The number zero (from Italian derived from the Arabic first transcribed in Italian) is a symbol marking an empty position in writing numbers in positional notation.

The number zero is an object mathematically possible to express an absence as a quantity (zero): the number of elements of the empty set. It is designed as the smallest of the integers. Special arithmetic properties, especially the impossibility of dividing by zero , sometimes involve handling of his case. It separates the real numbers into positive and negative, takes the place of origin to identify points on the real line.

More generally, zero denotes the neutral element for addition in most of abelian groups and particularly in the rings , body , vector spaces and algebras , sometimes under the name of zero element.

The Babylonians used the first, a little over 200 years BC, a form of zero within a number (eg 304) but never the right number, or left. It is the India that highlights the cultural heritage of the Greeks , the perfect count. She not only uses the zero rating as to how Babylonian, but also as a number with which to operate. Indian concept of zero rating and are then used by mathematicians Arab .

History

Zero as a number

It appeared three times in the history of numeration systems developed by different peoples and civilizations.

The first appearance of zero in Mesopotamia seems to go back to the ^third century BC. AD at the time of the Seleucids. However, it was not used in calculations and served only as a figure (marking an empty position in the system of numeration Babylonian ) ; While ignoring the Romans, he was picked up and used even more by Greek astronomers.

It was then rediscovered by the Chinese , who have failed other hand introduce zero. The inscriptions on bones and scales (jiaguwen) discovered in the region of Anyang, in today's Henan Province, in the late ^nineteenth century , we learn that from the ^fourteenth - XI ^centuriesBC. AD , the Chinese used a decimal type hybrid, combining ten signs fixed units 1 to 9, with special position markers for the tens, hundreds, thousands and myriads.

As the number

Its modern usage, as both figure and like many, is inherited from the invention Indian figures nagari to the ^fifth century. The Indian word denoting the zero was Sunyer (CuNi), which means "empty" space "or" vacant ". The Indian mathematician and astronomer Brahmagupta is the first to define the zero in his book Siddhanta Brahma. This word, first translated into Arabic as "sifr", which means "empty" and "grain", then gave in French the words figure and zero (sifr by translating into Italian Zephiro from which was formed zevero became zero). The graph of the zero, first a circle, is inspired by the representation of the heavens.

As indicated in the etymology , its introduction in the West is resulting from the translation of Arabic mathematics , including the work of al-Khwarizmi , around the ^eighth century. The numbers in India are imported from Spain into Christian Europe around the year one thousand by Gerbert of Aurillac , who became Pope Sylvester II. Zero does not generalize to all in everyday life, the Indian numerals are used primarily to mark tokens ... to chart from January to September!

Only with the return of the intensive trade row with the Crusades that Europeans generalize, the ^twelfth century , the use of zero. Curiosity about the works of Greek and Oriental takes birth at the same time ^{The two zeros of the Maya}

Zero is used by the Maya during the ^first millennium , as a number in their system of numeration position , as many and as ordinal in the calendar, where it corresponds to the introduction of months. By merging into a single transcript, Sylvanus Morley has masked that these two concepts and two different zeros . One corresponds to a zero ordinal dates, the other is a cardinal zero durations , never confused in their use by scribes .

current graph

The spelling "0" is not the only one used in the world, a number of scripts - especially those of Languages of the Indian subcontinent and Southeast Asia - use different spellings ...

Here the zero 7-segment display :

• 

  0

Uses

It is today the basis of our system of measuring temperature :

• 0 C: temperature of the portion of water from the solid state (ice) to liquid state at ambient pressure of 1 013 hPa ;
• 0 K : absolute zero, the lowest possible temperature (-273.15 C), for which the rovibrational energy and kinetics of molecules is zero.

There is no year zero in the Gregorian calendar. In fact, using the number 0 in Europe is after the creation of the Anno Domini by Dionysius Exiguus the ^sixth century. However, to simplify the calculations of ephemerides, the astronomers define a year 0 corresponds to the year -1 of historians, astronomers yr -1 corresponding to the year -2 historians and so on ...

Thus the ^third millennium and the twenty-first ^century began on ¹ January 2001.

Midnight may be noted 00:00.

Computer scientists have a habit of counting from 0 instead of 1. The reason is that the numbering of items stored continuously in a storage area (disk, memory, etc..) Is done by offset from a start address: first item is at the beginning of the zone ( + 0), the second element is the following (+ 1), etc.. This double standard counts from 0 and 1 (each system has its advantages and disadvantages) is the source of numerous errors in programming.

The zero rating as bases 2, 8, 10, 16 ...

In base ten that is used, the number furthest to the right shows the units , the second number indicates the tens , the third the hundreds , the fourth the thousands ...

Zero plays a special role in the positional arithmetic system , whatever the rest.

Recall that the use of base 10 from India, has emerged in France compared to other bases , such as 12 and 60 which were used in some cultures , the system vicsimal having left traces in the French language and the system duodecimal calculation methods in Britain.

When there are residual units, for example in thirty-two (32), the number of units (2) to understand that the other figure (3) shows the dozens.

If you have a whole number of tens (eg, three dozen, thirty), there is no residual unit. So you need a character to mark the 3 corresponds to the tens, and this character is 0, that is how we understand that "30" means "three decades".

We could have used any other character, such a point, so, two hundred and three of them would notice "2.3".

The use of a character "stopgap" dates from the Babylonian numeration , as shown above, but it is not the concept of "absence of quantity, it's just a convenience rating. In Roman numerals , this trick is not useful because the units (I, V), tens (X, L), hundreds (C, D) and thousands (M) are denoted with different characters. In return, the rating numbers above 8999 is problematic and recognition of structures for mental calculation much more tedious fast.

The absence of zero as the amount

Expressing the lack of quantity by a number is not evidence in itself. The absence of an object is expressed by the phrase "there is no" (or "plus").

The numbers are already abstraction : it is not interested in the quality of an object, but just its quantity, countability (the fact that objects are similar but distinct). With zero, we will deny until the quantity.

When we add or multiply two numbers, it was behind the image to combine two lots of similar objects, two herds. This image is no longer valid when handling zero.

The invention of zero has the invention of negative numbers.

Arithmetic and Algebraic Properties

As integer , zero is even.

For many real (or complex ) a:

• (0 is neutral element for addition)
• (0 is absorbing element for multiplication)
• if then
• ⁰ 0 is considered either as equal to 1 (algebra and set theory) , sometimes as undefined for the assessment of the limitations in analysis (it is an indeterminate form of computing limitations ).
• by extension of the factorial using the Gamma function ,
• 
• undefined (see clause division by zero )
• undefined, noting however that the calculation when the two values tend to zero is the basis of calculus.

Usage expanded from zero in mathematics

• Zero is the neutral element in an abelian group endowed with the law or the neutral element for addition in a ring.
• A zero of a function is a point in the domain of the function whose image by the function is zero, also known as root , especially in the case of a function polynomial. See zero (complex analysis).
• In geometry , the dimension of a point is 0.
• In topology , the topological dimension of the Cantor set is 0, although it has a Hausdorff dimension is not zero.
• In analytic geometry , 0 is called the origin , denoted as O (a case where the ambiguity is benign).
• 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 the only possible output value. A particular zero function is the zero morphism. A zero function is the identity in the additive group of functions.
• Zero is one of three possible return values of the Mbius function. If you enter an integer x ² x ² or y, the Mbius function returns zero.
• This is a Pell number.

References

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