Zero number definition
Zero number history
Zero number properties
Sets that contain zero

Zero number definition

Zero is a number used in mathematics to lớn describe no quantity ornull quantity.

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When there are 2 apples on the table và we take the 2 apples, wecan say that there are zero apples on the table.

The zero number is not positive number và not negative number.

The zero is also a placeholder digit in other numbers (e.g:40,103, 170).

Is zero a number?

Zero is a number. It is not positive nor negative number.

Zero digit

The zero digit is used as a placeholder when writing numbers.

For example:

204 = 2×100+0×10+4×1

Zero number history

Who invented the zero number?

The modern 0 symbol was invented in India in the 6-th century,used later by the Persians and Arabs và later in Europe.

Symbol of zero

The zero number is denoted with the0 symbol.

The Arabic numeral system uses the٠ symbol.

Zero number properties

x represents any number.


x + 0 = x

3 + 0 = 3
Subtractionx - 0 = x3 - 0 = 3
Multiplicationx × 0 = 05 × 0 = 0
Division0 ÷ x = 0, when x ≠ 00 ÷ 5 = 0
x ÷ 0 is undefined5 ÷ 0 is undefined
Exponentiation0 x= 005= 0
x 0= 150 =1
Root√0 =0
Logarithmlogb(0)is undefined
Factorial0! = 1
Sinesin 0º = 0
Cosinecos 0º = 1
Tangenttan 0º = 0


0" = 0

Integral∫ 0 dx = 0 + C
Zero addition
Addition of a number plus zero is equal to lớn the number:

x + 0 = x

For example:

5 + 0 = 5

Zero subtraction

Subtraction of a number minus zero is equal to lớn the number:

x - 0 = x

For example:

5 - 0 = 5

Multiplication by zero

Multiplication of a number times zero is equal to zero:

x × 0 = 0

For example:

5 × 0 = 0

Number divided by zero

Division of a number by zero is not defined:

x ÷ 0 is undefined

For example:

5 ÷ 0 is undefined

Zero divided by a number

Division of a zero by a number is zero:

0 ÷ x = 0

For example:

0 ÷ 5 = 0

Number to lớn the zero power

The power of a number raised by zero is one:

x0 = 1

For example:

50 = 1

Logarithm of zero

The base b logarithm of zero is undefined:

logb(0)is undefined

There is no number we can raise the base b with lớn get zero.

Only the limit of the base b logarithm of x, when x convergeszero is minus infinity:

Sets that contain zero

Zero is an element of the natural numbers, integer numbers, real numbersand complex numbers sets:

Set membership notation
Natural numbers (non negative)0 ∈ ℕ0
Integer numbers0 ∈ ℤ
Real numbers0 ∈ ℝ
Complex numbers0 ∈ ℂ
Rational numbers0 ∈ ℚ
Is zero even or odd number?

The set of even numbers is:

... ,-10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10, ...

The set of odd numbers is:

... ,-9, -7, -5, -3, -1, 1, 3, 5, 7, 9, ...

Zero is an integer multiple of 2:

0 × 2 = 0

Zero is a member of the even numbers set:

0 ∈ 2k,k∈ℤ

So zero is an even number và not an odd number.

Is zero a natural number?

There are two definitions for the natural numbers set.

The phối of non negative integers:

ℕ0= 0,1,2,3,4,5,6,7,8,...

The set of positive integers:

ℕ1= 1,2,3,4,5,6,7,8,...

Zero is a member of the mix of non negative integers:

0 ∈ ℕ0

Zero is not a thành viên of the mix of positive integers:

0 ∉ ℕ1

Is zero a whole number?

There are three definitions for the whole numbers:

The mix of integer numbers:

ℤ =0,1,2,3,4,5,6,7,8,...

The mix of non negative integers:

ℕ0= 0,1,2,3,4,5,6,7,8,...

The phối of positive integers:

ℕ1= 1,2,3,4,5,6,7,8,...

Zero is a thành viên of the phối of integer numbers & the mix of nonnegative integers:

0 ∈ ℤ

0 ∈ ℕ0

Zero is not a thành viên of the phối of positive integers:

0 ∉ ℕ1

Is zero an integer number?

The phối of integer numbers:

ℤ =0,1,2,3,4,5,6,7,8,...

Zero is a thành viên of the phối of integer numbers:

0 ∈ ℤ

So zero is an integer number.

Is zero a rational number?

A rational number is a number that can be expressed as thequotient of two integer numbers:

ℚ =n/m; n,m∈ℤ

Zero can be written as a quotient of two integer numbers.

For example:

0 = 0/3

So zero is a rational number.

Is zero a positive number?

A positive number is defined as a number that is greater thanzero:

Cardinal0 zero o/oh nought naught nilOrdinal0th zerothFactorization0displaystyle 0
DivisorsN/ARoman numeralN/ABinary0Octal0Duodecimal0Hexadecimal0

0 (zero) is both a number và a numerical digit used lớn represent that number in numerals. As a number, zero means nothing—an absence of other values. It plays a central role in mathematics as the identity element of the integers, real numbers, & many other algebraic structures. As a digit, zero is used as a placeholder in place value systems. Historically, it was the last digit khổng lồ come into use. In the English language, zero may also be called nil when a number, o/oh when a numeral, & nought/naught in either context.


4 History5 In mathematics6 In science7 In computer science

0 as a number

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

Zero is an integer which quantifies a count or an amount of null size; that is, if the number of your brothers is zero, that means the same thing as having no brothers, and if something has a weight of zero, it has no weight. If the difference between the number of pieces in two piles is zero, it means the two piles have an equal number of pieces. Before counting starts, the result can be assumed lớn be zero; that is the number of items counted before you count the first item and counting the first thắng lợi brings the result to one. & if there are no items khổng lồ be counted, zero remains the final result.

While mathematicians all accept zero as a number, some non-mathematicians would say that zero is not a number, arguing one cannot have zero of something. Others hold that if you have a bank balance of zero, you have a specific quantity of money in your account, namely none. It is that latter view which is accepted by mathematicians & most others.

Almost all historians omit the year zero from the proleptic Gregorian và Julian calendars, but astronomers include it in these same calendars. However, the phrase Year Zero may be used khổng lồ 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. In old-style fonts with text figures, 0 is usually the same height as a lowercase x.



On the seven-segment displays of calculators, watches, etc., 0 is usually written with six line segments, 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 using positional notation. Successive positions of digits have higher values, so the digit zero is used to lớn skip a position & give appropriate value lớn the preceding & following digits. A zero digit is not always necessary in a positional number system: bijective numeration provides a possible counterexample.


The word zero comes through the Arabic literal translation of the Sanskrit śūnya ( शून्य ), meaning void or empty, into ṣifr (صفر) meaning empty or vacant. 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 lớn mean a light breeze—"an almost nothing."<1> This became zefiro in Italian, which was contracted lớn zero in Venetian, giving the modern English word.

As the Hindu decimal zero & its new mathematics spread from the Arab world to Europe in the Middle Ages, words derived from sifr & zephyrus came to lớn refer lớn calculation, as well as to privileged knowledge and secret codes. According khổng lồ Ifrah, "in thirteenth-century Paris, a "worthless fellow" was called a… cifre en algorisme, i.e., an "arithmetical nothing.""<1> The Arabic root gave rise to lớn the modern French chiffre, which means digit, figure, or number; chiffrer, to calculate or compute; và chiffré, encrypted; as well as khổng lồ the English word cipher. Below are some more examples:

Arabic: SifrCzech/Slovak: cifra, digit; šifra, cypherDanish: ciffer, digitDutch: cijfer, digitFrench: zéro, zeroGerman: Ziffer, digit, figure, numeral, cypherHindi: shunyaHungarian: nullaItalian: cifra, digit, numeral, cypher; zero, zeroKannada: sonneNorwegian: siffer, digit, numeral, cypher; null, zeroPersian: SefrPolish: cyfra, digit; szyfrować, lớn encrypt; zero, zeroPortuguese: cifra, figure, numeral, cypher, code; zero, zeroRussian: цифра (tsifra), digit, numeral; шифр (shifr) cypher, codeSlovenian: cifra, digitSpanish: cifra, figure, numeral, cypher, code; cero, zeroSwedish: siffra, numeral, sum, digit; chiffer, cypherSerbian: цифра (tsifra), digit, numeral; шифра (shifra) cypher, code; нула (nula), zeroTurkish: SıfırUrdu: Sifer, Anda, Zero

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


Early history of zero

By the mid-second millennium B.C.E., the Babylonians had a sophisticated sexagesimal (base-60) positional numeral system. The lack of a positional value (or zero) was indicated by a space between sexagesimal numerals. By 300 B.C.E. A punctuation symbol (two slanted wedges) was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish (dating from perhaps as far back as 700 B.C.E.), the scribe Bêl-bân-aplu wrote his zeroes with three hooks, rather than two slanted wedges.<2>

The Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the over of a number. Thus numbers lượt thích 2 and 120 (2×60), 3 và 180 (3×60), 4 và 240 (4×60), etc. Looked the same because the larger numbers lacked a final sexagesimal placeholder. Only context could differentiate them.

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 lớn interesting philosophical and, by the Medieval period, religious arguments about the nature & existence of zero & 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.)

Early use of something like zero by the Indian scholar Pingala (circa 5th-2nd century B.C.E.), implied at first glance by his use of binary numbers, is only the modern binary representation using 0 & 1 applied to lớn Pingala"s binary system, which used short and long syllables (the latter equal in length to two short syllables), making it similar khổng lồ Morse code.<3><4> Nevertheless, he & other Indian scholars at the time used the Sanskrit word śūnya (the origin of the word zero after a series of transliterations và a literal translation) to lớn refer to lớn zero or void.<5>


History of zero

The Long Count calendar developed in south-central Mexico required the use of zero as a place-holder within its vigesimal (base-20) positional numeral system. A shell glyph—

—was used as a zero symbol for these Long Count dates, the earliest of which (on Stela 2 at Chiapa de Corzo, Chiapas) has a date of 36 B.C.E. Since the eight earliest Long Count dates appear outside the Maya homeland,<6> it is assumed that the use of zero in the Americas predated the Maya and was possibly the invention of the Olmecs. Indeed, many of the earliest Long Count dates were found within the Olmec heartland, although the fact that the Olmec civilization had come to an over by the fourth century B.C.E., several centuries before the earliest known Long Count dates, argues against the zero being an Olmec invention.

Although zero became an integral part of Maya numerals, it of course did not influence Old World numeral systems.

By 130 C.E., 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 just as a placeholder, this Hellenistic zero was perhaps the first documented use of a number zero in the Old World. However, the positions were usually limited to lớn the fractional part of a number (called minutes, seconds, thirds, fourths, etc.)—they were not used for the integral part of a number.

Another 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, as a zero symbol.

The oldest known text khổng lồ use zero is the Jain text from India entitled the Lokavibhaaga, dated 458 C.E.<1>

The first indubitable appearance of a symbol for zero appears in 876 in India on a stone tablet in Gwalior. Documents on copper plates, with the same small o in them, dated back as far as the sixth century C.E. Abound.<2>

Rules of Brahmagupta

The rules governing the use of zero appeared for the first time in the book Brahmasputha Siddhanta written in 628 by Brahmagupta (598-670). Here, Brahmagupta considers not only zero but also negative numbers, & the algebraic rules for the elementary operations of arithmetic with such numbers. In some instances, his rules differ from the modern standard. Brahamagupta"s rules are given below:<7>

The sum of two positive quantities is positive
The sum of two negative quantities is negative
The sum of zero & a negative number is negative
The sum of a positive number và zero is positive
The sum of zero và zero is zero
The sum of a positive & a negative is their difference; or, if they are equal, zero
In subtraction, the less is to be taken from the greater, positive from positive
In subtraction, the less is to be taken from the greater, negative from negative
When the greater however, is subtracted from the less, the difference is reversed
When positive is to be subtracted from negative, and negative from positive, they must be added together
The product of a negative quantity & a positive quantity is negative
The product of a negative quantity and a negative quantity is positive
The hàng hóa of two positive, is positive
Positive divided by positive or negative by negative is positive
Positive divided by negative is negative. Negative divided by positive is negative
A positive or negative number when divided by zero is a fraction with the zero as denominator
Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator và the finite quantity as denominator
Zero divided by zero is zero

In saying "zero divided by zero is zero," Brahmagupta differs from the modern position. Mathematicians normally vì chưng not assign a value, whereas computers and calculators will sometimes assign Na
N, which means "not a number." Moreover, non-zero positive or negative numbers when divided by zero are either assigned no value, or a value of unsigned infinity, positive infinity, or negative infinity. Once again, these assignments are not numbers, & are associated more with computer science than pure mathematics, where in most contexts no assignment is made. (See division by zero)

Zero as a decimal digit

Positional notation without the use of zero (using an empty space in tabular arrangements, or the word kha "emptiness") is known lớn have been in use in India from the sixth century. The earliest certain use of zero as a decimal positional digit dates to lớn the ninth century. The glyph for the zero digit was written in the shape of a dot, and consequently called bindu "dot."

The Hindu-Arabic numeral system reached Europe in the eleventh century, via the Iberian Peninsula through Spanish Muslims the Moors, together with knowledge of astronomy và instruments like the astrolabe, first imported by Gerbert of Aurillac (c. 940-1003). They came khổng lồ be known as "Arabic numerals." The Italian mathematician Leonardo of Pisa (c. 1170-1250), also called Fibonacci,* was instrumental in bringing the system into European mathematics in 1202. Here, Leonardo states:

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, & for it I realized that all its aspects were studied in Egypt, Syria, Greece, Sicily, and Provence, with their varying methods… But all this even, and the algorism, as well as the art of Pythagoras, I considered as almost a mistake in respect khổng lồ the method of the Hindus. (Modus Indorum)… The nine Indian figures are: 9 8 7 6 5 4 3 2 1. With these nine figures, và with the sign 0 … any number may be written.<8>

Here Leonardo of Pisa uses the word sign "0," indicating it is lượt thích a sign to bởi operations lượt thích addition or multiplication, but he did not recognize zero as a number in its own right.

In mathematics

Elementary algebra

Zero (0) is the lowest non-negative integer. The natural number following zero is one và 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 cardinality of the empty set: if one does not have any apples, then one has zero apples. Therefore, in some cases, zero is defined to lớn be the empty set.

Zero is neither positive nor negative, neither a prime number nor a composite number, nor is it a unit.

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 = 0 + x = x. (That is, 0 is an identity element with respect khổng lồ addition.)Subtraction: x − 0 = x & 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 sum of 0 numbers is 0, và the sản phẩm of 0 numbers is 1.

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.

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. When there are finitely many zeros these are called the roots of the function. See zero (complex analysis).The concept of "almost" impossible in probability. More generally, the concept of almost nowhere in measure theory. For instance: if one chooses a point on a unit line interval <0,1) at random, it is not impossible to lớn choose 0.5 exactly, but the probability that you will is zero.A zero function (or zero map) is a constant function with 0 as its only possible đầu ra value; i.e., f(x) = 0 for all x defined. A particular zero function is a zero morphism in category theory; e.g., a zero maps is the identity in the additive group of functions. The determinant on non-invertible square matrices is a zero map.Zero is one of three possible return values of the Möbius function. Passed an integer of the khung x2 or x2y (for x > 1), the Möbius function returns zero.Zero is the first Perrin number.

In science


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, whereas for others it is more or less arbitrarily chosen. For example, on the Kelvin temperature scale, zero is the coldest possible temperature (negative temperatures exist but are not actually colder), whereas on the Celsius scale, zero is arbitrarily defined khổng lồ be at the freezing point of water. Measuring sound intensity in decibels or phons, the zero level is arbitrarily phối at a reference value—for example, at a value for the threshold of hearing.


Zero has been proposed as the atomic number of the theoretical element tetraneutronium. It has been shown that a cluster of four neutrons may be stable enough to lớn be considered an atom in their own right. This would create an element with no protons và no charge on its nucleus.

As early as 1926 Professor Andreas von Antropoff coined the term neutronium for a conjectured khung 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. It is at the center of the Chemical Galaxy (2005).

In computer science

Numbering from 1 or 0?

The most common practice throughout human history has been to lớn start counting at one. Nevertheless, in computer science zero has become the standard starting point. For example, in almost all old programming languages, an array starts from 1 by default. As programming languages have developed, it has become more common that an array starts from zero by default, the "first" item in the array being thành phầm 0. In particular, the popularity of the programming language "C" in the 1980s has made this approach common.

One reason for this convention is that modular arithmetic normally describes a phối 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 lớn express in code unless the array starts at zero.

In certain cases, counting from zero improves the efficiency of various algorithms, such as in searching or sorting arrays. Improved efficiency means that the algorithm takes either less time, less resources, or both, to complete a given task.

This situation can lead khổng lồ 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." Therefore, an analogy from the ordinal numbers to lớn the quantity of objects numbered appears; the highest index of n objects will be (n-1) and referred to the n:th element. For this reason, the first element is often referred khổng lồ as the zeroth element to eliminate any possible doubt.

Null value

In databases a field can have a null value. This is equivalent khổng lồ 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.

Null pointer

A null pointer is a pointer in a computer program that does not point to any object or function, which means that when it appears in a program or code, it tells the computer lớn take no kích hoạt on the associated portion of the code.

Negative zero

In some signed number representations (but not the two"s complement representation predominant today) & most floating point number representations, zero has two distinct representations, one grouping it with the positive numbers & one with the negatives; this latter representation is known as negative zero. Representations with negative zero can be troublesome, because the two zeros will compare equal but may be treated differently by some operations.

Distinguishing zero from O


The oval-shaped zero and circular letter O together came into use on modern character displays. The zero with a dot in the center seems to lớn have originated as an option on IBM 3270 controllers (this has the problem that it looks lượt thích 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 because of its similarity to lớn the symbol ∅, representing the empty set, as well as 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 & 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 lớn 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 zero rather egg-shaped and the O more circular, but most of all by slitting open the zero on the upper right side, so the circle is not closed any more (as in German plates). The typeface chosen is called fälschungserschwerende Schrift (abbr.: fe Schrift), meaning "unfalsifiable script." note that those used in the United Kingdom bởi vì not differentiate between the two as there can never be any ambiguity if the design is correctly spaced.

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

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The importance of the creation of the zero mark can never be exaggerated. This giving lớn airy nothing, not merely a local habitation & a name, a picture, a symbol, but helpful power, is the characteristic of the Hindu race from whence it sprang. It is like coining the Nirvana into dynamos. No single mathematical creation has been more potent for the general on-go of intelligence và power. G. B. Halsted

…a profound & important idea which appears so simple lớn us now that we ignore its true merit. But its very simplicity & the great ease which it lent to lớn all computations put our arithmetic in the first rank of useful inventions. Pierre-Simon Laplace

The point about zero is that we vì not need khổng lồ use it in the operations of daily life. No one goes out to buy zero fish. It is in a way the most civilized of all the cardinals, & its use is only forced on us by the needs of cultivated modes of thought.Alfred North Whitehead

…a fine và wonderful refuge of the divine spirit—almost an amphibian between being and non-being. Gottfried Leibniz

In other fields


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