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. In the English
language, zero may also be called oh, null, nil or naught. 0 is
the integer preceding 1. In most systems, 0 was identified
before the idea of 'negative integers' was accepted. Zero is an
even number. 0 is neither positive nor negative. 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.
The word "zero" came via French zéro from Venetian zero, which
(together with cipher) came via Italian zefiro from Arabic
safira = "it was empty", sifr = "zero", "nothing", which was
used to translate Sanskrit śūnya, meaning void or empty.
As the Hindu 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
thirteenth-century 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 sifr, and cognates of sifr are common in
the languages of Europe and southwest Asia.
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. 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 India where 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 BC) 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.
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.
The oldest known text to use a decimal place-value system,
including a zero, is the Jain text from India entitled the
Lokavibhâga, dated 458 AD. This text uses Sanskrit numeral words
for the digits, with words such as the Sanskrit word for void
for zero (see also the section Etymology above). 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 AD, but their authenticity may
be doubted.
The Indian numerals and the positional number system were
introduced to the Islamic civilization by Al-Khwarizmi, the
founder of several branches and basic concepts of mathematics.
Al-Khwarizmi'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
Indian 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 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 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.
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
dot had been used in Greece during earlier ciphered numeral
periods.
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. For this
reason, the numerals came to be known in Europe as "Arabic
numerals".
The sum of 0 numbers is 0, and the product of 0 numbers is 1.
The factorial 0! evaluates to 1.
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 |