# 0 (number)

This page is about the number and numeral 0. For other uses of 0 or "zero", see 0 (disambiguation)
 Template:Numbers (digits) Cardinal 0 zero nought Ordinal 0th zeroth Factorization [itex] 0 [itex] Divisors N/A Roman numeral N/A Binary 0 Octal 0 Duodecimal 0 Hexadecimal 0

0 (zero) 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.

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## 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.

In certain calendars it is common usage to omit the year zero. This is true in particular in the proleptic Gregorian calendar and proleptic Julian calendar. However in modern English, 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

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Seven_segment_display_0_digit_16px_spacing.png

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.

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

.

## History

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

The Indus Valley Civilization peoples (c. 2600 BC) demonstrate the earliest known physical use of decimal fractions in an ancient weight system — 0.05, 0.1, 0.2, 0.5 and negative numbers. They also adopted a minuscule unit of measure equal to 1.704mm, the smallest division ever recorded on a scale of the Bronze Age. However, whether or not they recognized the "0" as a place holder in any sort of symbolic, written representation of these quantities is unknown.

### Using hooks for the numeral

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" ( (http://www-history.mcs.st-and.ac.uk/history/HistTopics/Zero.html) 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

The late Olmec had already begun to use a true zero (a shell glyph) several centuries before Ptolemy in the New World (possibly by the fourth century BC but certainly by 40 BC), which became an integral part of Maya numerals.

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 is the earliest known documented use of zero as a number 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 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.

The earliest known decimal digit zero is documented as having been introduced by Indian mathematicians about 300.

An early documented use of the zero by Brahmagupta dates to 628. He treated zero as a number and discussed operations involving this number. By this time (7th century) the concept had clearly reached Cambodia, and documentation shows the idea later spreading to China and the Islamic world, from where it is recorded to have reached Europe in the 12th century.

## 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 size 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. 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.
• 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), since 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:

[itex]a + s \times (i-1)[itex]

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

[itex]a + s \times i[itex]

This simpler expression can be more efficient to compute in certain situations.

This 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).

Note, however, that a language wishing to index arrays from 1 could simply adopt the convention that every "array address" is represented by [itex]a'=a-s[itex]; 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:

[itex]a' + s \times i[itex]

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.

### Null pointer

A null pointer is a pointer in a computer program that does not point to an object. In C it usually contains the memory address zero. However, it is not required to be zero. Some computer architectures use bit patterns other than zero as their null pointer.

### 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.

## Distinguishing zero from O

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Zero_o_comparison.PNG
A comparison of the letter O and the number 0.

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.

## References

• A Brief History of Zero (http://www.mediatinker.com/whirl/zero/zero.html) - Kristen McQuillin, July 1997 (revised January 2004)
• A history of Zero (http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Zero.html)
• Zero Saga (http://home.ubalt.edu/ntsbarsh/zero/ZERO.HTM)
• Charles Seife (2000). "Zero: The Biography of a Dangerous Idea". (http://www.amazon.com/exec/obidos/tg/detail/-/0140296476/qid=1111606043/sr=8-1/ref=sr_8_xs_ap_i1_xgl14/104-6166861-2891133?v=glance&s=books&n=507846) Publisher: Penguin USA (Paper). ISBN 0140296476
ca:Zero

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