0

Zero redirects here. For other meanings of that word, see also Zero (disambiguation).

Template:Numbers (digits)
Cardinal	0 zero nought
Ordinal	0th zeroth
Factorization	$0$
Divisors	N/A
Roman numeral	N/A
Binary	0
Octal	0
Duodecimal	0
Hexadecimal	0

Zero or Nought (0) is a number that precedes the positive one, and all positive numbers, and follows negative one, and all negative numbers.

Zero is a number introduced by Indian mathematicians, which means nothing, null, void or an absence of value. For example, if the number of your brothers is zero, then you have no brothers. If the difference between the number of pieces in two piles is zero, it means the two piles have the same number of pieces.

In certain calendars it is common usage to omit the year zero when extending the calendar to years prior to its introduction: see proleptic Gregorian calendar and proleptic Julian calendar.

History

The numeral or digit zero is used in numeral systems where the position of a digit signifies its value. Successive positions of digits have higher values, so the digit zero is used to skip a position and give appropriate value to the preceding and following digits.

By about 300 BC, the Babylonians had started to use a basic numeral system and were using two slanted wedges to mark an empty space. However, this symbol did not have any true function other than as a placeholder.

In ancient Greece, influenced by the Babylonians, compilers of astronomical almanacs were early adopters of the zero as a placeholder. An example was Ptolemy's Almagest of 130 AD. In general, zero did not have its own Roman numeral, but the concept of zero as a number was well known by all Christian medieval computists, who used it for calculating the date of Easter. They included zero (via the Latin word nullae meaning nothing) as one of nineteen epacts, or the age of the moon on March 22. The first three epacts were nullae, xi, and xxii (written in minuscule or lower case). An example was Dionysius Exiguus in 525, and the concept of zero was no doubt well known earlier. About 725, Bede or one of his colleagues used the letter N, the initial of nullae, in a table of epacts, all written in Roman numerals.

The use of zero as a number by itself was a relatively late addition to mathematics, thought to be introduced by Indian mathematicians. An early study of the zero by Brahmagupta dates to 628. By this time it was already known in Cambodia, and it later spread to China and the Islamic world, from where it reached Europe in the 12th century.

Zero was also used as a numeral in Pre-Columbian Mesoamerica, from as early as the 4th century BC. It was used by the Olmec and subsequent civiliations; see also: Maya numerals.

The word zero (as well as cipher) comes from Arabic sifr, meaning "empty".

In mathematics

Zero (0) is both a number and a numeral. The natural number following zero is one and no natural number precedes zero. Zero may or may not be counted as a natural number, depending on the definition of natural numbers.

In set theory, the number zero is the size of the empty set: if you do not have any apples, then you have zero apples. In fact, in certain axiomatic developments of mathematics from set theory, zero is defined to be the empty set.

The following are some basic rules for dealing with the number zero. These rules apply for any complex number x, unless otherwise stated.

Addition: x + 0 = x and 0 + x = x. (That is, 0 is an identity element with respect to addition.)
Subtraction: x - 0 = x and 0 - x = -x.
Multiplication: x × 0 = 0 and 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.
Exponentiation: x⁰ = 1, except that the case x = 0 may be left undefined in some contexts. For all positive real x, 0^x = 0.

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. See zero (complex analysis).
In geometry, the dimension of a point is 0.
In analytic geometry, 0 is the origin.
The concept of "almost" impossible in probability. More generally, the concept of almost nowhere in measure theory.
A zero function is a function with 0 as its only possible output value. A particular zero function is a zero morphism. A zero function is the identity in the additive group of functions.
The zero of a function is a preimage of zero, also called the root of a function.
Zero is one of three possible return values of the Möbius function. Passed an integer x² or x²y, the Möbius function returns zero.
It is the number of n×n magic squares for n = 2.
It is the number of n-queens problem solutions for n = 2,3.

In computer science

Counting from 1 or 0?

Human beings usually count things from one, not zero. Yet in computer science zero has become the popular indication for a starting point. For example, in almost all old programming languages, an array starts from 1 by default, which is natural for humans. As programming languages have developed, it has become more common that an array starts from zero by default. This is because, with a one-based index, one must be subtracted to obtain a correct offset for things like obtaining the location of a specific element.

Distinguishing zero from O

If your zero is centre-dotted and letter-O is not, or if letter-O looks almost rectangular but zero looks more like an American football stood on end (or the reverse), you're probably looking at a modern character display (though the dotted zero seems to have originated as an option on IBM 3270 controllers). If your zero is slashed but letter-O is not, you're probably looking at an old-style ASCII graphic set descended from the default typewheel on the venerable ASR-33 Teletype (which causes problems for Norwegians and Danish who use Ø as a letter).

If letter-O has a slash across it and the zero does not, your display is tuned for a very old convention used at IBM and a few other early mainframe makers (which 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. And yet another convention common on 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 rectangular, but most of all by opening the zero on the upper right side, so here the circle is not closed any more (as in German plates).

In paper writing one may not distinguish the 0 and O at all, or may add a slash across it in order to show the difference, although this sometimes causes ambiguity in regard to the symbol for the Null Set.

"Zero" as a verb

In computing, zero is a default digit, meaning none and initial value. To zero (or zeroise or zeroize) a set of data means to set every bit in the data to zero (or off). This is usually said of small pieces of data, such as bits or words (especially in the construction "zero out").

Zero means to erase, to discard all data from. This is often said of disks and directories, where "zeroing" need not involve actually writing zeroes throughout the area being zeroed. One may speak of something being "logically zeroed" rather than being "physically zeroed".

A Null pointer in C programming language usually contains the memory address of zero. However, it is not required to be zero. Some computer architectures use bit patterns other than zero as their null pointer.

References

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.