# 0 (number)

This page is about the number and numeral 0. For other uses of 0 or "zero", see 0 (disambiguation)
 0 10 20 30 40 50 60 70 80 90 >> Cardinal 0 zero nought aught Ordinal 0th zeroth Factorization $\displaystyle 0$ Divisors N/A Roman numeral N/A Binary 0 Octal 0 Duodecimal 0 Hexadecimal 0

0 (zero), alternatively called naught or nought, is both a number and a numeral. It was the last numeral to be created in most numerical systems, as it is not a counting number (which is to say, one begins counting at the number 1) and was in many eras and places represented only by a gap or mark very different from the other numerals.

## 0 as a number

0 is the integer that precedes the positive 1, and all positive integers, and follows -1, and all negative integers. In most (if not all) numerical systems, 0 was identified before the idea of 'negative integers' was accepted.

Zero is a number which means nothing, null, void or an absence of value. For example, if the number of one's brothers is zero, then that person has no brothers. If the difference between the number of pieces in two piles is zero, it means the two piles have an equal number of pieces.

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 virtually starts a new time reckoning.

## 0 as a numeral

The modern numeral 0 is normally written as a circle or (rounded) rectangle. On the seven-segment displays of calculators, watches, etc., 0 is usually written with six line segments (at right), though on some historical calculator models it was written with four line segments. This variant glyph has not caught on.

It is important to distinguish the number zero (as in the "zero brothers" example above) from the numeral or digit zero, 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. The Babylonian numeral system used two narrow slanting wedges, similar to \\, for the equivalent of a positional zero numeral starting in about 400BC.

A zero digit is not always necessary in a positional number system: decimal without a zero provides a possible counterexample.

In fonts with text figures, 0 is usually the same height as a lowercase X, for example, File:TextFigs036.png.

## History

### Etymology

The word zero comes ultimately from the Arabic sifr (صفر) meaning empty or vacant, a literal translation of the Sanskrit Template:IAST meaning void or empty. Through transliteration this became zephyr or zephyrus in Latin. The word zephyrus already meant "west wind" in Latin; the proper noun Zephyrus was the Roman god of the west wind (after the Greek god Zephyros). With its new use for the concept of zero, zephyr came to mean a light breeze—"an almost nothing" (Ifrah 2000; see References). The word zephyr survives with this meaning in English today. The Italian mathematician Fibonacci (c.1170-1250), who grew up in Arab North Africa and is credited with introducing the Arabic decimal system to Europe, used the term zephyrum. This became zefiro in Italian, which was contracted to zero in the Venetian dialect, giving the modern English word.

As the decimal zero and its new mathematics spread through a Europe that was still 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 (2000), "in thirteenth-century Paris, a 'worthless fellow' was called a... cifre en algorisme, i.e., an 'arithmetical nothing.' " (algorithm is also a borrowing from the Arabic, in this case from the name of the 9th-century mathematician al-Khwarizmi.) The Arabic root gave rise to the modern French chiffre, which means digit, figure, or number; chiffrer, to calculate or compute; and chiffré, encrypted; as well as to the English word cipher. Today, the word in Arabic is still sifr, and cognates of sifr are common throughout the languages of Europe. A few additional examples follow.

• Polish: cyfra, digit; szyfrować, to encrypt
• German: Ziffer, digit, figure, numeral, cypher
• French: zéro, zero
• Spanish: cifra, figure, numeral, cypher, code; cero, zero
• Swedish: siffra, numeral, sum, digit

Note that zero in Greek is translated as Μηδέν (Meithen).

### Babylonians and Greeks

By the mid second millennium BC, 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. However, "... a tablet found at Kish ... thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place" ([1] and natural number).

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 interesting 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 ancient Greeks even questioned that 1 was a number.)

### First use of the number

An early use of zero by the Indian mathematician Pingala (possibly 5th-3rd century BC), implied at first glance by Binary Numbers in Ancient India, is only the modern binary representation using 0 and 1 of Pingala's binary system, which used short and long syllables (the latter equal in length to two short syllables) as described in Math for Poets and Drummers (pdf), making it similar to Morse code. In Pingala's system, four short syllables meant one, not zero. Nevertheless, he does use the Sanskrit word Shunya to refer to the concept of void, which was fairly similar to the concept of zero [2].

The late Olmec people of south-central Mexico began to use a true zero (a shell glyph) in the New World possibly by the 4th century BC but certainly by 40 BC, which became an integral part of Maya numerals, but did not influence Old World numeral systems.

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 as just a placeholder, this Hellenistic zero was the first documented use of a true zero in the Old World. In later Byzantine manuscripts of his Syntaxis Mathematica (Almagest), the Hellenistic zero had morphed into the Greek letter omicron (otherwise meaning 70).

Another true 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 true zero symbol.

### Zero as a decimal digit

The earliest known decimal digit zero is thought to have been introduced by Indian mathematicians sometime around the 3rd century. It was written in the shape of a dot, and consequently called Template:IAST "dot". An early documented use of the zero by Brahmagupta dates to 628. He treated zero as a number and discussed operations involving it. By this time (7th century) the concept had clearly reached Cambodia, and documentation shows the idea later spreading to China and the Islamic world.

The Hindu-Arabic number system reached Europe in the late 11th century, via Andalusia, together with knowledge of astronomy and instruments like the astrolabe. The Italian mathematician Fibonacci was instrumental in bringing the system into European mathematics around 1200, though he spoke of the "sign" zero, not as a number. It was not until the 1600s that decimal notation began to come into widespread use in the Occident.

## 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. Zero is neither prime nor composite.

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.

The following are some basic rules for dealing with the number zero, first described in Brahmasphutasiddhanta. 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 · 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. The different quotients confirms that division by zero is undefined.
• 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 if f(x) and g(x) both approach 0 as x approaches some number, then f(x)/g(x) could approach any finite number or ∞ or −∞; it depends on which functions f and g are. See L'Hopital's rule.

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

## In physics

The value zero plays a special role for a large number of physical quantities. For some quantities, the zero level is naturally distinguished from all other levels, where as it for others is more or less arbitrarily chosen. For example, on the kelvin temperature scale, zero is the coldest possible temperature (so that negative temperatures are non-existent), where as 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, e.g. at a value for the threshold of hearing.

## In computer science

### Numbering from 1 or 0?

Human beings usually number things starting 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 (zero-based).

One reason for this convention is that modular arithmetic normally describes a set of N numbers as containing 0,1,2,...N-1 in order to contain the additive identity. Because of this, many arithmetic concepts (such as hash tables) are less elegant to express in code unless the array starts at zero.

Another reason to use zero-based array indices is that it 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:

$\displaystyle a + s \times (i-1)$

where s is the size of each element. In contrast, if the index numbers count from 0, the expression becomes this:

$\displaystyle a + s \times i$

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 $\displaystyle a'=a-s$ ; 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 1-based index would be the following:

$\displaystyle a' + s \times i$

Hence, the efficiency benefit of zero-based indexing is not inherent, but is an artifact of the decision to represent an array by the address of its first element.

This situation can lead to some confusion in terminology. In a zero-based indexing scheme, the first element is "element number zero"; likewise, the twelfth element is "element number eleven". For this reason, the first element is often referred to as the zeroth element to eliminate any possible doubt (though, strictly speaking, this is unnecessary and arguably incorrect, since the meanings of the ordinal numbers are not ambiguous).

### 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 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.

This is owing to the notion that records in a relational database are a set of key/value tuples. A null value, notionally, indicates not that the record has some particular value – "null" – for a given column, but rather that the record has no value at all for that particular column.

### 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), and has no particular association with zero.

### Negative zero

Template:Seemain In some signed number representations (but not the two's complement representation predominant 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. Representations with negative zero can be troublesome, because the two zeroes will compare equal but may be treated differently by some operations.

## Distinguishing zero from O

The oval-shaped zero (appearing like a rugby ball stood on end) 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 controllers (this has the problem that it looks like the Greek letter Theta). The slashed zero, looking identical to the letter O other than the slash, is used in old-style ASCII graphic sets descended from the default typewheel on the venerable ASR-33 Teletype. This format causes problems for certain Scandinavian languages which use Ø as a letter.

The convention which has the letter O with a slash and the zero without was used at IBM 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. 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 circular, 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.

## In other fields

• In some countries, 0 on a telephone calls for operator assistance. On the BlackBerry the 0 key also functions as a spacebar.
• 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".