0 (Definition and History)

• Definition 0 (zero) is both a number and a numerical digit used to 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, and many other algebraic structures. As a digit, zero is used as a placeholder in place value systems. Historically, it was the last digit to come into use. In the English language, zero may also be called nil when a number, o/oh when a numeral, and nought/naught in either context.

0 as a number

0 is the integer that comes before and after the positive number 1. Before the concept of ‘negative integers’ was recognized, 0 was recognized in most (if not all) number systems.

Zero is an integer that quantifies a count or a quantity of null size; for example, if your brother count is zero, it indicates you have no brothers, and if something has a weight of zero, it has no weight. If the difference in the number of pieces between two heaps is zero, the two piles have the same number of pieces. The result may be considered to be zero before counting begins; that is, the number of things counted before counting the first item, and counting the first item brings the result to one. If there are no items to count, the final result is zero.

While all mathematicians recognize zero as a number, some non-mathematicians argue that zero is not a number since you can’t have zero of anything. Others believe that a bank balance of zero indicates that you have a precise amount of money in your account, namely none. Mathematicians and the majority of others agree with the latter viewpoint.

The year zero is omitted by almost all historians from the proleptic Gregorian and Julian calendars, yet astronomers include it in these same calendars. However, the term “Year Zero” may refer to any event regarded so momentous that it effectively begins a new time reckoning.

0 as a numeral

Normally, the contemporary number 0 is written as a circle or (rounded) rectangle. 0 is generally the same height as a lowercase x in old-style typefaces with text figures.

0 is normally written with six line segments on seven-segment displays of calculators, clocks, and so on, while on certain vintage calculator models it was written with four line segments. This alternative glyph has not gained popularity.

It is critical to differentiate the number zero (as in the “zero brothers” example above) from the numeral or digit zero, which is employed in positional notation numeral systems. Because successive digit locations have larger values, the digit zero is used to skip a position and assign suitable values to the previous and subsequent digits. In a positional number system, a zero digit is not necessarily required: bijective numeration is one such counterexample.

Etymology

The term zero is derived from the Arabic literal translation of the Sanskrit nya (), which means void or empty, into ifr (), which means empty or unoccupied. In Latin, this became zephyr or zephyrus by transcription. In Latin, the term zephyrus already meant “west wind,” and the proper noun Zephyrus was the Roman deity of the wind (after the Greek god Zephyros). Zephyr came to indicate a faint breeze—”almost nothing” with its new meaning for the idea of zero. In Italian, this became zefiro, which was shortened to zero in Venetian, giving rise to the contemporary English term.

Words derived from sifr and zephyrus began to allude to computation, as well as privileged information and secret codes, when the Hindu decimal zero and its new mathematics moved from the Arab world to Europe in the Middle Ages. Ifrah writes that “in thirteenth-century Paris, a ‘worthless guy’ was dubbed a… chiffre en algorisme, i.e., a ‘arithmetical nothing.'” The Arabic root gave birth to the contemporary French words chiffre, which means digit, figure, or number; chiffrer, which means to calculate or compute; and chiffré, which means encrypted, as well as the English term cipher. Here are a few more examples:

• Arabic: Sifr
• Czech/Slovak: cifra, digit; šifra, cypher
• Danish: ciffer, digit
• Dutch: cijfer, digit
• French: zéro, zero
• German: Ziffer, digit, figure, numeral, cypher
• Hindi: shunya
• Hungarian: nulla
• Italian: cifra, digit, numeral, cypher; zero, zero
• Norwegian: siffer, digit, numeral, cypher; null, zero
• Persian: Sefr
• Polish: cyfra, digit; szyfrować, to encrypt; zero, zero
• Portuguese: cifra, figure, numeral, cypher, code; zero, zero
• Russian: цифра (tsifra), digit, numeral; шифр (shifr) cypher, code
• Slovenian: cifra, digit
• Spanish: cifra, figure, numeral, cypher, code; cero, zero
• Swedish: siffra, numeral, sum, digit; chiffer, cypher
• Serbian: цифра (tsifra), digit, numeral; шифра (shifra) cypher, code; нула (nula), zero
• Turkish: Sıfır
• Urdu: Sifer, Anda, Zero

It is worth noting that the Greek word for zero is v (Mèden).

History

Fun Fact: The final numerical digit to be used as 0 (zero).

Zero’s early history

The Babylonians possessed a sophisticated sexagesimal (base-60) positional numeric system by the mid-second millennium B.C.E. A gap between sexagesimal numerals signified the absence of a positional value (or zero). By 300 B.C.E., the same Babylonian system had adopted a punctuation mark (two slanted wedges) as a placeholder. Bêl-bân-aplu, a scribe from Kish, wrote his zeroes with three hooks rather than two slanted wedges on a tablet discovered there (dating maybe as far back as 700 B.C.E.).

Because it was not used alone, the Babylonian placeholder was not a real zero. It was also never used at the conclusion of a number. Because the bigger numbers lacked a final sexagesimal placeholder, figures like 2 and 120 (260), 3 and 180 (360), 4 and 240 (460), and so on seemed the same. Only context could tell them apart.

According to historical records, the ancient Greeks were unclear about the status of zero as a number: they questioned themselves, “How can nothing be something?” This sparked significant philosophical and, by the Medieval era, theological debates regarding the nature and existence of zero and the vacuum. Zeno of Elea’s paradoxes are heavily reliant on the unclear interpretation of zero. (The ancient Greeks even debated whether 1 was a number.)

The use of something like zero by the Indian scholar Pingala (circa 5th-2nd century B.C.E.) is only the modern binary representation using 0 and 1 applied to Pingala’s binary system, which used short and long syllables (the latter equal in length to two short syllables), making it similar to Morse code.

Nonetheless, he and other Indian intellectuals at the time used the Sanskrit term nya to refer to zero or empty (the origin of the word zero after a succession of transliterations and a literal translation).

History of zero

Within its vigesimal (base-20) positional numeric system, the Long Count calendar devised in south-central Mexico necessitated the use of zero as a place-holder. A shell glyph—MAYA-g-num-0-inc-v1.svg—was employed as a zero sign for these Long Count dates, the oldest of which had a date of 36 B.C.E. (on Stela 2 at Chiapa de Corzo, Chiapas). Because the first eight Long Count dates exist outside of the Maya homeland, it is considered that the use of zero in the Americas before the Maya and was perhaps invented by the Olmecs. Indeed, many of the oldest Long Count dates were discovered inside the Olmec heartland, yet the fact that the Olmec civilization had ended by the fourth century B.C.E., many centuries before the first known Long Count dates, suggests that the zero was not invented by the Olmecs.

Despite the fact that zero formed a fundamental element of Maya numbers, it had little impact on Old World numeric systems.

Ptolemy, inspired by Hipparchus and the Babylonians, was utilizing a sign for zero (a tiny circle with a long overbar) inside a sexagesimal numeric system that otherwise used alphabetic Greek numerals by 130 C.E. This Hellenistic zero was perhaps the earliest known usage of a number zero in the Old World since it was used alone, rather than as a placeholder. The positions, however, were normally confined to the fractional portion of a number (named minutes, seconds, thirds, fourths, and so on) and were not utilized for an integral part of a number.

By 525 (first recorded usage by Dionysius Exiguus), another zero was used in tables alongside Roman numerals, but as a word, nulla, meaning nothing, not as a sign. When division resulted in zero as a residue, the term nihil, which also means “nothing,” was employed. All subsequent medieval computists employed these medieval zeros (calculators of Easter). An isolated usage of their initial, N, was employed as a zero sign in a table of Roman numerals by Bede or a colleague about 725.

The earliest known manuscript to employ zero is the Lokavibhaaga, a Jain scripture from India dating from 458 C.E.

The earliest indubitable occurrence of a zero sign is in India in 876 on a stone inscription in Gwalior. There are documents on copper plates with the same little o in them dating back to the sixth century C.E.

Rules of Brahmagupta

The principles controlling the usage of zero were originally mentioned in Brahmagupta’s work Brahmasputha Siddhanta, which was published in 628. (598-670). Brahmagupta covers not just zero but also negative numbers, as well as the algebraic rules for the basic arithmetic operations with such numbers. In several cases, his guidelines deviate from the present practice. The following are Brahamagupta’s rules:

• The product of two positive numbers is positive.
• The product of two negative numbers is negative.
• The product of zero and a negative number is a negative number.
• A positive number plus zero is a positive number.
• The product of zero and zero equals zero.
• The difference between a positive and a negative is the total of their differences; if they are equal, the sum is zero.
• In subtraction, the less is subtracted from the larger, and the positive is subtracted from the positive.
• In subtraction, the less is subtracted from the larger, and the negative is subtracted from the negative.
• However, when the bigger is removed from the less, the difference is reversed.
• When subtracting positive from negative and subtracting negative from positive, they must be summed together.
• The sum of a negative and a positive amount is negative.
• A negative amount multiplied by another negative quantity yields a positive result.
• Positive is the result of two positives.
• Positive divided by positive, or negative divided by negative, equals positive.
• Positive minus negative equals negative. Negative is equal to negative when divided by positive.
• When a positive or negative number is divided by zero, the result is a fraction with zero as the denominator.
• When zero is divided by a negative or positive number, the result is either zero or a fraction with zero as the numerator and the finite amount as the denominator.
• Zero multiplied by zero equals zero.

Brahmagupta deviates from current thinking when he says “zero divided by zero equals zero.” Mathematicians seldom assign a value, although computers and calculators may sometimes offer NaN, which stands for “not a number.” Furthermore, when non-zero positive or negative integers are divided by zero, they are given a value of unsigned infinity, positive infinity, or negative infinity. Once again, these assignments are not numbers and are connected with computer science rather than pure mathematics, where no assignment is made in most instances. (See also a division by zero.)

As a decimal numeral, zero

Positional notation without the use of zero (using an empty space in tabular layouts or the term kha “emptiness”) has been documented in India from the sixth century. The usage of zero as a decimal positional numeral goes back to the ninth century. Bindu means “dot” because the glyph for the zero digit was written in the form of a dot.

The Hindu-Arabic number system arrived in Europe in the eleventh century, through the Iberian Peninsula via Spanish Muslims known as Moors, along with knowledge of astronomy and devices such as the astrolabe, which was originally introduced by Gerbert of Aurillac (c. 940-1003). They were dubbed “Arabic numerals.” Leonardo of Pisa (c. 1170-1250), commonly known as Fibonacci,* an Italian mathematician, was essential in introducing the method into European mathematics around 1202. Leonardo says here:

Following my introduction to the nine digits of the Hindus as a result of marvelous instruction in the art, 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… But all of this, as well as algorism and Pythagoras’ skill, I judged to be virtually a mistake in relation to the Hindu way. (Individual Modus)… The nine Indian digits are as follows: 9 8 7 6 5 4 3 2 1 Any number may be written using these nine numbers plus the symbol 0…

Leonardo of Pisa employs the word sign “0” here to indicate that it is used as a symbol to do operations such as addition and multiplication, but he did not recognize zero as a number in its own right.

In mathematics

Elementary algebra

The lowest non-negative integer is ero (0). The natural number that comes after zero is one, and there is no natural number that comes before zero. Depending on how natural numbers are defined, zero may or may not be considered a natural number.

In set theory, zero is the cardinality of the empty set: if there are no apples, there are zero apples. As a result, in certain circumstances, zero is defined as the empty set.

Zero is neither positive nor negative, a prime number or a composite number, and it is not a unit.

The following are some general guidelines for working with the number zero. Unless otherwise specified, these laws apply to any complex number x.

• x + 0 = x + x = x + x = x (In other words, 0 is an identity element in terms of addition.)
• Subtraction: x 0 = x and x 0 = x.
• x 0 = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0
• For nonzero x, division equals 0 / x = 0. However, x / 0 is undefined since 0 has no multiplicative inverse as a result of the preceding rule. When y in x / y approaches zero from positive values, the quotient approaches positive infinity; when y approaches zero from negative values, the quotient approaches negative infinity. The various quotients demonstrate that division by zero is indeterminate.
• Exponentiation: x0 = 1, except in rare cases when x = 0 is left undefined. 0x Equals 0 for any positive real x.
• The sum of 0 numbers equals zero, while the product of 0 numbers equals one.

The phrase “0/0” is referred to as an “indeterminate form.” This does not simply mean that it is undefined; rather, if f(x) and g(x) both approach 0 as x approaches some integer, then f(x)/g(x) might approach any finite number or; it depends on which functions f and g are. See the Rule of L’Hopital.

In mathematics, the number zero is used extensively

• The identity element in an additive group, or the additive identity of a ring, is zero.
• A function zero is a point in the function’s domain whose image under the function is zero. When there are a limited number of zeros, they are referred to as the function’s roots. Observe 0 (complex analysis).
• A point’s dimension in geometry is 0.
• In probability, the idea of “nearly” impossible. More broadly, the idea of appears nearly nowhere in measurement theory. For example, if one selects a point at random on a unit line interval [0,1], it is not impossible to choose 0.5 precisely, but the likelihood is zero.
• A zero function (or zero map) is a constant function with the single conceivable output value of 0; that is, f(x) = 0 for every x specified. In category theory, a zero morphism is a specific zero function; for example, a zero map is the identity in the additive group of functions. On non-invertible square matrices, the determinant is a zero map.
• The Möbius function has three potential return values, one of which is zero. The Möbius function returns zero when given an integer of the type x2 or x2y (for x > 1).
• The first Perrin number is zero.

In science

Physics

For a significant number of physical quantities, the value zero has a specific significance. For certain values, the zero level is inherently distinct from all other levels, but for others, it is selected more or less randomly. On the Kelvin temperature scale, for example, zero is the lowest feasible temperature (negative temperatures exist but are not truly colder), but on the Celsius temperature system, 0 is arbitrarily set to be at the freezing point of water. When measuring sound intensity in decibels or phons, the zero level is arbitrarily fixed at a reference value, such as the hearing threshold.

Chemistry

The theorized element tetraneutronium’s atomic number has been postulated as zero. A cluster of four neutrons has been found to be stable enough to be regarded an atom in its own right. This would result in a nucleus with no protons and no charge.

Professor Andreas von Antropoff developed the word neutronium in 1926 for a hypothetical form of matter composed of neutrons but no protons, which he put at the top of his new periodic table as the chemical element with atomic number zero. It was later put as a noble gas at the center of numerous spiral renderings of the periodic system for classifying chemical elements. It is located in the heart of the Chemical Galaxy (2005).

In computer science

Throughout human history, the most prevalent practice has been to begin numbering at one. Nonetheless, in computer science, zero has grown to be considered the conventional starting point. An array, for example, begins at 1 by default in practically all classic programming languages. As computer languages have evolved, it has become increasingly typical for an array to begin at zero by default, with item 0 being the “first” item in the array. This strategy became popular in the 1980s due to the prominence of the computer language “C.”

One reason for this approach is because in order to accommodate the additive identity, modular arithmetic generally characterizes a collection of N integers as including 0,1,2,… N-1. As a result, unless the array begins at zero, many arithmetic notions (such as hash tables) are less elegant to describe in code.

Counting from zero enhances the performance of different algorithms, such as finding or sorting arrays, in some circumstances. Improved efficiency implies that the algorithm uses less time, fewer resources, or both to execute a given job.

This condition may cause some terminology misunderstanding. The first element in a zero-based indexing system is “element number zero,” and the twelfth element is “element number eleven.” As a result, a connection between ordinal numbers and the number of items counted develops; the highest index of n objects is (n-1) and is referred to as the n:th element. As a result, to avoid any confusion, the initial element is often referred to as the zeroth element.

Null value

A field in a database may have a null value. This is equal to the field being empty. It is not the value zero for numeric fields. This is neither blank nor the empty string for text fields. The existence of null values causes three-valued reasoning to emerge. A condition may no longer be either true or false; instead, it might be uncertain. Any calculation that includes a null value yields a null outcome. Requesting all records with value 0 or value not equal to 0 will not return all records since records with value null are omitted.

Null pointer

A null pointer is a pointer in a computer program that does not refer to any object or function, therefore when it occurs in a program or code, it instructs the computer to perform no action on the related piece of the code.

Negative zero

In certain signed number representations (but not the two’s complement form used today) and most floating-point number representations, zero has two separate representations, one with positive values and one with negatives; the latter is known as negative zero. Negative zero representations may be tricky since the two zeros will compare equally but may be regarded differently by various operations.

Distinguishing zero from O

On current character displays, the oval-shaped zero and circular letter O were used simultaneously. The zero with a dot in the middle seems to have started as a feature on IBM 3270 controllers (this has the problem that it looks like the Greek letter Theta). The slashed zero, which looks similar to the letter O except for the slash, is used in old-style ASCII graphic sets developed from the ASR-33 Teletype’s default typewheel. This format poses issues due to its resemblance to the sign, which represents the empty set, as well as for some Scandinavian languages that use as a letter.

The convention of the letter O with a slash and the zero without was employed by IBM and a few other early mainframe manufacturers; this is especially troublesome for Scandinavians since it implies two of their letters will clash. A zero with a reversed slash is shown on certain Burroughs/Unisys equipment. Another early line printer habit was to leave zero unadorned but add a tail or hook to the letter-O so that it resembled an inverted Q or cursive capital letter-O.

The font used on certain European automobile license plates distinguishes the two symbols by making the zero more egg-shaped and the O more circular, but most importantly by slicing open the zero on the top right side, so the circle is no longer closed (as in German plates). The font used is called fälschungserschwerende Schrift (abbr.: FE Schrift), which translates as “unfalsifiable script.” It is worth noting that those used in the United Kingdom do not distinguish between the two since there can never be any misunderstanding if the design is properly spaced.

In paper writing, one may not separate the 0 and O at all, or one may put a slash across it to highlight the difference, albeit this sometimes creates uncertainty in reference to the sign for the null set.

Quotes

The significance of the invention of the zero mark cannot be overstated. This giving to airy nothing, not only a local residence and a name, but also an image, a symbol, and helping power, is a feature of the Hindu race from which it sprung. It’s like turning Nirvana into dynamos. No one mathematical innovation has been more influential in the evolution of intellect and power. Halsted, G. B.

…a deep and vital concept that looks so easy to us today that we overlook its ultimate value. However, its simplicity and the ease with which it made all calculations placed our arithmetic at the top of the list of valuable innovations. Laplace, Pierre-Simon

The idea of zero is that we don’t need to utilize it in our everyday lives. Nobody goes fishing with the intention of catching nothing. It is, in some ways, the most civilized of the cardinals, and its usage is only compelled by the demands of developed habits of mind. Whitehead, Alfred North

a magnificent and wonderful holy spirit’s refuge—almost an amphibian between being and non-being. Gottfried Wilhelm Leibniz

In other areas

• Dialing 0 on a phone in certain countries initiates a call for operator help.
• The number 0 has the same dot structure as the letter J in Braille.
• DVDs that may be played in any region are referred to as “region 0” at times.
• In classical music, the number 0 is relatively seldom employed as a compositional number; the only two instances on the periphery of the conventional repertory are possibly Anton Bruckner’s Symphony No. 0 in D minor and Alfred Schnittke’s Symphony No. 0 in E minor.
• The Fool is represented by card number 0 in the tarot.

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