A history of Zero

6:14 PM | BY ZeroDivide EDIT
Wikipedia article is at the bottom

One of the commonest questions which the readers of this archive ask is: Who discovered zero? Why then have we not written an article on zero as one of the first in the archive? The reason is basically because of the difficulty of answering the question in a satisfactory form. If someone had come up with the concept of zero which everyone then saw as a brilliant innovation to enter mathematics from that time on, the question would have a satisfactory answer even if we did not know which genius invented it. The historical record, however, shows quite a different path towards the concept. Zero makes shadowy appearances only to vanish again almost as if mathematicians were searching for it yet did not recognise its fundamental significance even when they saw it.
The first thing to say about zero is that there are two uses of zero which are both extremely important but are somewhat different. One use is as an empty place indicator in our place-value number system. Hence in a number like 2106 the zero is used so that the positions of the 2 and 1 are correct. Clearly 216 means something quite different. The second use of zero is as a number itself in the form we use it as 0. There are also different aspects of zero within these two uses, namely the concept, the notation, and the name. (Our name "zero" derives ultimately from the Arabic sifr which also gives us the word "cipher".)
Neither of the above uses has an easily described history. It just did not happen that someone invented the ideas, and then everyone started to use them. Also it is fair to say that the number zero is far from an intuitive concept. Mathematical problems started as 'real' problems rather than abstract problems. Numbers in early historical times were thought of much more concretely than the abstract concepts which are our numbers today. There are giant mental leaps from 5 horses to 5 "things" and then to the abstract idea of "five". If ancient peoples solved a problem about how many horses a farmer needed then the problem was not going to have 0 or -23 as an answer.
One might think that once a place-value number system came into existence then the 0 as an empty place indicator is a necessary idea, yet the Babylonians had a place-value number system without this feature for over 1000 years. Moreover there is absolutely no evidence that the Babylonians felt that there was any problem with the ambiguity which existed. Remarkably, original texts survive from the era of Babylonian mathematics. The Babylonians wrote on tablets of unbaked clay, using cuneiform writing. The symbols were pressed into soft clay tablets with the slanted edge of a stylus and so had a wedge-shaped appearance (and hence the name cuneiform). Many tablets from around 1700 BC survive and we can read the original texts. Of course their notation for numbers was quite different from ours (and not based on 10 but on 60) but to translate into our notation they would not distinguish between 2106 and 216 (the context would have to show which was intended). It was not until around 400 BC that the Babylonians put two wedge symbols into the place where we would put zero to indicate which was meant, 216 or 21 '' 6.
The two wedges were not the only notation used, however, and on a tablet found at Kish, an ancient Mesopotamian city located east of Babylon in what is today south-central Iraq, a different notation is used. This tablet, 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. There is one common feature to this use of different marks to denote an empty position. This is the fact that it never occured at the end of the digits but always between two digits. So although we find 21 '' 6 we never find 216 ''. One has to assume that the older feeling that the context was sufficient to indicate which was intended still applied in these cases.
If this reference to context appears silly then it is worth noting that we still use context to interpret numbers today. If I take a bus to a nearby town and ask what the fare is then I know that the answer "It's three fifty" means three pounds fifty pence. Yet if the same answer is given to the question about the cost of a flight from Edinburgh to New York then I know that three hundred and fifty pounds is what is intended.
We can see from this that the early use of zero to denote an empty place is not really the use of zero as a number at all, merely the use of some type of punctuation mark so that the numbers had the correct interpretation.
Now the ancient Greeks began their contributions to mathematics around the time that zero as an empty place indicator was coming into use in Babylonian mathematics. The Greeks however did not adopt a positional number system. It is worth thinking just how significant this fact is. How could the brilliant mathematical advances of the Greeks not see them adopt a number system with all the advantages that the Babylonian place-value system possessed? The real answer to this question is more subtle than the simple answer that we are about to give, but basically the Greek mathematical achievements were based on geometry. Although Euclid's Elements contains a book on number theory, it is based on geometry. In other words Greek mathematicians did not need to name their numbers since they worked with numbers as lengths of lines. Numbers which required to be named for records were used by merchants, not mathematicians, and hence no clever notation was needed.
Now there were exceptions to what we have just stated. The exceptions were the mathematicians who were involved in recording astronomical data. Here we find the first use of the symbol which we recognise today as the notation for zero, for Greek astronomers began to use the symbol O. There are many theories why this particular notation was used. Some historians favour the explanation that it is omicron, the first letter of the Greek word for nothing namely "ouden". Neugebauer, however, dismisses this explanation since the Greeks already used omicron as a number - it represented 70 (the Greek number system was based on their alphabet). Other explanations offered include the fact that it stands for "obol", a coin of almost no value, and that it arises when counters were used for counting on a sand board. The suggestion here is that when a counter was removed to leave an empty column it left a depression in the sand which looked like O.
Ptolemy in the Almagest written around 130 AD uses the Babylonian sexagesimal system together with the empty place holder O. By this time Ptolemy is using the symbol both between digits and at the end of a number and one might be tempted to believe that at least zero as an empty place holder had firmly arrived. This, however, is far from what happened. Only a few exceptional astronomers used the notation and it would fall out of use several more times before finally establishing itself. The idea of the zero place (certainly not thought of as a number by Ptolemy who still considered it as a sort of punctuation mark) makes its next appearance in Indian mathematics.
The scene now moves to India where it is fair to say the numerals and number system was born which have evolved into the highly sophisticated ones we use today. Of course that is not to say that the Indian system did not owe something to earlier systems and many historians of mathematics believe that the Indian use of zero evolved from its use by Greek astronomers. As well as some historians who seem to want to play down the contribution of the Indians in a most unreasonable way, there are also those who make claims about the Indian invention of zero which seem to go far too far. For example Mukherjee in [6] claims:-
... the mathematical conception of zero ... was also present in the spiritual form from 17 000 years back in India.
What is certain is that by around 650AD the use of zero as a number came into Indian mathematics. The Indians also used a place-value system and zero was used to denote an empty place. In fact there is evidence of an empty place holder in positional numbers from as early as 200AD in India but some historians dismiss these as later forgeries. Let us examine this latter use first since it continues the development described above.
In around 500AD Aryabhata devised a number system which has no zero yet was a positional system. He used the word "kha" for position and it would be used later as the name for zero. There is evidence that a dot had been used in earlier Indian manuscripts to denote an empty place in positional notation. It is interesting that the same documents sometimes also used a dot to denote an unknown where we might use x. Later Indian mathematicians had names for zero in positional numbers yet had no symbol for it. The first record of the Indian use of zero which is dated and agreed by all to be genuine was written in 876.
We have an inscription on a stone tablet which contains a date which translates to 876. The inscription concerns the town of Gwalior, 400 km south of Delhi, where they planted a garden 187 by 270 hastas which would produce enough flowers to allow 50 garlands per day to be given to the local temple. Both of the numbers 270 and 50 are denoted almost as they appear today although the 0 is smaller and slightly raised.
We now come to considering the first appearance of zero as a number. Let us first note that it is not in any sense a natural candidate for a number. From early times numbers are words which refer to collections of objects. Certainly the idea of number became more and more abstract and this abstraction then makes possible the consideration of zero and negative numbers which do not arise as properties of collections of objects. Of course the problem which arises when one tries to consider zero and negatives as numbers is how they interact in regard to the operations of arithmetic, addition, subtraction, multiplication and division. In three important books the Indian mathematicians Brahmagupta, Mahavira and Bhaskara tried to answer these questions.
Brahmagupta attempted to give the rules for arithmetic involving zero and negative numbers in the seventh century. He explained that given a number then if you subtract it from itself you obtain zero. He gave the following rules for addition which involve zero:-
The sum of zero and a negative number is negative, the sum of a positive number and zero is positive, the sum of zero and zero is zero.
Subtraction is a little harder:-
A negative number subtracted from zero is positive, a positive number subtracted from zero is negative, zero subtracted from a negative number is negative, zero subtracted from a positive number is positive, zero subtracted from zero is zero.
Brahmagupta then says that any number when multiplied by zero is zero but struggles when it comes to division:-
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 and the finite quantity as denominator. Zero divided by zero is zero.
Really Brahmagupta is saying very little when he suggests that n divided by zero is n/0. Clearly he is struggling here. He is certainly wrong when he then claims that zero divided by zero is zero. However it is a brilliant attempt from the first person that we know who tried to extend arithmetic to negative numbers and zero.
In 830, around 200 years after Brahmagupta wrote his masterpiece, Mahavira wrote Ganita Sara Samgraha which was designed as an updating of Brahmagupta's book. He correctly states that:-
... a number multiplied by zero is zero, and a number remains the same when zero is subtracted from it.
However his attempts to improve on Brahmagupta's statements on dividing by zero seem to lead him into error. He writes:-
A number remains unchanged when divided by zero.
Since this is clearly incorrect my use of the words "seem to lead him into error" might be seen as confusing. The reason for this phrase is that some commentators on Mahavira have tried to find excuses for his incorrect statement.
Bhaskara wrote over 500 years after Brahmagupta. Despite the passage of time he is still struggling to explain division by zero. He writes:-
A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.
So Bhaskara tried to solve the problem by writing n/0 = ∞. At first sight we might be tempted to believe that Bhaskara has it correct, but of course he does not. If this were true then 0 times ∞ must be equal to every number n, so all numbers are equal. The Indian mathematicians could not bring themselves to the point of admitting that one could not divide by zero. Bhaskara did correctly state other properties of zero, however, such as 02 = 0, and √0 = 0.
Perhaps we should note at this point that there was another civilisation which developed a place-value number system with a zero. This was the Maya people who lived in central America, occupying the area which today is southern Mexico, Guatemala, and northern Belize. This was an old civilisation but flourished particularly between 250 and 900. We know that by 665 they used a place-value number system to base 20 with a symbol for zero. However their use of zero goes back further than this and was in use before they introduced the place-valued number system. This is a remarkable achievement but sadly did not influence other peoples.

You can see a separate article about Mayan mathematics.

The brilliant work of the Indian mathematicians was transmitted to the Islamic and Arabic mathematicians further west. It came at an early stage for al-Khwarizmi wrote Al'Khwarizmi on the Hindu Art of Reckoning which describes the Indian place-value system of numerals based on 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. This work was the first in what is now Iraq to use zero as a place holder in positional base notation. Ibn Ezra, in the 12th century, wrote three treatises on numbers which helped to bring the Indian symbols and ideas of decimal fractions to the attention of some of the learned people in Europe. The Book of the Number describes the decimal system for integers with place values from left to right. In this work ibn Ezra uses zero which he calls galgal (meaning wheel or circle). Slightly later in the 12th century al-Samawal was writing:-
If we subtract a positive number from zero the same negative number remains. ... if we subtract a negative number from zero the same positive number remains.
The Indian ideas spread east to China as well as west to the Islamic countries. In 1247 the Chinese mathematician Ch'in Chiu-Shao wrote Mathematical treatise in nine sections which uses the symbol O for zero. A little later, in 1303, Zhu Shijie wrote Jade mirror of the four elements which again uses the symbol O for zero.
Fibonacci was one of the main people to bring these new ideas about the number system to Europe. As the authors of [12] write:-
An important link between the Hindu-Arabic number system and the European mathematics is the Italian mathematician Fibonacci.
In Liber Abaci he described the nine Indian symbols together with the sign 0 for Europeans in around 1200 but it was not widely used for a long time after that. It is significant that Fibonacci is not bold enough to treat 0 in the same way as the other numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 since he speaks of the "sign" zero while the other symbols he speaks of as numbers. Although clearly bringing the Indian numerals to Europe was of major importance we can see that in his treatment of zero he did not reach the sophistication of the Indians Brahmagupta, Mahavira and Bhaskara nor of the Arabic and Islamic mathematicians such as al-Samawal.
One might have thought that the progress of the number systems in general, and zero in particular, would have been steady from this time on. However, this was far from the case. Cardan solved cubic and quartic equations without using zero. He would have found his work in the 1500's so much easier if he had had a zero but it was not part of his mathematics. By the 1600's zero began to come into widespread use but still only after encountering a lot of resistance.
Of course there are still signs of the problems caused by zero. Recently many people throughout the world celebrated the new millennium on 1 January 2000. Of course they celebrated the passing of only 1999 years since when the calendar was set up no year zero was specified. Although one might forgive the original error, it is a little surprising that most people seemed unable to understand why the third millennium and the 21st century begin on 1 January 2001. Zero is still causing problems!

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0 (zero; BrE: /ˈzɪərəʊ/ or AmE: /ˈziːroʊ/) is both a number[1] and the numerical digit used to represent that number in numerals. It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraicstructures. As a digit, 0 is used as a placeholder in place value systems. In the English language, 0 may be called zero,nought or (US) naught /ˈnɔːt/, nil, or — in contexts where at least one adjacent digit distinguishes it from the letter "O" — ohor o /ˈoʊ/. Informal or slang terms for zero include zilch and zip.[2] Ought or aught /ˈɔːt/ has also been used historically.[3] (SeeNames for the number 0 in English.)

Etymology

Main articles: Names for the number 0 and Names for the number 0 in English
The word zero came via French zéro from Venetian zero, which (together with cypher) came via Italian zefiro from Arabic صفر, ṣafira = "it was empty", ṣifr = "zero", "nothing". The first known English use was in 1598.[4][5][6][7]
In AD 976 the Persian encyclopedist Muhammad ibn Ahmad al-Khwarizmi, in his "Keys of the Sciences", remarked that if, in a calculation, no number appears in the place of tens, then a little circle should be used "to keep the rows". This circle was called صفر (ṣifr, "empty") in Arabic language. That was the earliest mention of the name ṣifr that eventually became zero.[8]
Italian zefiro already meant "west wind" from Latin and Greek zephyrus; this may have influenced the spelling when transcribing Arabic ṣifr.[9] The Italian mathematicianFibonacci (c.1170–1250), who grew up in North Africa and is credited with introducing the decimal system to Europe, used the term zephyrum. This became zefiro in Italian, which was contracted to zero in Venetian.
As the decimal zero and its new mathematics spread from the Arabic world to Europe in the Middle Ages, words derived from ṣifr and zephyrus came to refer to calculation, as well as to privileged knowledge and secret codes. According to Ifrah, "in thirteenth-century Paris, a 'worthless fellow' was called a '... cifre en algorisme', i.e., an 'arithmetical nothing'."[9] From ṣifr also came French chiffre = "digit", "figure", "number", chiffrer = "to calculate or compute", chiffré = "encrypted". Today, the word in Arabic is still ṣifr, and cognates of ṣifr are common in the languages of Europe and southwest Asia.
Modern usage
There are different words used for the number or concept of zero depending on the context. For the simple notion of lacking, the words nothing and none are often used, while nought, naught and aught[10] are archaic and poetic forms with the same meaning. Several sports have specific words for zero, such as nil in football, love in tennisand a duck in cricket. In British English, it is often called oh in the context of telephone numbers. Slang words for zero include zip, zilch, nada, scratch and even duck egg orgoose egg.[11]

History

Egypt

nfr
 		heart with trachea
beautiful, pleasant, good
F35
Ancient Egyptian numerals were base 10. They used hieroglyphs for the digits and were not positional. By 1740 BCE, the Egyptians had a symbol for zero in accounting texts. The symbol nfr, meaning beautiful, was also used to indicate the base level in drawings of tombs and pyramids and distances were measured relative to the base line as being above or below this line.[12]

Mesopotamia

By the middle of the 2nd millennium BC, the Babylonian mathematics 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. In a tablet unearthed at Kish (dating from about 700 BC), the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges.[13]
The Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the end of a number. Thus numbers like 2 and 120 (2×60), 3 and 180 (3×60), 4 and 240 (4×60), looked the same because the larger numbers lacked a final sexagesimal placeholder. Only context could differentiate them.

India

Statue of Aryabhata
The concept of zero as a number and not merely a symbol or an empty space for separation is attributed to India, where, by the 9th century AD, practical calculations were carried out using zero, which was treated like any other number, even in case of division.[14][15] The Indian scholar Pingala (circa 5th–2nd century BC) used binary numbers in the form of short and long syllables (the latter equal in length to two short syllables), making it similar to Morse code.[16][17] He and his contemporary Indian scholars used the Sanskrit word śūnya to refer to zero orvoid.
In 498 AD, Indian mathematician and astronomer Aryabhata stated that "sthānāt sthānaṁ daśaguṇaṁ syāt;"[18] i.e., "from place to place each is ten times the preceding,"[18][19] which is the origin of the modern decimal-based place value notation.[20][21]
The oldest known text to use a decimal place-value system, including a zero, is the Jain text from India entitled the Lokavibhāga, dated 458 AD, where śūnya ("void" or "empty") was employed for this purpose.[22] The first known use of special glyphs for the decimal digits that includes the indubitable appearance of a symbol for the digit zero, a small circle, appears on a stone inscription found at the Chaturbhuja Temple at Gwalior in India, dated 876 AD.[23][24] There are many documents on copper plates, with the same small o in them, dated back as far as the sixth century AD, but their authenticity may be doubted.[13]
Rules of Brahmagupta
The rules governing the use of zero appeared for the first time in Brahmagupta's book Brahmasputha Siddhanta (The Opening of the Universe),[25] written in 628 AD. Here Brahmagupta considers not only zero, but negative numbers, and the algebraic rules for the elementary operations of arithmetic with such numbers. In some instances, his rules differ from the modern standard. Here are the rules of Brahmagupta:[25]
  • The sum of zero and a negative number is negative.
  • The sum of zero and a positive number is positive.
  • The sum of zero and zero is zero.
  • The sum of a positive and a negative is their difference; or, if their absolute values are equal, zero.
  • 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 and 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 do not assign a value to this, whereas computers and calculators sometimes assign NaN, 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.

China

This is a depiction of zero expressed in Chinese counting rods, based on the example provided by A History of Mathematics. An empty space is used to represent zero.[26]
The Sunzi Suanjing, of unknown date but estimated to be dated from the 1st to 5th centuries, and Japanese records dated from the eighteenth century, describe how counting rods were used for calculations. According to A History of Mathematics, the rods "gave the decimal representation of a number, with an empty space denoting zero."[26] The counting rod system is considered a positional notation system.[27]
Zero was not treated as a number at that time, but as a "vacant position", unlike the Indian mathematicians who developed the numerical zero.[28] Ch’in Chu-shao's 1247 Mathematical Treatise in Nine Sections is the oldest surviving Chinese mathematical text using a round symbol for zero.[29] Chinese authors had been familiar with the idea of negative numbers by the Han Dynasty (2nd century CE), as seen in the The Nine Chapters on the Mathematical Art,[30] much earlier than the fifteenth century when they became well established in Europe.[29]

Islamic world

The Arabic-language inheritance of science was largely Greek, a legacy of the rich Hellenistic tradition in Egypt and Syria. [31] In 773, at Al-Mansur's behest, translations were made of many ancient treatises including Greek, Latin, Indian, and others.
The Arabic numerals and the positional number system were introduced around 500 AD, and about 825 AD, it was introduced by a Persian scientist, al-Khwārizmī,[8] in his book on arithmetic, which was translated into Latin in the 12th century under the title Algoritmi de numero Indorum. This title means "Algoritmi on the numbers of the Indians", where "Algoritmi" was the translator's Latinization of Al-Khwarizmi's name.[32] This book synthesized Greek and Hindu knowledge and also contained his own fundamental contribution to mathematics and science including an explanation of the use of zero. It was only centuries later, in the 12th century, that the Arabic numeral system was introduced to the Western world through Latin translations of his treatise Arithmetic.

Greeks and Romans

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 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.
Example of the early Greek symbol for zero (lower right corner) from a 2nd-century papyrus
By 130 AD, 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 the fractional part of a number (called minutes, seconds, thirds, fourths, etc.)—they were not used for the integral part of a number. In later Byzantine manuscripts of Ptolemy'sSyntaxis Mathematica (also known as the Almagest), the Hellenistic zero had morphed into the Greek letter omicron(otherwise meaning 70).
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). The initial "N" was used as a zero symbol in a table of Roman numerals by Bede or his colleague around 725.

Medieval Europe – zero as a decimal digit

See also: History of the Hindu-Arabic numeral system
Positional notation without the use of zero (using an empty space in tabular arrangements, or the word kha "emptiness") is known to have been in use in India from the 6th century. The earliest certain use of zero as a decimal positional digit dates to the 5th century mention in the text Lokavibhaga. The glyph for the zero digit was written in the shape of a dot, and consequently called bindu ("dot"). The dot had been used in Greece during earlier ciphered numeral periods.
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 ofastronomy 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 1202, 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: 9 8 7 6 5 4 3 2 1. With these nine figures, and with the sign 0 ... any number may be written.[33][34]
Here Leonardo of Pisa uses the phrase "sign 0", indicating it is like a sign to do operations like addition or multiplication. From the 13th century, manuals on calculation (adding, multiplying, extracting roots, etc.) became common in Europe where they were called algorismus after the Persian mathematician al-Khwārizmī. The most popular was written by Johannes de Sacrobosco, about 1235 and was one of the earliest scientific books to be printed in 1488. Until the late 15th century, Hindu-Arabic numerals seem to have predominated among mathematicians, while merchants preferred to use the Roman numerals. In the 16th century, they became commonly used in Europe.

The Americas

The back of Olmec stela C fromTres Zapotes, the second oldest Long Count date discovered. The numerals 7.16.6.16.18 translate to September, 32 BC (Julian). The glyphs surrounding the date are thought to be one of the few surviving examples of Epi-Olmec script.
The Mesoamerican Long Count calendar developed in south-central Mexico and Central America required the use of zero as a place-holder within its vigesimal (base-20) positional numeral system. Many different glyphs, including this partial quatrefoil—MAYA-g-num-0-inc-v1.svg—were 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 BC.[35] Since the eight earliest Long Count dates appear outside the Maya homeland,[36] it is assumed that the use of zero in the Americas predated the Maya and was possibly the invention of the Olmecs. Many of the earliest Long Count dates were found within the Olmec heartland, although the Olmec civilization ended by the 4th century BC, several centuries before the earliest known Long Count dates.
Although zero became an integral part of Maya numerals, with a different, empty tortoise-like "shell shape" used for many depictions of the "zero" numeral, it did not influence Old World numeral systems. Quipu, a knotted cord device, used in the Inca Empire and its predecessor societies in the Andean region to record accounting and other digital data, is encoded in a base ten positional system. Zero is represented by the absence of a knot in the appropriate position.

In mathematics

See also: parity of zero
0 is the integer immediately preceding 1. Zero is an even number,[37] because it is divisible by 2. 0 is neither positive nor negative. By most definitions[38] 0 is a natural number, and then the only natural number not to be positive. 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 (quantities) that go lower 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 02. In some instances, a leading zero may be used to distinguish a number.

Elementary algebra

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 a whole number 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 appears in the middle of 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 by multiplying prime numbers (0 must always be one of the factors).[39] Zero is, however, even (see parity of zero).
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.
  • Addition: x + 0 = 0 + x = x. That is, 0 is an identity element (or neutral 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 (no real number multiplied by 0 produces 1), a consequence of the previous rule; see division by zero.
  • Exponentiation: x0 = x/x = 1, except that the case x = 0 may be left undefined in some contexts; see Zero to the zero power. For all positive real x, 0x = 0.
The expression 0⁄0, which may be obtained in an attempt to determine the limit of an expression of the form f(x)⁄g(x) as a result of applying the lim operator independently to both operands of the fraction, is a so-called "indeterminate form". That does not simply mean that the limit sought is necessarily undefined; rather, it means that the limit off(x)⁄g(x), if it exists, must be found by another method, such as l'Hôpital's rule.
The sum of 0 numbers is 0, and the product of 0 numbers is 1. The factorial 0! evaluates to 1.

Other branches of mathematics

Related mathematical terms

  • A zero of a function f is a point x in the domain of the function such that f(x) = 0. When there are finitely many zeros these are called the roots of the function. See alsozero (complex analysis) for zeros of a holomorphic function.
  • The zero function (or zero map) on a domain D is the constant function with 0 as its only possible output value, i.e., the function f defined by f(x) = 0 for all x in D. A particular zero function is a zero morphism in category theory; e.g., a zero map is the identity in the additive group of functions. The determinant on non-invertible square matrices is a zero map.
  • Several branches of mathematics have zero elements, which generalise either the property 0 + x = x, or the property 0 × x = 0, or both.

In science

Physics

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 Kelvin) 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 thefreezing 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.

Chemistry

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 1926, Professor 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.

In computer science

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 1950s 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. For example, the elements of an array are numbered starting from 0 in C, so that for an array of n items the sequence of array indices runs from 0 to n−1. 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 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.
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 timewhen 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).
In mathematics −0 = +0 = 0, both −0 and +0 represent exactly the same number, i.e., there is no "negative zero" distinct from zero. In some signed number representations(but not the two's complement representation used to represent integers in most computers today) and most floating point number representations, zero has two distinct representations, one grouping it with the positive numbers and one with the negatives; this latter representation is known as negative zero.

In other fields

  • In telephony, pressing 0 is often used for dialling out of a company network or to a different city or region, and 00 is used for dialling abroad. In some countries, dialling 0 places a call for operator assistance.
  • DVDs that can be played in any region are sometimes referred to as being "region 0"
  • Roulette wheels usually feature a "0" space (and sometimes also a "00" space), whose presence is ignored when calculating payoffs (thereby allowing the house to win in the long run).
  • In Formula One, if the reigning World Champion no longer competes in Formula One in the year following their victory in the title race, 0 is given to one of the drivers of the team that the reigning champion won the title with. This happened in 1993 and 1994, with Damon Hill driving car 0, due to the reigning World Champion (Nigel Mansell and Alain Prost respectively) not competing in the championship.

Symbols and representations of zero

Main article: Symbols for zero
Text figures 036.svg
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.[40] Typewriters originally made no distinction in shape between O and 0; some models did not even have a separate key for the digit 0. The distinction came into prominence on modern character displays.[40]
A slashed zero can be used to distinguish the number from the letter. The digit 0 with a dot in the center seems to have originated as an option on IBM 3270 displays and has continued with some modern computer typefaces such as Andalé Mono, and in some airline reservation systems. 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.

Zero as a year label

Main article: 0 (year)
In the BC calendar era, the year 1 BC is the first year before AD 1; no room is reserved for a year zero. By contrast, in astronomical year numbering, the year 1 BC is numbered 0, the year 2 BC is numbered −1, and so on.[41]

See also

Notes

  1. Jump up^ Russell, Bertrand (1942). Principles of mathematics (2 ed.). Forgotten Books. p. 125. ISBN 1-4400-5416-9., Chapter 14, page 125
  2. Jump up^ Soanes, Catherine; Waite, Maurice; Hawker, Sara, eds. (2001). The Oxford Dictionary, Thesaurus and Wordpower Guide (Hardback) (2nd ed.). New York: Oxford University Press. ISBN 978-0-19-860373-3.
  3. Jump up^ aught at etymonline.com
  4. Jump up^ Menninger, Karl (1992). Number words and number symbols: a cultural history of numbers. Courier Dover Publications. p. 401. ISBN 0-486-27096-3.
  5. Jump up^ ""zero, n.". OED Online. December 2011. Oxford University Press. (accessed March 04, 2012).". Archived from the original on 6 March 2012. Retrieved 2012-03-04.
  6. Jump up^ "cipher | cypher, n.". OED Online. December 2011. Oxford University Press. (accessed 4 March 2012).
  7. Jump up^ Merriam Webster online Dictionary
  8. ^ Jump up to:a b Will Durant, The Story of Civilization, Volume 4, The Age of Faith, p. 241
  9. ^ Jump up to:a b Ifrah, Georges (2000). The Universal History of Numbers: From Prehistory to the Invention of the Computer. Wiley. ISBN 0-471-39340-1.
  10. Jump up^ 'Aught' definition, Dictionary.com – Reterieved April 2013.
  11. Jump up^ 'Aught' synonyms, Thesaurus.com – Reterieved April 2013.
  12. Jump up^ George Gheverghese Joseph (2011). The Crest of the Peacock: Non-European Roots of Mathematics (Third Edition). Princeton. p. 86. ISBN 978-0-691-13526-7.
  13. ^ Jump up to:a b Kaplan, Robert. (2000). The Nothing That Is: A Natural History of Zero. Oxford: Oxford University Press.
  14. Jump up^ Bourbaki, Nicolas (1998). Elements of the History of Mathematics. Berlin, Heidelberg, and New York: Springer-Verlag. 46. ISBN 3-540-64767-8.
  15. Jump up^ Britannica Concise Encyclopedia (2007), entry algebra
  16. Jump up^ Binary Numbers in Ancient India
  17. Jump up^ Math for Poets and Drummers (pdf, 145KB)
  18. ^ Jump up to:a b Aryabhatiya of Aryabhata, translated by Walter Eugene Clark.
  19. Jump up^ Agarwal, M.K. (23 May 2012). From Bharata to India: Chrysee the Golden. iUniverse. p. 206. ISBN 9781475907650.
  20. Jump up^ O'Connor, Robertson, J.J., E. F. "Aryabhata the Elder". School of Mathematics and Statistics University of St Andrews, Scotland. Retrieved 26 May 2013.
  21. Jump up^ The Britannica Guide to Numbers and Measurement (Math Explained). The Rosen Publishing Group. 15 August 2010. pp. 97–98. ISBN 9781615301089.
  22. Jump up^ Ifrah, Georges (2000), p. 416.
  23. Jump up^ Bill Casselman (University of British Columbia), American Mathematical Society, "All for Nought"
  24. Jump up^ Ifrah, Georges (2000), p. 400.
  25. ^ Jump up to:a b Algebra with Arithmetic of Brahmagupta and Bhaskara, translated to English by Henry Thomas Colebrooke, London1817
  26. ^ Jump up to:a b Luke Hodgkin (2 June 2005). A History of Mathematics : From Mesopotamia to Modernity: From Mesopotamia to Modernity. Oxford University Press. p. 85.ISBN 978-0-19-152383-0.
  27. Jump up^ Crossley, Lun. 1999, p.12 "the ancient Chinese system is a place notation system"
  28. Jump up^ Kang-Shen Shen; Hui Liu; Anthony W. C. Lun (1999). The Nine Chapters on the Mathematical Art: Companion and Commentary. Oxford University Press. p. 35.ISBN 978-0-19-853936-0. "zero was regarded as a number in India... whereas the Chinese employed a vacant position"
  29. ^ Jump up to:a b Mathematics in the Near and Far East, p. 262
  30. Jump up^ Struik, Dirk J. (1987). A Concise History of Mathematics. New York: Dover Publications. pp. 32–33. "In these matrices we find negative numbers, which appear here for the first time in history."
  31. Jump up^ Pannekoek, A. (1961). A History of Astronomy. George Allen & Unwin. p. 165.
  32. Jump up^ Brezina, Corona (2006). Al-Khwarizmi: The Inventor Of Algebra. The Rosen Publishing Group. ISBN 978-1-4042-0513-0.
  33. Jump up^ Sigler, L., Fibonacci's Liber Abaci. English translation, Springer, 2003.
  34. Jump up^ Grimm, R.E., "The Autobiography of Leonardo Pisano", Fibonacci Quarterly 11/1 (February 1973), pp. 99–104.
  35. Jump up^ No long count date actually using the number 0 has been found before the 3rd century AD, but since the long count system would make no sense without some placeholder, and since Mesoamerican glyphs do not typically leave empty spaces, these earlier dates are taken as indirect evidence that the concept of 0 already existed at the time.
  36. Jump up^ Diehl, p. 186
  37. Jump up^ Lemma B.2.2, The integer 0 is even and is not odd, in Penner, Robert C. (1999).Discrete Mathematics: Proof Techniques and Mathematical Structures. World Scientific. p. 34. ISBN 981-02-4088-0.
  38. Jump up^ Bunt, Lucas Nicolaas Hendrik; Jones, Phillip S.; Bedient, Jack D. (1976). The historical roots of elementary mathematics. Courier Dover Publications. pp. 254–255.ISBN 0-486-13968-9., Extract of pages 254–255
  39. Jump up^ Reid, Constance (1992). From zero to infinity: what makes numbers interesting(4th ed.). Mathematical Association of America. p. 23. ISBN 978-0-88385-505-8.
  40. ^ Jump up to:a b Bemer, R. W. (1967). "Towards standards for handwritten zero and oh: much ado about nothing (and a letter), or a partial dossier on distinguishing between handwritten zero and oh". Communications of the ACM 10 (8): 513–518.doi:10.1145/363534.363563.
  41. Jump up^ Steel, Duncan (2000). Marking time: the epic quest to invent the perfect calendar. John Wiley & Sons. p. 113. ISBN 0-471-29827-1. "In the B.C./A.D. scheme there is no year zero. After 31 December 1 BC came AD 1 January 1. ... If you object to that no-year-zero scheme, then don't use it: use the astronomer's counting scheme, with negative year numbers."

References

This article is based on material taken from the Free On-line Dictionary of Computing prior to 1 November 2008 and incorporated under the "relicensing" terms of the GFDL, version 1.3 or later.
  • Barrow, John D. (2001) The Book of Nothing, Vintage. ISBN 0-09-928845-1.
  • Diehl, Richard A. (2004) The Olmecs: America's First Civilization, Thames & Hudson, London.
  • Ifrah, Georges (2000) The Universal History of Numbers: From Prehistory to the Invention of the Computer, Wiley. ISBN 0-471-39340-1.
  • Kaplan, Robert (2000) The Nothing That Is: A Natural History of Zero, Oxford: Oxford University Press.
  • Seife, Charles (2000) Zero: The Biography of a Dangerous Idea, Penguin USA (Paper). ISBN 0-14-029647-6.
  • Bourbaki, Nicolas (1998). Elements of the History of Mathematics. Berlin, Heidelberg, and New York: Springer-Verlag. ISBN 3-540-64767-8.
  • Isaac Asimov (1978). Article "Nothing Counts" in Asimov on Numbers. Pocket Books.

External links

Wikimedia Commons has media related to 0 (number).
Look up zero in Wiktionary, the free dictionary.
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