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 (/oʊ/). 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" (/oʊ/), 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]

The Italian mathematician Fibonacci (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, and was then contracted to zero in Venetian. The Italian word zefiro was already in existence (meaning "west wind" from Latin and Greek zephyrus) and may have influenced the spelling when transcribing Arabic ṣifr.[9]

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]

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.

Pre-Columbian Americas

Estela C de Tres Zapotes
The back of Epi-Olmec stela C from Tres Zapotes, the second oldest Long Count date discovered. The numerals 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 quatrefoilsmall illustration of a partial quatrefoil in right half, whitespace in left half—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.

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 is assumed to have not influenced 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.

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.

P. Lund, Inv. 35a
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) 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).

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.[18] When division produced zero as a remainder, nihil, also meaning "nothing", was used. These medieval zeros were used by all future medieval calculators of Easter. The initial "N" was used as a zero symbol in a table of Roman numerals by Bede or his colleagues around 725.


Zero in Rod Calculus
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.[19]

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]


Khmer Numerals - 605 from the Sambor inscriptions
The number 605 in Khmer numerals, from the Sambor inscription (Saka era 605 corresponds to AD 683). The earliest known material use of zero as a decimal figure.

There are numerous copper plate inscriptions, with the same small o in them, some of them possibly dated to the 6th century, but their date or authenticity may be open to doubt.[13]

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]

Muhammad ibn Ahmad al-Khwarizmi, in 976, stated that if no number appears in the place of tens in a calculation, a little circle should be used "to keep the rows". This circle was called ṣifr.[45]

Transmission to Europe

The Hindu–Arabic numeral system (base 10) reached Europe in the 11th century, via Al-Andalus through Spanish Muslims, the Moors, together with knowledge of astronomy and instruments like the astrolabe, first imported by Gerbert of Aurillac. For this reason, the numerals came to be known in Europe as "Arabic numerals". The Italian mathematician Fibonacci or Leonardo of Pisa was instrumental in bringing the system into European mathematics in 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.[46][47]

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.


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.

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 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 (as well as being a multiple of any other integer, rational, or real number).

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.
  • Exponentiation: x0 = x/x = 1, except that the case x = 0 may be left undefined in some contexts. 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 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

  • 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. This is related to 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. The zero function is the only function that is both even and odd. 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 generalize either the property 0 + x = x, or the property 0 · x = 0, or both.


The value zero plays a special role for many physical quantities. For some quantities, the zero level is naturally distinguished from all other levels, whereas for others it is more or less arbitrarily chosen. For example, for an absolute temperature (as measured in kelvins) zero is the lowest possible value (negative temperatures are defined, but negative-temperature systems are not actually colder). This is in contrast to for example temperatures on the Celsius scale, where zero is arbitrarily defined to be at the freezing point of water. Measuring sound intensity in decibels or phons, the zero level is arbitrarily set at a reference value—for example, at a value for the threshold of hearing. In physics, the zero-point energy is the lowest possible energy that a quantum mechanical physical system may possess and is the energy of the ground state of the system.


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

As early as 1926, 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.

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.[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.

A null pointer is a pointer in a computer program that does not point to any object or function. In C, the integer constant 0 is converted into the null pointer at compile time when it appears in a pointer context, and so 0 is a standard way to refer to the null pointer in code. However, the internal representation of the null pointer may be any bit pattern (possibly different values for different data types).

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

  • 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.
  • 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

Text figures 036

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]

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.

Year label

In the BC calendar era, the year 1 BC is the first year before AD 1; there is not 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.[59]

See also


  1. ^ 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.


  1. ^ Matson, John (21 August 2009). "The Origin of Zero". Scientific American. Springer Nature. Retrieved 24 April 2016.
  2. ^ 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. ^ "aught, Also ought" in Webster's Collegiate Dictionary (1927), Third Edition, Springfield, MA: G. & C. Merriam.
  4. ^ "cipher", in Webster's Collegiate Dictionary (1927), Third Edition, Springfield, MA: G. & C. Merriam.
  5. ^ aught at etymonline.com
  6. ^ See:
    • Douglas Harper (2011), Zero, Etymology Dictionary, Quote="figure which stands for naught in the Arabic notation," also "the absence of all quantity considered as quantity," c. 1600, from French zéro or directly from Italian zero, from Medieval Latin zephirum, from Arabic sifr "cipher," translation of Sanskrit sunya-m "empty place, desert, naught";
    • 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.
  10. ^ 'Aught' definition, Dictionary.com – Retrieved April 2013.
  11. ^ 'Aught' synonyms, Thesaurus.com – Retrieved April 2013.
  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.
  13. ^ a b Kaplan, Robert. (2000). The Nothing That Is: A Natural History of Zero. Oxford: Oxford University Press.
  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.
  17. ^ O'Connor, John J.; Robertson, Edmund F., "A history of Zero", MacTutor History of Mathematics archive, University of St Andrews.
  18. ^ "Zero and Fractions". Know the Romans. Retrieved 21 September 2016.
  19. ^ a b Hodgkin, Luke (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.
  20. ^ Crossley, Lun. 1999, p. 12 "the ancient Chinese system is a place notation system"
  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). grmath4.phpnet.us. p. 262.
  23. ^ 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."
  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.
  26. ^ "Math for Poets and Drummers" (pdf). people.sju.edu.
  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.
  28. ^ Ifrah, Georges (2000), p. 416.
  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). books.google.com.my. The Rosen Publishing Group. pp. 97–98. ISBN 978-1-61530-108-9.
  36. ^ Bourbaki, Nicolas Elements of the History of Mathematics (1998), p. 46. Britannica Concise Encyclopedia (2007), entry "Algebra"
  37. ^ Algebra with Arithmetic of Brahmagupta and Bhaskara, translated to English by Henry Thomas Colebrooke (1817) London
  38. ^ Cœdès, Georges, "A propos de l'origine des chiffres arabes," Bulletin of the School of Oriental Studies, University of London, Vol. 6, No. 2, 1931, pp. 323–328. Diller, Anthony, "New Zeros and Old Khmer," The Mon-Khmer Studies Journal, Vol. 25, 1996, pp. 125–132.
  39. ^ Casselman, Bill. "All for Nought". ams.org. University of British Columbia), American Mathematical Society.
  40. ^ Ifrah, Georges (2000), p. 400.
  41. ^ "Much ado about nothing: ancient Indian text contains earliest zero symbol". The Guardian. Retrieved 14 September 2017.
  42. ^ Pannekoek, A. (1961). A History of Astronomy. George Allen & Unwin. p. 165.
  43. ^ a b c Will Durant (1950), The Story of Civilization, Volume 4, The Age of Faith: Constantine to Dante – A.D. 325–1300, Simon & Schuster, ISBN 978-0-9650007-5-8, p. 241, Quote = "The Arabic inheritance of science was overwhelmingly Greek, but Hindu influences ranked next. In 773, at Mansur's behest, translations were made of the Siddhantas – Indian astronomical treatises dating as far back as 425 BC; these versions may have the vehicle through which the "Arabic" numerals and the zero were brought from India into Islam. In 813, al-Khwarizmi used the Hindu numerals in his astronomical tables."
  44. ^ Brezina, Corona (2006). Al-Khwarizmi: The Inventor Of Algebra. The Rosen Publishing Group. ISBN 978-1-4042-0513-0.
  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."
  46. ^ Sigler, L., Fibonacci's Liber Abaci. English translation, Springer, 2003.
  47. ^ Grimm, R.E., "The Autobiography of Leonardo Pisano", Fibonacci Quarterly 11/1 (February 1973), pp. 99–104.
  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". mathworld.wolfram.com. 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
  51. ^ 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.
  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
  58. ^ 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.
  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.


  • Amir D. Aczel (2015) Finding Zero, New York City: Palgrave Macmillan. ISBN 978-1-137-27984-2
  • 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.
  • 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.
  • Chris Woodford (2006), Digital Technology, Evans Brothers, ISBN 978-0-237-52725-9

External links

2018–19 Premier League

The 2018–19 Premier League was the 27th season of the Premier League, the top English professional league for association football clubs, since its establishment in 1992. The season started on 10 August 2018 and concluded on 12 May 2019. Fixtures for the 2018–19 season were announced on 14 June 2018. The league was contested by the top 17 teams from the 2017–18 season as well as Wolverhampton Wanderers, Cardiff City and Fulham, who joined as the promoted clubs from the 2017–18 EFL Championship. They replaced West Bromwich Albion, Swansea City and Stoke City who were relegated to the 2018–19 EFL Championship.Defending champions Manchester City won their fourth Premier League title, and sixth English top-flight title overall. They won their last 14 league games and retained the league title on the final day of the season, finishing on 98 points. Liverpool finished runners-up with 97 points – the highest total in English top-flight history for a second-placed team. This was Manchester City’s second step in becoming the first English team to complete a domestic treble, as they earlier won the 2018–19 EFL Cup and would later win the 2018–19 FA Cup.

2018–19 Serie A

The 2018–19 Serie A is the 117th season of top-tier Italian football, the 87th in a round-robin tournament, and the 9th since its organization under a league committee separate from Serie B. Juventus were the seven-time defending champions and defended their title after winning against Fiorentina on 20 April 2019. The season is scheduled to run from 18 August 2018 to 26 May 2019.

2018–19 UEFA Champions League

The 2018–19 UEFA Champions League is the 64th season of Europe's premier club football tournament organised by UEFA, and the 27th season since it was renamed from the European Champion Clubs' Cup to the UEFA Champions League.

The final will be played at the Wanda Metropolitano in Madrid, Spain, between Tottenham Hotspur and Liverpool. It will be the second all-English final after the 2008 final, which was contested between Manchester United and Chelsea in Moscow. The winners of the 2018–19 UEFA Champions League will earn the right to play against the winners of the 2018–19 UEFA Europa League in the 2019 UEFA Super Cup. They will also automatically qualify for the 2019–20 UEFA Champions League group stage. As both finalists have already qualified for the group stage through their league performance, the berth reserved will be given to the champions of the 2018–19 Austrian Bundesliga, the 11th-ranked association according to next season's access list.For the first time, the video assistant referee (VAR) system was used in the competition from the round of 16 onward.Real Madrid were the defending champions, having won the title for three successive seasons in 2015–16, 2016–17 and 2017–18. They were eliminated by Ajax in the round of 16.

2018–19 UEFA Europa League

The 2018–19 UEFA Europa League is the 48th season of Europe's secondary club football tournament organised by UEFA, and the 10th season since it was renamed from the UEFA Cup to the UEFA Europa League.

The final will be played at the Olympic Stadium in Baku, Azerbaijan, between English sides Arsenal and Chelsea — which it marked for the first time ever in UEFA history that both the finals of UEFA Champions League and UEFA Europa League this season will be all played by teams from one country. The winners of the 2018–19 UEFA Europa League will earn the right to play against the winners of the 2018–19 UEFA Champions League in the 2019 UEFA Super Cup. They will also automatically qualify for the 2019–20 UEFA Champions League group stage, and if they have already qualified through their league performance, the berth reserved will be given to the third-placed team of the 2018–19 Ligue 1, the 5th-ranked association according to next season's access list.For the first time, the video assistant referee (VAR) system will be used in the competition, where it will be implemented in the final.As the title holders of Europa League, Atlético Madrid qualified for the 2018–19 UEFA Champions League, although they had already qualified before the final through their league performance. They were unable to defend their title as they advanced to the Champions League knockout stage.

2019 FIFA Women's World Cup

The 2019 FIFA Women's World Cup will be the eighth edition of the FIFA Women's World Cup, the quadrennial international football championship contested by the women's national teams of the member associations of the Fédération Internationale de Football Association (FIFA) between 7 June and 7 July 2019. In March 2015, France won the right to host the event; the first time the country will host the tournament, and the third time a European nation will. Matches are planned for nine cities across France. The United States enters the competition as defending champions. It will also be the first Women's World Cup to use the video assistant referee (VAR) system.

2019 Philippine general election

The 2019 Philippine general election was conducted on May 13, 2019. A midterm election, those elected therein will take office on June 30, 2019, midway through the term of President Rodrigo Duterte.

The following positions were contested:

12 seats in the Senate of the Philippines

All seats in the House of Representatives of the Philippines

All provincial-level elected positions in the provinces of the Philippines

All city-level elected positions in the cities of the Philippines

All municipal-level elected positions in the municipalities of the PhilippinesUnder the Local Government Code and the 1987 constitution, all terms start on June 30, 2019, and end on June 30, 2022, except for elected senators, whose terms shall end on June 30, 2025. The Commission on Elections administered the election.

Creative Commons license

A Creative Commons (CC) license is one of several public copyright licenses that enable the free distribution of an otherwise copyrighted "work". A CC license is used when an author wants to give other people the right to share, use, and build upon a work that he or she (that author) has created. CC provides an author flexibility (for example, he or she might choose to allow only non-commercial uses of a given work) and protects the people who use or redistribute an author's work from concerns of copyright infringement as long as they abide by the conditions that are specified in the license by which the author distributes the work.There are several types of Creative Commons licenses. The licenses differ by several combinations that condition the terms of distribution. They were initially released on December 16, 2002 by Creative Commons, a U.S. non-profit corporation founded in 2001. There have also been five versions of the suite of licenses, numbered 1.0 through 4.0. As of December 2018, the 4.0 license suite is the most current.

In October 2014 the Open Knowledge Foundation approved the Creative Commons CC BY, CC BY-SA and CC0 licenses as conformant with the "Open Definition" for content and data.

Elections in the Philippines

Philippine elections are of several types. The president, vice-president, and the senators are elected for a six-year term, while the members of the House of Representatives, governors, vice-governors, members of the Sangguniang Panlalawigan (provincial board members), mayors, vice-mayors, members of the Sangguniang Panlungsod/members of the Sangguniang Bayan (city/municipal councilors), barangay officials, and the members of the Sangguniang Kabataan (youth councilors) are elected to serve for a three-year term.

The Congress or Kongreso has two chambers. The House of Representatives or Kapulungan ng mga Kinatawan has 292 seats as of 2013, of which 80% are contested in single seat electoral districts and 20% are allotted to party-lists according to a modified Hare quota with remainders disregarded and a three-seat cap. These party list seats are only accessible to marginalized and under-represented groups and parties, local parties, and sectoral wings of major parties that represent the marginalized. The Constitution of the Philippines allows the House of Representatives to have more than 250 members by statute without a need for a constitutional amendment. The Senate or Senado has 24 members which are elected on a nationwide at-large basis; they do not represent any geographical district. Half of the Senate is renewed every three years.

The Philippines has a multi-party system, with numerous parties in which no one party often has a chance of gaining power alone, and parties must work with each other to form a coalition government. The Commission on Elections (COMELEC) is responsible for running the elections.

Under the Constitution, elections for the members of Congress and local positions (except barangay officials) occur every second Monday of May every third year after May 1992, and presidential and vice presidential elections occur every second Monday of May every sixth year after May 1992. All elected officials, except those at the barangay level, start (and end) their terms of office on June 30 of the election year.

Human Development Index

The Human Development Index (HDI) is a statistic composite index of life expectancy, education, and per capita income indicators, which are used to rank countries into four tiers of human development. A country scores a higher HDI when the lifespan is higher, the education level is higher, and the gross national income GNI (PPP) per capita is higher. It was developed by Pakistani economist Mahbub ul Haq, with help from Gustav Ranis of Yale University and Meghnad Desai of the London School of Economics, and was further used to measure a country's development by the United Nations Development Program (UNDP)'s Human Development Report Office.The 2010 Human Development Report introduced an Inequality-adjusted Human Development Index (IHDI). While the simple HDI remains useful, it stated that "the IHDI is the actual level of human development (accounting for inequality)", and "the HDI can be viewed as an index of 'potential' human development (or the maximum IHDI that could be achieved if there were no inequality)". The index does not take into account several factors, such as the net wealth per capita or the relative quality of goods in a country. This situation tends to lower the ranking for some of the most advanced countries, such as the G7 members and others.The index is based on the human development approach, developed by Amartya Sen, often framed in terms of whether people are able to "be" and "do" desirable things in life. Examples include—Being: well fed, sheltered, healthy; Doings: work, education, voting, participating in community life. The freedom of choice is central—someone choosing to be hungry (as during a religious fast) is quite different from someone who is hungry because they cannot afford to buy food, or because the country is in a famine.


JPEG ( JAY-peg) is a commonly used method of lossy compression for digital images, particularly for those images produced by digital photography. The degree of compression can be adjusted, allowing a selectable tradeoff between storage size and image quality. JPEG typically achieves 10:1 compression with little perceptible loss in image quality.JPEG compression is used in a number of image file formats. JPEG/Exif is the most common image format used by digital cameras and other photographic image capture devices; along with JPEG/JFIF, it is the most common format for storing and transmitting photographic images on the World Wide Web. These format variations are often not distinguished, and are simply called JPEG.

The term "JPEG" is an initialism/acronym for the Joint Photographic Experts Group, which created the standard. The MIME media type for JPEG is image/jpeg, except in older Internet Explorer versions, which provides a MIME type of image/pjpeg when uploading JPEG images. JPEG files usually have a filename extension of .jpg or .jpeg.

JPEG/JFIF supports a maximum image size of 65,535×65,535 pixels, hence up to 4 gigapixels for an aspect ratio of 1:1.

List of countries and dependencies by area

This is a list of the world's countries and their dependent territories by area, ranked by total area.

Entries in this list include, but are not limited to, those in the ISO 3166-1 standard, which includes sovereign states and dependent territories. Largely unrecognised states not in ISO 3166-1 are included in the list in ranked order, but are not given a rank number. The areas of such largely unrecognised states are in most cases also included in the areas of the more widely recognised states that claim the same territory; see the notes in the "notes" column for each country for clarification.

Not included in the list are individual country claims to parts of the continent of Antarctica, entities such as the European Union that have some degree of sovereignty but do not consider themselves to be sovereign countries or dependent territories, and unrecognised micronations such as the Principality of Sealand.

This list includes three measurements of area:

Total area: the sum of land and water areas within international boundaries and coastlines.

Land area: the aggregate of all land within international boundaries and coastlines, excluding water area.

Water area: the sum of the surface areas of all inland water bodies (lakes, reservoirs, and rivers) within international boundaries and coastlines. Coastal internal waters (some small bays) may be included. Territorial waters are not included unless otherwise noted. Contiguous zones and exclusive economic zones are not included.Data is taken from the United Nations Statistics Division unless otherwise noted.

List of countries and dependencies by population

This is a list of countries and dependent territories by population. It includes sovereign states, inhabited dependent territories and, in some cases, constituent countries of sovereign states, with inclusion within the list being primarily based on the ISO standard ISO 3166-1. For instance, the United Kingdom is considered as a single entity, while the constituent countries of the Kingdom of the Netherlands are considered separately. In addition, this list includes certain states with limited recognition not found in ISO 3166-1.

Also given in percent is each country's population compared with the population of the world, which the United Nations estimates at 7.71 billion as of today.

List of countries by Human Development Index

This is a full list of all the countries by the Human Development Index as included in a United Nations Development Programme's Human Development Report. The latest report was released on 14 September 2018 and is based on data collected in 2017.In the 2010 Human Development Report, a further Inequality-adjusted Human Development Index (IHDI) was introduced. It stated that while the HDI remains useful, "the IHDI is the actual level of human development (accounting for inequality)" and "the HDI can be viewed as an index of "potential" human development (or the maximum IHDI that could be achieved if there were no inequality)". The index does not take into account several factors, such as the net wealth per capita or the relative quality of goods in a country. This situation tends to lower the ranking for some of the most advanced countries, such as the G7 members and others.

Native Americans in the United States

Native Americans, also known as American Indians, Indigenous Americans and other terms, are the indigenous peoples of the United States, except Hawaii. There are over 500 federally recognized tribes within the US, about half of which are associated with Indian reservations. The term "American Indian" excludes Native Hawaiians and some Alaska Natives, while Native Americans (as defined by the US Census) are American Indians, plus Alaska Natives of all ethnicities. Native Hawaiians are not counted as Native Americans by the US Census, instead being included in the Census grouping of "Native Hawaiian and other Pacific Islander".

The ancestors of modern Native Americans arrived in what is now the United States at least 15,000 years ago, possibly much earlier, from Asia via Beringia. A vast variety of peoples, societies and cultures subsequently developed. Native Americans were greatly affected by the European colonization of the Americas, which began in 1492, and their population declined precipitously mainly due to introduced diseases as well as warfare, territorial confiscation and slavery. After the founding of the United States, many Native American peoples were subjected to warfare, removals and one-sided treaties, and they continued to suffer from discriminatory government policies into the 20th century. Since the 1960s, Native American self-determination movements have resulted in changes to the lives of Native Americans, though there are still many contemporary issues faced by Native Americans. Today, there are over five million Native Americans in the United States, 78% of whom live outside reservations.

When the United States was created, established Native American tribes were generally considered semi-independent nations, as they generally lived in communities separate from British settlers. The federal government signed treaties at a government-to-government level until the Indian Appropriations Act of 1871 ended recognition of independent native nations, and started treating them as "domestic dependent nations" subject to federal law. This law did preserve the rights and privileges agreed to under the treaties, including a large degree of tribal sovereignty. For this reason, many (but not all) Native American reservations are still independent of state law and actions of tribal citizens on these reservations are subject only to tribal courts and federal law.

The Indian Citizenship Act of 1924 granted U.S. citizenship to all Native Americans born in the United States who had not yet obtained it. This emptied the "Indians not taxed" category established by the United States Constitution, allowed natives to vote in state and federal elections, and extended the Fourteenth Amendment protections granted to people "subject to the jurisdiction" of the United States. However, some states continued to deny Native Americans voting rights for several decades. Bill of Rights protections do not apply to tribal governments, except for those mandated by the Indian Civil Rights Act of 1968.

Normal distribution

In probability theory, the normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. A random variable with a Gaussian distribution is said to be normally distributed and is called a normal deviate.

The normal distribution is useful because of the central limit theorem. In its most general form, under some conditions (which include finite variance), it states that averages of samples of observations of random variables independently drawn from independent distributions converge in distribution to the normal, that is, they become normally distributed when the number of observations is sufficiently large. Physical quantities that are expected to be the sum of many independent processes (such as measurement errors) often have distributions that are nearly normal. Moreover, many results and methods (such as propagation of uncertainty and least squares parameter fitting) can be derived analytically in explicit form when the relevant variables are normally distributed.

The normal distribution is sometimes informally called the bell curve. However, many other distributions are bell-shaped (such as the Cauchy, Student's t-, and logistic distributions).

The probability density of the normal distribution is


Portland, Oregon

Portland is the largest and most populous city in the U.S. state of Oregon and the seat of Multnomah County. It is a major port in the Willamette Valley region of the Pacific Northwest, at the confluence of the Willamette and Columbia rivers. As of 2018, Portland had an estimated population of 653,115, making it the 25th most populated city in the United States, and the second-most populous in the Pacific Northwest (after Seattle). Approximately 2.4 million people live in the Portland metropolitan statistical area (MSA), making it the 25th most populous MSA in the United States. Its Combined Statistical Area (CSA) ranks 19th-largest with a population of around 3.2 million. Approximately 60% of Oregon's population resides within the Portland metropolitan area.Named after Portland, Maine, the Oregon settlement began to be populated in the 1830s near the end of the Oregon Trail. Its water access provided convenient transportation of goods, and the timber industry was a major force in the city's early economy. At the turn of the 20th century, the city had a reputation as one of the most dangerous port cities in the world, a hub for organized crime and racketeering. After the city's economy experienced an industrial boom during World War II, its hard-edged reputation began to dissipate. Beginning in the 1960s, Portland became noted for its growing progressive political values, earning it a reputation as a bastion of counterculture.The city operates with a commission-based government guided by a mayor and four commissioners as well as Metro, the only directly elected metropolitan planning organization in the United States. The city government is notable for its land-use planning and investment in public transportation. Portland is frequently recognized as one of the world's most environmentally conscious cities because of its high walkability, large community of bicyclists, farm-to-table dining, expansive network of public transportation options, and over 10,000 acres (4,000 hectares) of public parks. As a result, Portland consistently ranks highly in quality of life in the United States. Its climate is marked by warm, dry summers and cool, rainy winters. This climate is ideal for growing roses, and Portland has been called the "City of Roses" for over a century.


Universal Serial Bus (USB) is an industry standard that establishes specifications for cables and connectors and protocols for connection, communication and power supply between computers, peripheral devices and other computers. Released in 1996, the USB standard is currently maintained by the USB Implementers Forum (USB IF). There have been three generations of USB specifications: USB 1.x, USB 2.0 and USB 3.x; the fourth called USB4 is scheduled to be published in the middle of 2019.


Unicode is a computing industry standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems. The standard is maintained by the Unicode Consortium, and as of May 2019 the most recent version, Unicode 12.1, contains a repertoire of 137,994 characters covering 150 modern and historic scripts, as well as multiple symbol sets and emoji. The character repertoire of the Unicode Standard is synchronized with ISO/IEC 10646, and both are code-for-code identical.

The Unicode Standard consists of a set of code charts for visual reference, an encoding method and set of standard character encodings, a set of reference data files, and a number of related items, such as character properties, rules for normalization, decomposition, collation, rendering, and bidirectional display order (for the correct display of text containing both right-to-left scripts, such as Arabic and Hebrew, and left-to-right scripts).Unicode's success at unifying character sets has led to its widespread and predominant use in the internationalization and localization of computer software. The standard has been implemented in many recent technologies, including modern operating systems, XML, Java (and other programming languages), and the .NET Framework.

Unicode can be implemented by different character encodings. The Unicode standard defines UTF-8, UTF-16, and UTF-32, and several other encodings are in use. The most commonly used encodings are UTF-8, UTF-16, and UCS-2 (without full support for Unicode), a precursor of UTF-16; GB18030 is standardized in China and implements Unicode fully, while not an official Unicode standard.

UTF-8, the dominant encoding on the World Wide Web (used in over 93% of websites), uses one byte for the first 128 code points, and up to 4 bytes for other characters. The first 128 Unicode code points are the ASCII characters, which means that any ASCII text is also a UTF-8 text.

UCS-2 uses two bytes (16 bits) for each character but can only encode the first 65,536 code points, the so-called Basic Multilingual Plane (BMP). With 1,114,112 code points on 17 planes being possible, and with over 137,000 code points defined so far, UCS-2 is only able to represent less than half of all encoded Unicode characters. Therefore, UCS-2 is outdated, though still widely used in software. UTF-16 extends UCS-2, by using the same 16-bit encoding as UCS-2 for the Basic Multilingual Plane, and a 4-byte encoding for the other planes. As long as it contains no code points in the reserved range U+D800–U+DFFF, a UCS-2 text is a valid UTF-16 text.

UTF-32 (also referred to as UCS-4) uses four bytes for each character. Like UCS-2, the number of bytes per character is fixed, facilitating character indexing; but unlike UCS-2, UTF-32 is able to encode all Unicode code points. However, because each character uses four bytes, UTF-32 takes significantly more space than other encodings, and is not widely used.

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