this may have influenced
the spelling when transcribing Arabic şifr.[3] The Italian mathematician Fibonacci (c.1170-1250),
who grew up in Arab North Africa and is credited with introducing the Hindu decimal system to
Europe, used the term zephyrum. This became zefiro in Italian, which was contracted to zero in
Venetian, giving the modern English word.

As the Hindu decimal zero and its new mathematics spread from the Arab world to Europe in the
Middle Ages, words derived from sifr 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"."[3] (Algorithm is also a
borrowing from the Arabic, in this case from the name of the 9th century mathematician al-
Khwarizmi.) From şifr also came French chiffre = "digit", "figure", "number", chiffrer = "to calculate
or compute", chiffré= "encrypted". Today, the word in Arabic is still sifr, and cognates of sifr are
common throughout the languages of Europe and southwest Asia.
History

By the mid 2nd millennium BC, the Babylonians 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 perhaps as far back as 700
BC), the scribe Bêl-bân-aplu wrote his zeroes with three hooks, rather than two slanted wedges.[4]

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), et al.,
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.

Early use of something like zero by the Indian scholar Pingala (circa 5th-2nd century BC), implied at
first glance by his use of binary numbers, is only the modern binary representation using 0 and 1
applied to Pingala's binary system, which used short and long syllables (the latter equal in length to
two short syllables), making it similar to Morse code.[5][6] Nevertheless, he and other Indian scholars
at the time used the Sanskrit word śūnya (the origin of the word zero after a series of transliterations
and a literal translation) to refer to zero or void.

History of zero

The use of a blank on a counting board to represent 0 dated

back in India to 4th century BC[8]. The Mesoamerican Long
Count calendar developed in south-central Mexico required
the use of zero as a place-holder within its vigesimal (base-
20) positional numeral system. A shell glyph— —was
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.[9] Since the eight earliest Long Count
dates appear outside the Maya homeland,[10] it is assumed
that the use of zero in the Americas predated the Maya and
was possibly the invention of the Olmecs. Indeed, many of
the earliest Long Count dates were found within the Olmec
heartland, although the fact that the Olmec civilization had
come to an end by the 4th century BC, several centuries
before the earliest known Long Count dates, argues against
the zero being an Olmec discovery.

Although zero became an integral part of Maya numerals, it,

of course, did not influence Old World numeral systems.

In China, counting rods were used for calculation since the 4th century BCE and Chinese
mathematicians understood negative numbers and zero, though they had no symbol for the latter. [11]
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."[12]
By 130, 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 his Syntaxis
Mathematica (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 their initial, N, was used in a table of
Roman numerals by Bede or a colleague about 725, a zero symbol.

In 498 AD, 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 decimal based
place value notation.[13]

The oldest known text to use zero is the Jain text from India entitled the Lokavibhaaga, dated 458
AD.[14] it was first introduced to the world centuries later by Al-Khwarizmi, a Persian mathematician,
astronomer and geographer[citation needed]. He was the founder of several branches and basic concepts of
mathematics. In the words of Philip Hitti, Al Khawarizmi's contribution to mathematics influenced
mathematical thought to a greater extent. His work on algebra initiated the subject in a systematic
form and also developed it to the extent of giving analytical solutions of linear and quadratic
equations, which established him as the founder of Algebra. The very name Algebra has been
derived from his famous book Al-Jabr wa-al-Muqabilah.

His arithmetic synthesized Greek and Hindu knowledge and also contained his own contribution of
fundamental importance to mathematics and science. Thus, he explained the use of zero, a numeral
of fundamental importance developed by the Indians. And 'algorithm' or 'algorizm' is named after
him.

The first apparent appearance of a symbol for zero appears in 876 in India on a stone tablet in
Gwalior. Documents on copper plates, with the same small o in them, dated back as far as the sixth
century AD, abound.

Zero as a decimal digit

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 9th century. The glyph for the zero digit
was written in the shape of a dot, and consequently called bindu ("dot").

The Indian 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. So in Europe they came to be known 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.

Here Leonardo of Pisa uses the word sign "0", indicating it is like a sign to do operations like
addition or multiplication, but he did not recognize zero as a number in its own right. 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 al-Khwarizmi. The most
popular was written by John of Sacrobosco about 1235 and was one of the earliest scientific books to
be printed in 1488. Hindu-Arabic numerals until the late 15th century seem to have predominated
among mathematicians, while merchants preferred to use the abacus. It was only from the 16th
century that they became common knowledge in Europe.

In Mathematics
A) Elementary algebra

Zero (0) is the least non-negative integer. The natural number following zero is one and no natural
number precedes zero. Zero 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).

In set theory, the number zero is the cardinality of the empty set: if one does not have any apples,
then one has zero apples. In fact, in certain axiomatic developments of mathematics from set theory,
zero 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 zero elements.

Zero 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). If zero is excluded from the rational numbers, the real
numbers or the complex numbers, the remaining numbers form an abelian group under
multiplication.

The following are some basic (elementary) rules for dealing with the number zero. 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,
a consequence of the previous rule. For positive x, as y in x/y approaches zero from positive
values, its quotient increases toward positive infinity, but as y approaches zero from negative
 values, the quotient increases toward negative infinity. It is also said that x/0 equals unsigned
infinity, see division by zero.
• Exponentiation: x0 = 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 is an "indeterminate form". That does not simply mean that it is undefined; rather,
it means that the limit of f(x)/g(x) is determined by the particular functions f and g as they both
approach 0. As x approaches some number, the limit may approach any finite number, 0, ∞, or −∞,
depending on the specific behavior of the functions. See l'Hôpital's rule.

The sum of 0 numbers is 0, and the product of 0 numbers is 1.

B) Extended use of zero in mathematics

• Zero is the identity element in an additive group or the additive identity of a ring.
• A zero of a function is a point in the domain of the function whose image under the function
is zero. When there are finitely many zeros these are called the roots of the function. See zero
(complex analysis).
• In geometry, the dimension of a point is 0.
• The concept of "almost" impossible in probability. More generally, the concept of almost
nowhere in measure theory. For instance: if one chooses a point on a unit line interval [0,1) at
random, it is not impossible to choose 0.5 exactly, but the probability that you will get is
zero.
• A zero function (or zero map) is a constant function with 0 as its only possible output value;
i.e., f(x) = 0 for all x defined. 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.
• Zero is one of three possible return values of the Möbius function. Passed an integer of the
form x² or x²y (for x > 1, x and y are both integers), the Möbius function returns zero.
• Zero is the first Perrin number.

Importance

The importance of the creation of the zero mark can never be exaggerated. This giving to airy
nothing, not merely a local habitation and a name, a picture, a symbol, but helpful power, is the
characteristic of the Hindu race from whence it sprang. It is like coining the Nirvana into dynamos.
No single mathematical creation has been more potent for the general on-go of intelligence and
power. G.B. Halsted

Dividing by zero...allows you to prove, mathematically, anything in the universe. You can prove that
1+1=42, and from there you can prove that J. Edgar Hoover is a space alien, that William
Shakespeare came from Uzbekistan, or even that the sky is polka-dotted. (See appendix A for a proof
that Winston Churchill was a carrot.) Charles Seife, from: Zero: The Biography of a Dangerous Idea

...a profound and important idea which appears so simple to us now that we ignore its true merit. But
its very simplicity and the great ease which it lent to all computations put our arithmetic in the first
rank of useful inventions. Pierre-Simon Laplace

The point about zero is that we do not need to use it in the operations of daily life. No one goes out to
buy zero fish. It is in a way the most civilized of all the cardinals, and its use is only forced on us by
the needs of cultivated modes of thought. Alfred North Whitehead

...a fine and wonderful refuge of the divine spirit--almost an amphibian between being and non-
being. Gottfried Leibniz
When a person is pointed out by others as a fool or a '0' he must take it in that sense that there is this
vast empty space within him which he has ignored or which he has found a little hard to fill i.e a wee
bit harder than the rest that he has filled.. He must be self-effacing though as it will do him good
because the biggest zeroes in life later come up from the marsh as fascinating white horses..
Shiksha . S . Suvarna

In Science
A) 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. See also Zero-point energy.

B) 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 their 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).

Rules of Brahmagupta

The rules governing the use of zero appeared for the first time in Brahmagupta's book Brahmasputha
Siddhanta, 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 Brahamagupta:[16]

• 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 they 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, whereas computers and calculators will 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. 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