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0 (zero) is both a number[1] and the numerical digit used to represent that number in numerals. The number 0 fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems. Names for the number 0 in English include zero, nought (UK), naught (US) (/nɔːt/), nil, or—in contexts where at least one adjacent digit distinguishes it from the letter "O"—oh or o (//). Informal or slang terms for zero include zilch and zip.[2]Ought and aught (/ɔːt/),[3] as well as cipher,[4] have also been used historically.[5]

← −1 0 1 →
-1 0 1 2 3 4 5 6 7 8 9
Cardinal0, zero, "oh" (//), nought, naught, nil
OrdinalZeroth, noughth
DivisorsAll numbers
Base 36036
Arabic & Kurdish٠
Hindu Numerals
Chinese零, 〇
Japanese零, 〇



The word zero came into the English language via French zéro from Italian zero, Italian contraction of Venetian zevero form of Italian zefiro via ṣafira or ṣifr.[6] In pre-Islamic time the word ṣifr (Arabic صفر) had the meaning "empty".[7] Sifr evolved to mean zero when it was used to translate śūnya (Sanskrit: शून्य) from India.[7] The first known English use of zero was in 1598.[8]


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. Sometimes the words nought, naught and aught[10] are used. Several sports have specific words for zero, such as nil in association football (soccer), love in tennis, and duck in cricket. It is often called oh in the context of telephone numbers. Slang words for zero include zip, zilch, nada, and scratch. Duck egg and goose egg are also slang for zero.[11]


Ancient Near East

heart with trachea
beautiful, pleasant, good

Ancient Egyptian numerals were base 10. They used hieroglyphs for the digits and were not positional. By 1770 BC, 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]



Pre-Columbian Americas


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 —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.[a]

Since the eight earliest Long Count dates appear outside the Maya homeland,[14] it is generally believed that the use of zero in the Americas predated the Maya and was possibly the invention of the Olmecs.[15] 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.



Classical antiquity

The ancient Greeks had no symbol for zero (μηδέν), and did not use a digit placeholder for it.[16] They 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.


By 130 AD, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for zero (a small circle with a long overbar) in his work on mathematical astronomy called the Syntaxis Mathematica, also known as the Almagest. The way in which it is used can be seen in his table of chords in that book. Ptolemy's zero was used 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 earliest documented use of a numeral representing zero in the Old World.[17] 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, indicating a concept perhaps better expressed as "none", rather than "zero" in the modern sense. In later Byzantine manuscripts of Ptolemy's Almagest, the Hellenistic zero had morphed into the Greek letter omicron (otherwise meaning 70).




The Sūnzĭ Suànjīng, of unknown date but estimated to be dated from the 1st to 5th centuries AD, and Japanese records dated from the 18th century, describe how the c. 4th century BC Chinese counting rods system enables one to perform decimal calculations. According to A History of Mathematics, the rods "gave the decimal representation of a number, with an empty space denoting zero."[19] The counting rod system is considered a positional notation system.[20]

In AD 690, Empress Wǔ promulgated Zetian characters, one of which was "〇". The symbol 0 for denoting zero is a variation of this character.

Zero was not treated as a number at that time, but as a "vacant position".[21] Qín Jiǔsháo's 1247 Mathematical Treatise in Nine Sections is the oldest surviving Chinese mathematical text using a round symbol for zero.[22] Chinese authors had been familiar with the idea of negative numbers by the Han Dynasty (2nd century AD), as seen in The Nine Chapters on the Mathematical Art.[23]


Pingala (c. 3rd/2nd century BC[24]), a Sanskrit prosody scholar,[25] used binary numbers in the form of short and long syllables (the latter equal in length to two short syllables), a notation similar to Morse code.[26] Pingala used the Sanskrit word śūnya explicitly to refer to zero.[27]

It was considered that the earliest text to use a decimal place-value system, including a zero, is the Lokavibhāga, a Jain text on cosmology surviving in a medieval Sanskrit translation of the Prakrit original, which is internally dated to AD 458 (Saka era 380). In this text, śūnya ("void, empty") is also used to refer to zero.[28]

A symbol for zero, a large dot likely to be the precursor of the still-current hollow symbol, is used throughout the Bakhshali manuscript, a practical manual on arithmetic for merchants.[29] In 2017 three samples from the manuscript were shown by radiocarbon dating to come from three different centuries: from 224-383 AD, 680-779 AD, and 885-993 AD, making it the world's oldest recorded use of the zero symbol. It is not known how the birch bark fragments from different centuries that form the manuscript came to be packaged together.[30][31][32]

The origin of the modern decimal-based place value notation can be traced to the Aryabhatiya (c. 500), which states sthānāt sthānaṁ daśaguṇaṁ syāt "from place to place each is ten times the preceding."[33][33][34][35] The concept of zero as a digit in the decimal place value notation was developed in India, presumably as early as during the Gupta period (c. 5th century), with the oldest unambiguous evidence dating to the 7th century.[36]

The rules governing the use of zero appeared for the first time in Brahmagupta's Brahmasputha Siddhanta (7th century). This work considers not only zero, but also 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, specifically the definition of the value of zero divided by zero as zero.[37]




A stone tablet found in the ruins of a temple near Sambor on the Mekong, Kratié Province, Cambodia, includes the inscription of "605" in Khmer numerals (a set of numeral glyphs for the Hindu–Arabic numeral system). The number is the year of the inscription in the Saka era, corresponding to a date of AD 683.[38]

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.[39][40] Zero is also used as a placeholder in the Bakhshali manuscript, portions of which date from AD 224–383.[41]

Middle Ages

Transmission to Islamic culture

The Arabic-language inheritance of science was largely Greek,[42] followed by Hindu influences.[43] In 773, at Al-Mansur's behest, translations were made of many ancient treatises including Greek, Roman, Indian, and others.

In AD 813, astronomical tables were prepared by a Persian mathematician, Muḥammad ibn Mūsā al-Khwārizmī, using Hindu numerals;[43] and about 825, he published a book synthesizing Greek and Hindu knowledge and also contained his own contribution to mathematics including an explanation of the use of zero.[44] This book was later translated into Latin in the 12th century under the title Algoritmi de numero Indorum. This title means "al-Khwarizmi on the Numerals of the Indians". The word "Algoritmi" was the translator's Latinization of Al-Khwarizmi's name, and the word "Algorithm" or "Algorism" started meaning any arithmetic based on decimals.[43]


Transmission to Europe





0 is the integer immediately preceding 1. Zero is an even number[48] because it is divisible by 2 with no remainder. 0 is neither positive nor negative.[49] Many definitions[50] include 0 as 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, or quantities less than zero, was accepted.


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 an integer, and hence a rational number and a real number (as well as an algebraic number and a complex number).

The number 0 is neither positive nor negative and is usually displayed as the central number in a number line. It is neither a prime number nor a composite number. It cannot be prime because it has an infinite number of factors, and cannot be composite because it cannot be expressed as a product of prime numbers (0 must always be one of the factors).[51] Zero is, however, even (i.e. a multiple of 2, as well as being a multiple of any other integer, rational, or real number).


  • 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.

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 of f(x)/g(x), if it exists, must be found by another method, such as l'Hôpital's rule.

The sum of 0 numbers (the empty sum) is 0, and the product of 0 numbers (the empty product) is 1. The factorial 0! evaluates to 1, as a special case of the empty product.

Other branches of mathematics

Related mathematical terms

  • Several branches of mathematics have zero elements, which generalize either the property 0 + x = x, or the property 0 · x = 0, or both.






Computer science


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.[citation needed]

In databases, it is possible for a field not to have a value. It is then said to have a null value.[52] 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.[citation needed]


In mathematics −0 = +0 = 0; both −0 and +0 represent exactly the same number, i.e., there is no "positive zero" or "negative zero" distinct from zero. However, in some computer hardware signed number representations, zero has two distinct representations, a positive one grouped with the positive numbers and a negative one grouped with the negatives; this kind of dual representation is known as signed zero, with the latter form sometimes called negative zero. These representations include the signed magnitude and one's complement binary integer representations (but not the two's complement binary form used in most modern computers), and most floating point number representations (such as IEEE 754 and IBM S/390 floating point formats).

In binary, 0 represents the value for "off", which means no electricity flow.[53]

Zero is the value of false in many programming languages.

The Unix epoch (the date and time associated with a zero timestamp) begins the midnight before the first of January 1970.[54][55][56]

The MacOS epoch and Palm OS epoch (the date and time associated with a zero timestamp) begins the midnight before the first of January 1904.[57]

Many APIs and operating systems that require applications to return an integer value as an exit status typically use zero to indicate success and non-zero values to indicate specific error or warning conditions.

Other fields

  • On the U.S. Interstate Highway System, in most states exits are numbered based on the nearest milepost from the highway's western or southern terminus within that state. Several that are less than half a mile (800 m) from state boundaries in that direction are numbered as Exit 0.

Symbols and representations

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.[58] 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.[58]


Year label


See also




  5. ^ aught at
  6. ^ See:
    • Menninger, Karl (1992). Number words and number symbols: a cultural history of numbers. Courier Dover Publications. pp. 399–404. ISBN 978-0-486-27096-8.;
    • "zero, n." OED Online. Oxford University Press. December 2011. Archived from the original on 7 March 2012. Retrieved 4 March 2012. French zéro (1515 in Hatzfeld & Darmesteter) or its source Italian zero, for *zefiro, < Arabic çifr
  7. ^ a b See:
    • Smithsonian Institution, Oriental Elements of Culture in the Occident, p. 518, at Google Books, Annual Report of the Board of Regents of the Smithsonian Institution; Harvard University Archives, Quote="Sifr occurs in the meaning of "empty" even in the pre-Islamic time. ... Arabic sifr in the meaning of zero is a translation of the corresponding India sunya.";
    • Jan Gullberg (1997), Mathematics: From the Birth of Numbers, W.W. Norton & Co., ISBN 978-0-393-04002-9, p. 26, Quote = Zero derives from Hindu sunya – meaning void, emptiness – via Arabic sifr, Latin cephirum, Italian zevero.;
    • Robert Logan (2010), The Poetry of Physics and the Physics of Poetry, World Scientific, ISBN 978-981-4295-92-5, p. 38, Quote = "The idea of sunya and place numbers was transmitted to the Arabs who translated sunya or "leave a space" into their language as sifr."
  8. ^ Zero, Merriam Webster online Dictionary
  9. ^ Ifrah, Georges (2000). The Universal History of Numbers: From Prehistory to the Invention of the Computer. Wiley. ISBN 978-0-471-39340-5.
  12. ^ Joseph, George Gheverghese (2011). The Crest of the Peacock: Non-European Roots of Mathematics (Third Edition). Princeton UP. p. 86. ISBN 978-0-691-13526-7.
  14. ^ Diehl, p. 186
  15. ^ Mortaigne, Véronique (28 November 2014). "The golden age of Mayan civilisation – exhibition review". The Guardian. Archived from the original on 28 November 2014. Retrieved 10 October 2015.
  16. ^ Wallin, Nils-Bertil (19 November 2002). "The History of Zero". YaleGlobal online. The Whitney and Betty Macmillan Center for International and Area Studies at Yale. Archived from the original on 25 August 2016. Retrieved 1 September 2016.
  21. ^ Kang-Shen Shen; John N. Crossley; Anthony W.C. Lun; Hui Liu (1999). The Nine Chapters on the Mathematical Art: Companion and Commentary. Oxford UP. p. 35. ISBN 978-0-19-853936-0. zero was regarded as a number in India ... whereas the Chinese employed a vacant position
  22. ^ "Mathematics in the Near and Far East" (pdf). p. 262.
  24. ^ Kim Plofker (2009). Mathematics in India. Princeton UP. pp. 55–56. ISBN 978-0-691-12067-6.
  25. ^ Vaman Shivaram Apte (1970). Sanskrit Prosody and Important Literary and Geographical Names in the Ancient History of India. Motilal Banarsidass. pp. 648–649. ISBN 978-81-208-0045-8.
  27. ^ Kim Plofker (2009), Mathematics in India, Princeton University Press, ISBN 978-0-691-12067-6, pp. 54–56. Quote – "In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, [ ...] Pingala's use of a zero symbol [śūnya] as a marker seems to be the first known explicit reference to zero." Kim Plofker (2009), Mathematics in India, Princeton University Press, ISBN 978-0-691-12067-6, 55–56. "In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, there are five questions concerning the possible meters for any value "n". [ ...] The answer is (2)7 = 128, as expected, but instead of seven doublings, the process (explained by the sutra) required only three doublings and two squarings – a handy time saver where "n" is large. Pingala's use of a zero symbol as a marker seems to be the first known explicit reference to zero.
  29. ^ Weiss, Ittay (20 September 2017). "Nothing matters: How India's invention of zero helped create modern mathematics". The Conversation.
  30. ^ Devlin, Hannah (13 September 2017). "Much ado about nothing: ancient Indian text contains earliest zero symbol". The Guardian. ISSN 0261-3077. Retrieved 14 September 2017.
  31. ^ Revell, Timothy (14 September 2017). "History of zero pushed back 500 years by ancient Indian text". New Scientist. Retrieved 25 October 2017.
  32. ^ "Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol 'zero'". Bodleian Library. 14 September 2017. Retrieved 25 October 2017.
  33. ^ a b Aryabhatiya of Aryabhata, translated by Walter Eugene Clark.
  34. ^ O'Connor, Robertson, J.J., E.F. "Aryabhata the Elder". School of Mathematics and Statistics University of St Andrews, Scotland. Retrieved 26 May 2013.
  35. ^ William L. Hosch, ed. (15 August 2010). The Britannica Guide to Numbers and Measurement (Math Explained). The Rosen Publishing Group. pp. 97–98. ISBN 978-1-61530-108-9.
  40. ^ Ifrah, Georges (2000), p. 400.
  41. ^ "Much ado about nothing: ancient Indian text contains earliest zero symbol". The Guardian. Retrieved 14 September 2017.
  45. ^ Will Durant (1950), The Story of Civilization, Volume 4, The Age of Faith, Simon & Schuster, ISBN 978-0-9650007-5-8, p. 241, Quote = "In 976, Muhammad ibn Ahmad, in his Keys of the Sciences, remarked that if, in a calculation, no number appears in the place of tens, a little circle should be used "to keep the rows". This circle the Mosloems called ṣifr, "empty" whence our cipher."
  48. ^ 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 978-981-02-4088-2.
  49. ^ W., Weisstein, Eric. "Zero". Retrieved 4 April 2018.
  50. ^ Bunt, Lucas Nicolaas Hendrik; Jones, Phillip S.; Bedient, Jack D. (1976). The historical roots of elementary mathematics. Courier Dover Publications. pp. 254–255. ISBN 978-0-486-13968-5., Extract of pp. 254–255
  52. ^ Wu, X.; Ichikawa, T.; Cercone, N. (25 October 1996). Knowledge-Base Assisted Database Retrieval Systems. World Scientific. ISBN 978-981-4501-75-0.
  53. ^ Chris Woodford 2006, p. 9.
  54. ^ Paul DuBois. "MySQL Cookbook: Solutions for Database Developers and Administrators" 2014. p. 204.
  55. ^ Arnold Robbins; Nelson Beebe. "Classic Shell Scripting". 2005. p. 274
  56. ^ Iztok Fajfar. "Start Programming Using HTML, CSS, and JavaScript". 2015. p. 160.
  57. ^ Darren R. Hayes. "A Practical Guide to Computer Forensics Investigations". 2014. p. 399
  59. ^ Steel, Duncan (2000). Marking time: the epic quest to invent the perfect calendar. John Wiley & Sons. p. 113. ISBN 978-0-471-29827-4. 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.


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