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Muhammad ibn Ahmad al-Khwarizmi , in , stated that if no number appears in the place of tens in a calculation, a little circle should be used "to keep the rows".

The Hindu—Arabic 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 Italian mathematician Fibonacci or Leonardo of Pisa was instrumental in bringing the system into European mathematics in , stating:.

After my father's appointment by his homeland as state official in the customs house of Bugia for the Pisan merchants who thronged to it, he took charge; and in view of its future usefulness and convenience, had me in my boyhood come to him and there wanted me to devote myself to and be instructed in the study of calculation for some days.

There, following my introduction, as a consequence of marvelous instruction in the art, to the nine digits of the Hindus, the knowledge of the art very much appealed to me before all others, and for it I realized that all its aspects were studied in Egypt, Syria, Greece, Sicily, and Provence, with their varying methods; and at these places thereafter, while on business.

I pursued my study in depth and learned the give-and-take of disputation. But all this even, and the algorism, as well as the art of Pythagoras, I considered as almost a mistake in respect to the method of the Hindus Modus Indorum.

Therefore, embracing more stringently that method of the Hindus, and taking stricter pains in its study, while adding certain things from my own understanding and inserting also certain things from the niceties of Euclid's geometric art.

I have striven to compose this book in its entirety as understandably as I could, dividing it into fifteen chapters.

Almost everything which I have introduced I have displayed with exact proof, in order that those further seeking this knowledge, with its pre-eminent method, might be instructed, and further, in order that the Latin people might not be discovered to be without it, as they have been up to now.

If I have perchance omitted anything more or less proper or necessary, I beg indulgence, since there is no one who is blameless and utterly provident in all things.

The nine Indian figures are: With these nine figures, and with the sign Here Leonardo of Pisa uses the phrase "sign 0", indicating it is like a sign to do operations like addition or multiplication.

The most popular was written by Johannes de Sacrobosco , about and was one of the earliest scientific books to be printed in Zero is an even number [48] because it is divisible by 2 with no remainder.

Zero is a number which quantifies a count or an amount of null size. In most cultures , 0 was identified before the idea of negative things, or quantities less than zero, was accepted.

The value, or number , zero is not the same as the digit zero, used in numeral systems using positional notation.

Successive positions of digits have higher weights, so inside a numeral the digit zero is used to skip a position and give appropriate weights to the preceding and following digits.

A zero digit is not always necessary in a positional number system, for example, in the number In some instances, a leading zero may be used to distinguish a number.

The number 0 is the smallest non-negative integer. The natural number following 0 is 1 and no natural number precedes 0.

The number 0 may or may not be considered a natural number , but it is an integer, and hence a rational number and a real number as well as an algebraic number and a complex number.

The number 0 is neither positive nor negative and is usually displayed as the central number in a number line. It is neither a prime number nor a composite number.

It cannot be prime because it has an infinite number of factors , and cannot be composite because it cannot be expressed as a product of prime numbers 0 must always be one of the factors.

The following are some basic elementary rules for dealing with the number 0. These rules apply for any real or complex number x , unless otherwise stated.

The sum of 0 numbers the empty sum is 0, and the product of 0 numbers the empty product is 1. The value zero plays a special role for many physical quantities.

For some quantities, the zero level is naturally distinguished from all other levels, whereas for others it is more or less arbitrarily chosen.

For example, for an absolute temperature as measured in kelvins zero is the lowest possible value negative temperatures are defined, but negative-temperature systems are not actually colder.

This is in contrast to for example temperatures on the Celsius scale, where 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—for example, at a value for the threshold of hearing.

In physics , the zero-point energy is the lowest possible energy that a quantum mechanical physical system may possess and is the energy of the ground state of the system.

Zero has been proposed as the atomic number of the theoretical element tetraneutron. It has been shown that a cluster of four neutrons may be stable enough to be considered an atom in its own right.

This would create an element with no protons and no charge on its nucleus. As early as , Andreas von Antropoff coined the term neutronium for a conjectured form of matter made up of neutrons with no protons, which he placed as the chemical element of atomic number zero at the head of his new version of the periodic table.

It was subsequently placed as a noble gas in the middle of several spiral representations of the periodic system for classifying the chemical elements.

The most common practice throughout human history has been to start counting at one, and this is the practice in early classic computer science programming languages such as Fortran and COBOL.

However, in the late s LISP introduced zero-based numbering for arrays while Algol 58 introduced completely flexible basing for array subscripts allowing any positive, negative, or zero integer as base for array subscripts , and most subsequent programming languages adopted one or other of these positions.

This permits an array element's location to be calculated by adding the index directly to address of the array, whereas 1-based languages precalculate the array's base address to be the position one element before the first.

There can be confusion between 0- and 1-based indexing, for example Java's JDBC indexes parameters from 1 although Java itself uses 0-based indexing.

In databases, it is possible for a field not to have a value. It is then said to have a null value. 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. 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. However, in some computer hardware signed number representations , zero has two distinct representations, a positive one grouped with the positive numbers and a negative one grouped with the negatives; this kind of dual representation is known as signed zero , with the latter form sometimes called negative zero.

In binary, 0 represents the value for "off", which means no electricity flow. The Unix epoch the date and time associated with a zero timestamp begins the midnight before the first of January The MacOS epoch and Palm OS epoch the date and time associated with a zero timestamp begins the midnight before the first of January Many APIs and operating systems that require applications to return an integer value as an exit status typically use zero to indicate success and non-zero values to indicate specific error or warning conditions.

The modern numerical digit 0 is usually written as a circle or ellipse. Traditionally, many print typefaces made the capital letter O more rounded than the narrower, elliptical digit 0.

The distinction came into prominence on modern character displays. A slashed zero can be used to distinguish the number from the letter. One variation uses a short vertical bar instead of the dot.

Some fonts designed for use with computers made one of the capital-O—digit-0 pair more rounded and the other more angular closer to a rectangle.

A further distinction is made in falsification-hindering typeface as used on German car number plates by slitting open the digit 0 on the upper right side.

Sometimes the digit 0 is used either exclusively, or not at all, to avoid confusion altogether. From Wikipedia, the free encyclopedia. Redirected from 0 number.

This article is about the number and digit 0. For other uses, see 0 disambiguation and Zero disambiguation.

For the Stolen Babies album, see Naught album. List of numbers — Integers. Names for the number 0 and Names for the number 0 in English.

History of the Hindu—Arabic numeral system. Retrieved 24 April Number words and number symbols: Archived from the original on 7 March Arabic sifr in the meaning of zero is a translation of the corresponding India sunya.

From the Birth of Numbers, W. The Universal History of Numbers: From Prehistory to the Invention of the Computer. The Crest of the Peacock: The Nothing That Is: A Natural History of Zero.

Similarly, to support division of any integer by any other, the realm of numbers must expand to the rational numbers.

During this gradual expansion of the number system, care is taken to ensure that the "extended operations", when applied to the older numbers, do not produce different results.

Loosely speaking, since division by zero has no meaning is undefined in the whole number setting, this remains true as the setting expands to the real or even complex numbers.

As the realm of numbers to which these operations can be applied expands there are also changes in how the operations are viewed. For instance, in the realm of integers, subtraction is no longer considered a basic operation since it can be replaced by addition of signed numbers.

In keeping with this change of viewpoint, the question, "Why can't we divide by zero? Answering this revised question precisely requires close examination of the definition of rational numbers.

In the modern approach to constructing the field of real numbers, the rational numbers appear as an intermediate step in the development that is founded on set theory.

First, the natural numbers including zero are established on an axiomatic basis such as Peano's axiom system and then this is expanded to the ring of integers.

The next step is to define the rational numbers keeping in mind that this must be done using only the sets and operations that have already been established, namely, addition, multiplication and the integers.

This relation is shown to be an equivalence relation and its equivalence classes are then defined to be the rational numbers. It is in the formal proof that this relation is an equivalence relation that the requirement that the second coordinate is not zero is needed for verifying transitivity.

The above explanation may be too abstract and technical for many purposes, but if one assumes the existence and properties of the rational numbers, as is commonly done in elementary mathematics, the "reason" that division by zero is not allowed is hidden from view.

Nevertheless, a non-rigorous justification can be given in this setting. It follows from the properties of the number system we are using that is, integers, rationals, reals, etc.

The concept that explains division in algebra is that it is the inverse of multiplication. In general, a single value can't be assigned to a fraction where the denominator is 0 so the value remains undefined.

A compelling reason for not allowing division by zero is that, if it were allowed, many absurd results i. When working with numerical quantities it is easy to determine when an illegal attempt to divide by zero is being made.

For example, consider the following computation. The fallacy here is the assumption that dividing by 0 is a legitimate operation with the same properties as dividing by any other number.

A formal calculation is one carried out using rules of arithmetic, without consideration of whether the result of the calculation is well-defined. This infinity can be either positive, negative, or unsigned, depending on context.

As with any formal calculation, invalid results may be obtained. A logically rigorous as opposed to formal computation would assert only that.

Since the one-sided limits are different, the two-sided limit does not exist in the standard framework of the real numbers. It is the natural way to view the range of the tangent function and cotangent functions of trigonometry: This definition leads to many interesting results.

However, the resulting algebraic structure is not a field , and should not be expected to behave like one. This set is analogous to the projectively extended real line, except that it is based on the field of complex numbers.

While this makes division defined in more cases than usual, subtraction is instead left undefined in many cases, because there are no negative numbers.

Although division by zero cannot be sensibly defined with real numbers and integers, it is possible to consistently define it, or similar operations, in other mathematical structures.

In the hyperreal numbers and the surreal numbers , division by zero is still impossible, but division by non-zero infinitesimals is possible.

Any number system that forms a commutative ring —for instance, the integers, the real numbers, and the complex numbers—can be extended to a wheel in which division by zero is always possible; however, in such a case, "division" has a slightly different meaning.

The concepts applied to standard arithmetic are similar to those in more general algebraic structures, such as rings and fields. In a field, every nonzero element is invertible under multiplication; as above, division poses problems only when attempting to divide by zero.

This is likewise true in a skew field which for this reason is called a division ring. However, in other rings, division by nonzero elements may also pose problems.

Since the field axioms only guarantee the existence of such inverses for nonzero elements, this expression has no meaning when b is zero.

In the zero ring, division by zero is possible, which shows that the other field axioms are not sufficient to exclude division by zero in a field.

The IEEE floating-point standard , supported by almost all modern floating-point units , specifies that every floating point arithmetic operation, including division by zero, has a well-defined result.

The standard supports signed zero , as well as infinity and NaN not a number. There are two zeroes: The justification for this definition is to preserve the sign of the result in case of arithmetic underflow.

Integer division by zero is usually handled differently from floating point since there is no integer representation for the result. Some processors generate an exception when an attempt is made to divide an integer by zero, although others will simply continue and generate an incorrect result for the division.

The result depends on how division is implemented, and can either be zero, or sometimes the largest possible integer.

Because of the improper algebraic results of assigning any value to division by zero, many computer programming languages including those used by calculators explicitly forbid the execution of the operation and may prematurely halt a program that attempts it, sometimes reporting a "Divide by zero" error.

In these cases, if some special behavior is desired for division by zero, the condition must be explicitly tested for example, using an if statement.

Some programs especially those that use fixed-point arithmetic where no dedicated floating-point hardware is available will use behavior similar to the IEEE standard, using large positive and negative numbers to approximate infinities.

In some programming languages, an attempt to divide by zero results in undefined behavior. The graphical programming language Scratch 2 used in many schools returns Infinity or -Infinity depending on the sign of the dividend.

From Wikipedia, the free encyclopedia. This article is about the concept in mathematics and exception in computing.

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