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Glagolitic script

The Glagolitic script (, , glagolitsa) is the oldest known Slavic alphabet. It is generally agreed that it was created in the 9th century for the purpose of translating liturgical texts into Old Church Slavonic by Saint Cyril, a monk from Thessalonica. He and his brother Saint Methodius were sent by the Byzantine Emperor Michael III in 863 to Great Moravia to spread Christianity there. After the deaths of Cyril and Methodius, their disciples were expelled and they moved to the First Bulgarian Empire instead. The Cyrillic alphabet, which developed gradually in the Preslav Literary School by Greek alphabet scribes who incorporated some Glagolitic letters, gradually replaced Glagolitic in that region. Glagolitic remained in use alongside Latin in the Kingdom of Croatia and alongside Cyrillic until the 14th century in the Second Bulgarian Empire and the Serbian Empire, and later mainly for cryptographic purposes.

Glagolitic also spread to the Kievan Rus' and the Kingdom of Bohemia, though its use declined there in the 12th century, although some manuscripts in the territory of the former retained Glagolitic inclusions for centuries. It had also spread to Duklja and Zachlumia, from which it reached the March of Verona where the Investiture Controversy afforded it refuge from the opposition of Latin rite prelates, and allowed it to entrench itself in Istria, spreading from there to nearby lands.

It survived there and as far south as Dalmatia without interruption into the 20th century for Church Slavonic in addition to its use as a secular script in parts of its range, which at times extended into Bosnia, Slavonia, and Carniola, in addition to 14th-15th century exclaves in Prague and Kraków, and a 16th century exclave in Putna.

Its authorship by Cyril was forgotten, having been replaced with an attribution to St. Jerome by the early Benedictine adopters of Istria in a bid to secure the approval of the papacy. The bid was ultimately successful, though sporadic restrictions and repressions from individual bishops continued even after its official recognition by Pope Innocent IV. These had little effect on the vitality of the script, which evolved from its original Rounded Glagolitic form into an Angular Glagolitic form, in addition to a cursive form developed for notary purposes.

But the Ottoman conquests left the script without most of its continental population, and as a result of the Counter-Reformation its use was restricted in Istria and the Diocese of Zagreb, and the only active printing press with a Glagolitic type was confiscated, leading to a shift towards Latinic and Cyrillic literacy when coupled with the Tridentine requirement that priests be educated at seminaries. The result was its gradual death as a written script in most of its continental range, but also the unusually late survival of medieval scribal tradition for the reproduction of Glagolitic texts in isolated areas like the island of Krk and the Zadar Archipelago. Although the Propaganda Fide would eventually resume printing Glagolitic books, very few titles were published, so the majority of Glagolitic literary works continued to be written and copied by hand well into the 18th century. Of the major European scripts, only the Arabic script is comparable in this regard.

In the early 19th century, the policies of the First French Empire and Austrian Empire left the script without legal status and its last remaining centers of education were abolished, concurrent with the weakening of the script in the few remaining seminaries that used the cursive form in instruction, resulting in a rapid decline. But the when Slavicists discovered the script and established it as the original script devised by Cyril, Glagolitic gained new niche applications in certain intellectual circles, while a small number of priests fought to keep its liturgical use alive, encountering difficulties but eventually succeeding to the point that its area expanded in the early 20th century.

Latinic translations and transliterations of the matter of the missal in this period led to its decline in the decades before Vatican II, whose promulgation of the vernacular had the effect of confining regular use of Glagolitic to a few monasteries and academic institutions, in addition to a small population of enthusiasts, whose numbers grew and shrank with the prevalence of the script in literature, but grew exponentially in pious and nationalist circles in the years leading up to and following Independence of Croatia, and again more broadly with the Internet.

Name and etymology
The word glagolitic comes from Neo-Latin glagoliticus and Croatian glagoljica, from Old Church Slavonic ⰳⰾⰰⰳⱁⰾⱏ (glagolŭ), meaning "utterance" or "word".

The name glagolitsa is speculated to have developed in Croatia, around the 14th century, and was derived from the word glagoljati, literally "verb (glagol) using (jati)", meaning to say Mass in Old Church Slavonic liturgy.

In the languages now spoken in the places where Glagolitic script was once used, the script is known as глаголица (romanized as glagolitsa or glagolica, depending on which language) in Bulgarian, Macedonian and Russian; glagoljica in Croatian and Serbian; глаголиця (hlaholytsia) in Ukrainian; hlaholice in Czech; hlaholika in Slovak; głagolica in Polish; and glagolica in Slovene.

Origins

The creation of the characters is popularly attributed to Saints Cyril and Methodius, who may have created them to facilitate the introduction of Christianity. It is believed that the original letters were fitted to Slavic dialects in geographical Macedonia specifically (the Byzantine theme of Thessalonica). The words of that language could not be easily written by using either the Greek or Latin alphabets.

The number of letters in the original Glagolitic alphabet is not known, but it may have been close to its presumed Greek model. The 41 letters known today include letters for non-Greek sounds, which may have been added by Saint Cyril, as well as ligatures added in the 12th century under the influence of Cyrillic, as Glagolitic lost its dominance. In later centuries, the number of letters dropped dramatically, to fewer than 30 in modern Croatian and Czech recensions of the Church Slavic language. Twenty-four of the 41 original Glagolitic letters (see table below) probably derive from graphemes of the medieval cursive Greek small alphabet but have been given an ornamental design.

The source of the other consonantal letters is unknown. If they were added by Cyril, it is likely that they were taken from an alphabet used for Christian scripture. It is frequently proposed that the letters sha, tsi , and cherv were taken from the letters shin ש and tsadi צ of the Hebrew alphabet, and that Ⰶ zhivete derives from Coptic janja Ϫ. However, Cubberley suggests that if a single prototype were presumed, the most likely source would be Armenian. Other proposals include the Samaritan alphabet, which Cyril learned during his journey to the Khazars in Cherson.

For writing numbers, the Glagolitic numerals use letters with a numerical value assigned to each based on their native alphabetic order. This differs from Cyrillic numerals, which inherited their numeric value from the corresponding Greek letter (see Greek numerals).

The two brothers from Thessaloniki, who were later canonized as Saints Cyril and Methodius, were sent to Great Moravia in 862 by the Byzantine emperor at the request of Prince Rastislav, who wanted to weaken the dependence of his country on East Frankish priests. The Glagolitic alphabet, however it originated, was used between 863 and 885 for government and religious documents and books and at the Great Moravian Academy (Veľkomoravské učilište) founded by the missionaries, where their followers were educated. The Kiev Missal, found in the 19th century in Jerusalem, was dated to the 10th century.

In 885, Pope Stephen V issued a papal bull to restrict spreading and reading Christian services in languages other than Latin or Greek. Around the same time, Svatopluk I, following the interests of the Frankish Empire, persecuted the students of Cyril and Methodius and expelled them from Great Moravia.

In 886, an East Frankish bishop of Nitra named Wiching banned the script and jailed 200 followers of Methodius, mostly students of the original academy. They were then dispersed or, according to some sources, sold as slaves by the Franks. However, many of them, including Saints Naum, Clement, Angelar, Sava and Gorazd, reached the First Bulgarian Empire and were commissioned by Boris I of Bulgaria to teach and instruct the future clergy of the state in the Slavic language. After the adoption of Christianity in Bulgaria in 865, religious ceremonies and Divine Liturgy were conducted in Greek by clergy sent from the Byzantine Empire, using the Byzantine rite. Fearing growing Byzantine influence and weakening of the state, Boris viewed the introduction of the Slavic alphabet and language into church use as a way to preserve the independence of the Bulgarian Empire from Byzantine Constantinople. As a result of Boris' measures, two academies, one in Ohrid and one in Preslav, were founded.

Spread of the script
From there, the students travelled to other places and spread the use of their alphabet. Students of the two apostles who were expelled from Great Moravia in 886, notably Clement of Ohrid and Saint Naum, brought the Glagolitic alphabet to the First Bulgarian Empire on Balkans and were received and accepted officially by Boris I of Bulgaria. This led to the establishment of the two literary schools: the Preslav Literary School and the Ohrid Literary School. Some went to Croatia (Dalmatia), where the squared variant arose and where Glagolitic remained in use for a long time. In 1248, Pope Innocent IV granted the Croatians of southern Dalmatia the unique privilege of using their own language and this script in the Roman Rite liturgy. Formally granted to bishop Philip of Senj, permission to use the Glagolitic liturgy (the Roman Rite conducted in the Slavic language instead of Latin, not the Byzantine rite), actually extended to all Croatian lands, mostly along the Adriatic coast. The Holy See had several Glagolitic missals published in Rome. Authorization for the use of this language was extended to some other Slavic regions between 1886 and 1935. In missals, the Glagolitic script was eventually replaced with the Latin alphabet, but the use of the Slavic language in the Mass continued, until replaced by modern vernacular languages.

At the end of the 9th century, one of these students of Methodius – Saint Naum, one of the founders of the Pliska Literary School (commonly known as the Preslav Literary School, where the Bulgarian capital, along with the school, was transferred to in 893) – is often credited, at least by supporters of glagolitic precedence, for the "creation" or wider adoption of the Cyrillic script, which almost entirely replaced Glagolitic during the Middle Ages. The Cyrillic alphabet is derived from the Greek alphabet used at that time, with some additional letters for sounds peculiar to Slavic languages (like ⟨ш⟩, ⟨ц⟩, ⟨ч⟩, ⟨ъ⟩, ⟨ь⟩, ⟨ѣ⟩), likely derived from the Glagolitic alphabet. The decision by a great assembly of notables summoned by Boris in the year 893 in favor of Cyrillic created an alphabetical difference between the two literary centres of the Bulgarian state in Pliska and Ohrid. In the western part the Glagolitic alphabet remained dominant at first. However, subsequently in the next two centuries, mostly after the fall of the First Bulgarian Empire to the Byzantines, Glagolitic gradually ceased to be used there at all. Nevertheless, particular passages or words written with the Glagolitic alphabet appeared in Bulgarian Cyrillic manuscripts till the end of the 14th century. Some students of the Ohrid academy went to Bohemia where the alphabet was used in the 10th and 11th centuries, along with other scripts. It is not clear whether the Glagolitic alphabet was used in the Duchy of Kopnik before the Wendish Crusade, but it was certainly used in Kievan Rus'. Another use of Glagolitic is presumed in now southern Poland (Duchy of Vistula/White Croats state) and the Transcarpathia region.

Survival and use in Croatia
In Croatia, from the 12th century, Glagolitic inscriptions appeared mostly in littoral areas: Istria, Primorje, Kvarner, and Kvarner islands, notably Krk, Cres, and Lošinj; in Dalmatia, on the islands of Zadar, but there were also findings in inner Lika and Krbava, reaching to Kupa river, and even as far as Međimurje and Slovenia. Hrvoje's Missal from 1404 was illuminated in Split, and it is considered one of the most beautiful Croatian Glagolitic books. The 1483 Missale Romanum Glagolitice was the first printed Croatian Glagolitic book.

It was believed that Glagolitsa in Croatia was present only in those areas. But, in 1992, the discovery of Glagolitic inscriptions in churches along the Orljava river in Slavonia totally changed the picture (churches in Brodski Drenovac, Lovčić, and some others), showing that use of the Glagolitic alphabet was spread from Slavonia also.

Sporadic instances aside, Glagolitic survived beyond the 12th century as a primary script in Croatian lands alone, although from there a brief attempt at reintroduction was made in the West Slavic area in the 14th century through the Emmaus Benedictine Monastery in Prague, where it survived well into the 15th century, the last manuscript with Glagolitic script dating to 1450–1452. Its use for special applications continued in some Cyrillic areas, for example in the Bologna Psalter (1230-1241), the Sinodalna 895 Menaion (1260), the RPK 312 Gospel (13th), the Karakallou Epistolary (13th), the NBKM 933 Triodion (13th), the Skopje 1511 Octoechos (13th), the BRAN 4.9.39 Miscellany (13th), the Hilandar Chrysorrhoas (13th/14th), the Mazurin 1698 Pandects (13th/14th), the Sofia Psalter (1337), the SANU 55 Epistolary (1366–1367), the RNB F.п.I.2 Psalter (14th), the Čajniče Gospel (late 14th), the Radosav Miscellany (1444–1461), the Prague NM IX.F.38 Psalter (18th) and in the initials of many manuscripts of the Prophets with Commentary dating to the late 15th and early 16th centuries from Muscovy and Russia. Most later use in the Cyrillic world was for cryptographic purposes, such as in the Krushedol Miscellany (15th), the RNB F.п.I.48 Prologue (1456), the Piskarev 59 Isaac (1472), the Shchukin 511 Miscellany (1511) and the Hludov Gospel (17th/18th).

The early development of the Glagolitic minuscule script alongside the increasingly square majuscule is poorly documented, but a mutual relationship evolved between the two varieties; the majuscule being used primarily for inscriptions and higher liturgical uses, and the minuscule being used in both religious and secular documents. Ignoring the problematic early Slavonian inscriptions, the use of the Glagolitic script at its peak before the Croatian-Ottoman wars corresponded roughly to the area that spoke the Chakavian dialect at the time, in addition to some adjacent Kajkavian regions within the Zagreb bishopric. As a result, vernacular impact on the liturgical language and script largely stems from Chakavian sub-dialects, although South Chakavian speakers mostly used Cyrillic, with Glagolitic only in certain parishes as a high liturgical script until a Glagolitic seminary was opened in Split in the 18th century, aside from a period of time in the parish of Kučiće-Vinišće.

Decline in Croatia
The Ottoman Empire's repeated incursions into Croatia in the 15th and 16th centuries posed the first major existential threat to the script's survival. The Counter-Reformation, alongside other factors, led to the suppression of Glagolitic in Istria in the 16th–17th centuries as well as in the Zagreb archdiocese. The Latinisation of the coastal cities and islands took much longer, where the script continued to be used by the notaries of Krk into the first decade of the 19th century, with education by rural chapters on that island ensuring the survival of the script until well after their abolition by the Napoleon administration in the second decade of the 19th century. Novitiates continued to be educated primarily in the Glagolitic script as late as the third decade of the 19th century. But without centres of education, Latin script and Italian rapidly took over, so that very little was written in the script after the third quarter of the 19th century except for ceremonial purposes, and soon very few could read the cursive script apart from a few scholars.

Academic debates
The exact nature of relationship between the Glagolitic alphabet and the Early Cyrillic alphabet, their order of development, and influence on each other has been a matter of great study, controversy, and dispute in Slavic studies since the 19th century.

Versions of authorship and name
A once common belief was that the Glagolitic was created or used in the 4th century by St. Jerome, hence the alphabet was sometimes named "Hieronymian".

It has also acrophonically been called azbuka from the names of its first two letters, on the same model as "alpha" + "beta" (the same name can also refer to Cyrillic and in some modern languages it simply means "alphabet" in general). The Slavs of Great Moravia (present-day Slovakia and Moravia), Hungary, Slovenia and Slavonia were called Slověne at that time, which gives rise to the name "Slovenish" for the alphabet. Some other, rarer, names for this alphabet are Bukvitsa (from common Slavic word "bukva" meaning "letter", and a suffix "-itsa") and "Illyrian" (presumably similar to using the same anachronistic name for the Illyrian (Slavic) language).

In the Middle Ages, Glagolitsa was also known as "St. Jerome's script" due to a popular mediaeval legend (created by Croatian scribes in the 13th century) ascribing its invention to St. Jerome (342–429). The legend was partly based on the saint's place of birth on the border of Dalmatia and Pannonia. He was viewed as a "compatriot" and anachronistically as belonging to the same ethnic group; this helped the spread of the cult of the saint in Dalmatia and was later used to support the idea of the presence of Slavic communities in the Eastern Adriatic Coast from ancient times, but the legend was probably firstly introduced for other reasons, like giving a more solid religious justification for the use of this script and Slavic liturgy. The theory nevertheless gained much popularity and spread to other countries before being resolutely disproven.

"Until the end of the 18th century, a strange but widespread opinion dominated that the Glagolitic writing system, which was in use in Dalmatia and Istria along with neighboring islands, including the translation of the Holy Scripture, owe their existence to the famous church father St. Jerome. Knowing him as the author of the Latin Vulgate, considering him – by his own words, born on the border between Dalmatia and Pannonia (remembering that the Dalmatian borders extended well into Istria at that time) – presumed to be an Illyrian, the self-styled Slavic intellectuals in Dalmatia very early began to ascribe to him the invention of glagolitsa, possibly with the intention of more successfully defending both Slavic writing and the Slavic holy service against prosecutions and prohibitions from Rome's hierarchy, thus using the opinion of the famous Latin Father of the Church to protect their church rituals which were inherited not from the Greeks Cyril and Methodius but unknown. We do not know who was the first to put in motion this unscientifically-based tradition about Jerome's authorship of the Glagolitic script and translation of the Holy Scripture, but in 1248 this version came to the knowledge of Pope Innocent IV. <...> The belief in Jerome as an inventor of the Glagolitic lasted many centuries, not only in his homeland, i.e. in Dalmatia and Croatia, not only in Rome, due to Slavs living there... but also in the West. In the 14th century, Croatian monks brought the legend to the Czechs, and even the Emperor Charles IV believed them."

The epoch of traditional attribution of the script to Jerome ended probably in 1812. In modern times, only certain marginal authors share this view, usually "re-discovering" one of the already-known mediaeval sources.

Characteristics
The phonetic values of many of the letters are thought to have been displaced under Cyrillic influence or to have become confused through the early spread to different dialects, so the original values are not always clear. For instance, the letter yu Ⱓ is thought to have perhaps originally had the sound /u/ but was displaced by the adoption of the ligature Ⱆ under the influence of later Cyrillic oѵ, mirroring the Greek ου. Other letters were late creations after a Cyrillic model. It should also be noted that Ⱑ corresponds to two different Cyrillic letters (Ѣ and Я), present even in older manuscripts, and not to different later variants of the same Cyrillic letter in different times or places.

The following table lists each letter in its modern order, showing its Unicode representation, images of the letter in both the round and angular/squared variant forms, the corresponding modern Cyrillic letter, the approximate sound transcribed with the IPA, the name, and suggestions for its origin. The Old Church Slavonic names follow the scientific transliteration, while the mostly similar Church Slavonic ones follow an approach more familiar to a generic English speaking reader. Several letters have no modern counterpart. The column for the angular variant, sometimes referred to as Croatian Glagolitic, is not complete as some of the letters were not used following the Croatian recension of Old Church Slavonic.

In older texts, uk and three out of four yuses  also can be written as digraphs, in two separate parts.

The order of izhe and i  varies from source to source, as does the order of the various forms of yus. Correspondence between Glagolitic izhe and i  with Cyrillic И and І is unknown.

The Proto-Slavic language did not have the phoneme /f/, and the letters fert and fita  were used for transcribing words of Greek origin, and so was izhitsa  for the Greek upsilon.

Unicode
The Glagolitic alphabet was added to the Unicode Standard in March 2005 with the release of version 4.1.

The Unicode block for Glagolitic is U+2C00–U+2C5F.

The Glagolitic combining letters for Glagolitic Supplement block (U+1E000–U+1E02F) was added to the Unicode Standard in June, 2016 with the release of version 9.0:

Pre-Glagolitic Slavic writing systems
A hypothetical pre-Glagolitic writing system is typically referred to as cherty i rezy (strokes and incisions) – but no material evidence of the existence of any pre-Glagolitic Slavic writing system has been found, except for a few brief and vague references in old chronicles and "lives of the saints". All artifacts presented as evidence of pre-Glagolitic Slavic inscriptions have later been identified as texts in known scripts and in known non-Slavic languages, or as fakes. The well-known Chernorizets Hrabar's strokes and incisions are usually considered to be a reference to a kind of property mark or alternatively fortune-telling signs. Some "Ruthenian letters" found in one version of St. Cyril's life are explainable as misspelled "Syrian letters" (in Slavic, the roots are very similar: rus- vs. sur- or syr-), etc.

In popular culture
Glagolitic script is the writing system used in the world of The Witcher books and video game series. It is also featured, in various uses, in several of the point and click adventure games made by Cateia Games, a Croatian game studio.

In the 2023 PS5 game Forspoken, Athian script, the written language of the Athian continent and cultures, seems to be based upon Glagolitic script.

It is also featured on 1 euro cent, 2 euro cent and 5 euro cent coins minted in Croatia.

Literature

 * Branko Franolić, and Mateo Žagar: A Historical Outline of Literary Croatian and The Glagolitic Heritage of Croatian Culture. Erasmus, Zagreb 2008. ISBN 978-953-6132-80-5
 * Fučić, Branko: Glagoljski natpisi. Croatian Academy of Sciences and Arts, Zagreb 1982.
 * Fullerton, Sharon Golke: Paleographic Methods Used in Dating Cyrillic and Glagolitic Manuscripts. Ohio State University, Columbus 1971.
 * Jagić, Vatroslav: Gramatika jezika hèrvatskoga. A. Jakić, Zagreb 1864.
 * Marko Japundžić: Hrvatska glagoljica. Hrvatska uzdanica, Zagreb 1998.
 * Marko Japundžić: Tragom hrvatskog glagolizma. Kršćanska sadašnjost, Zagreb 1995.
 * Heinz Miklas, Sylvia Richter, and Velizar Sadovski : Glagolitica. Österreichische Akademie der Wissenschaften, Vienna 2000. ISBN 3700128959
 * Josef Vajs: Abecedarium palaeoslovenicum in usum glagolitarum. Staroslavenska akademija, Krk 1917.
 * Josef Vajs: Rukověť hlaholské paleografie. Orbis, Prague 1932.
 * Darko Žubrinić: Crtice iz povijesti glagoljice. Hrvatsko književno društvo sv. Jeronima, Zagreb 1994. ISBN 953-6111-15-2

title: Glagolitic script


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Greatest common divisor

In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers $x$, $y$, the greatest common divisor of $x$ and $y$ is denoted $$\gcd (x,y)$$. For example, the GCD of 8 and 12 is 4, that is, $gcd(8, 12) = 4$.

In the name "greatest common divisor", the adjective "greatest" may be replaced by "highest", and the word "divisor" may be replaced by "factor", so that other names include highest common factor, etc. Historically, other names for the same concept have included greatest common measure.

This notion can be extended to polynomials (see Polynomial greatest common divisor) and other commutative rings (see  below).

Definition
The greatest common divisor (GCD) of integers $a$ and $b$, at least one of which is nonzero, is the greatest positive integer $d$ such that $d$ is a divisor of both $a$ and $b$; that is, there are integers $e$ and $f$ such that $a = de$ and $b = df$, and $d$ is the largest such integer. The GCD of $a$ and $b$ is generally denoted $gcd(a, b)$.

When one of $(a, b)$ and $(a, b)$ is zero, the GCD is the absolute value of the nonzero integer: $g.c.d.(a, b)$. This case is important as the terminating step of the Euclidean algorithm.

The above definition is unsuitable for defining $(a, b)$, since there is no greatest integer $a$ such that $b$. However, zero is its own greatest divisor if greatest is understood in the context of the divisibility relation, so $gcd(a, 0) = gcd(0, a) = |a|$ is commonly defined as $gcd(0, 0)$. This preserves the usual identities for GCD, and in particular Bézout's identity, namely that $n$ generates the same ideal as $0 × n = 0$. This convention is followed by many computer algebra systems. Nonetheless, some authors leave $gcd(0, 0)$ undefined.

The GCD of $a$ and $b$ is their greatest positive common divisor in the preorder relation of divisibility. This means that the common divisors of $a$ and $b$ are exactly the divisors of their GCD. This is commonly proved by using either Euclid's lemma, the fundamental theorem of arithmetic, or the Euclidean algorithm. This is the meaning of "greatest" that is used for the generalizations of the concept of GCD.

Example
The number 54 can be expressed as a product of two integers in several different ways:
 * $$ 54 \times 1 = 27 \times 2 = 18 \times 3 = 9 \times 6.$$

Thus the complete list of divisors of 54 is 1, 2, 3, 6, 9, 18, 27, 54. Similarly, the divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. The numbers that these two lists have in common are the common divisors of 54 and 24, that is,
 * $$ 1, 2, 3, 6. $$

Of these, the greatest is 6, so it is the greatest common divisor:
 * $$ \gcd(54,24) = 6. $$

Computing all divisors of the two numbers in this way is usually not efficient, especially for large numbers that have many divisors. Much more efficient methods are described in .

Coprime numbers
Two numbers are called relatively prime, or coprime, if their greatest common divisor equals $0$. For example, 9 and 28 are coprime.

A geometric view
For example, a 24-by-60 rectangular area can be divided into a grid of: 1-by-1 squares, 2-by-2 squares, 3-by-3 squares, 4-by-4 squares, 6-by-6 squares or 12-by-12 squares. Therefore, 12 is the greatest common divisor of 24 and 60. A 24-by-60 rectangular area can thus be divided into a grid of 12-by-12 squares, with two squares along one edge ($gcd(a, b)$) and five squares along the other ($\{a, b\}$).

Reducing fractions
The greatest common divisor is useful for reducing fractions to the lowest terms. For example, $gcd(0, 0)$, therefore,
 * $$\frac{42}{56}=\frac{3 \cdot 14 }{ 4 \cdot 14}=\frac{3 }{ 4}.$$

Least common multiple
The least common multiple of two integers that are not both zero can be computed from their greatest common divisor, by using the relation
 * $$\operatorname{lcm}(a,b)=\frac{|a\cdot b|}{\operatorname{gcd}(a,b)}.$$

Using prime factorizations
Greatest common divisors can be computed by determining the prime factorizations of the two numbers and comparing factors. For example, to compute $1$, we find the prime factorizations 48 = 24 · 31 and 180 = 22 · 32 · 51; the GCD is then 2min(4,2) · 3min(1,2) · 5min(0,1) = 22 · 31 · 50 = 12 The corresponding LCM is then 2max(4,2) · 3max(1,2) · 5max(0,1) = 24 · 32 · 51 = 720.

In practice, this method is only feasible for small numbers, as computing prime factorizations takes too long.

Euclid's algorithm
The method introduced by Euclid for computing greatest common divisors is based on the fact that, given two positive integers $a$ and $b$ such that $24/12 = 2$, the common divisors of $a$ and $b$ are the same as the common divisors of $60/12 = 5$ and $b$.

So, Euclid's method for computing the greatest common divisor of two positive integers consists of replacing the larger number with the difference of the numbers, and repeating this until the two numbers are equal: that is their greatest common divisor.

For example, to compute $gcd(42, 56) = 14$, one proceeds as follows:
 * $$\begin{align}\gcd(48,18)\quad&\to\quad \gcd(48-18, 18)= \gcd(30,18)&&\to \quad \gcd(30-18, 18)= \gcd(12,18)\\

&\to \quad \gcd(12,18-12)= \gcd(12,6)&&\to \quad \gcd(12-6,6)= \gcd(6,6).\end{align}$$ So $gcd(48, 180)$.

This method can be very slow if one number is much larger than the other. So, the variant that follows is generally preferred.

Euclidean algorithm

A more efficient method is the Euclidean algorithm, a variant in which the difference of the two numbers $a$ and $b$ is replaced by the remainder of the Euclidean division (also called division with remainder) of $a$ by $b$.

Denoting this remainder as $a > b$, the algorithm replaces $a – b$ with $gcd(48,18)$ repeatedly until the pair is $gcd(48, 18) = 6$, where $d$ is the greatest common divisor.

For example, to compute gcd(48,18), the computation is as follows:
 * $$\begin{align}\gcd(48,18)\quad&\to\quad \gcd(18, 48\bmod 18)= \gcd(18, 12)\\

&\to \quad \gcd(12, 18\bmod 12)= \gcd(12,6)\\ &\to \quad \gcd(6,12\bmod 6)= \gcd(6,0).\end{align}$$ This again gives $a mod b$.

Binary GCD algorithm
The binary GCD algorithm is a variant of Euclid's algorithm that is specially adapted to the binary representation of the numbers, which is used in most computers.

The binary GCD algorithm differs from Euclid's algorithm essentially by dividing by two every even number that is encountered during the computation. Its efficiency results from the fact that, in binary representation, testing parity consists of testing the right-most digit, and dividing by two consists of removing the right-most digit.

The method is as follows, starting with $(a, b)$ and $(b, a mod b)$ that are the two positive integers whose GCD is sought.
 * 1) If $(d, 0)$ and $gcd(48, 18) = 6$ are both even, then divide both by two until at least one of them becomes odd; let $d$ be the number of these paired divisions.
 * 2) If $a$ is even, then divide it by two until it becomes odd.
 * 3) If $b$ is even, then divide it by two until it becomes odd.
 * Now, $a$ and $b$ are both odd and will remain odd until the end of the computation
 * 1) While $a$ do
 * 2) * If $b$, then replace $a$ with $a$ and divide the result by two until $a$ becomes odd (as $b$ and $a ≠ b$ are both odd, there is, at least, one division by 2).
 * 3) * If $a > b$, then replace $b$ with $a – b$ and divide the result by two until $b$ becomes odd.
 * 4) Now, $a$, and the greatest common divisor is  $$2^d a.$$

Step 1 determines $d$ as the highest power of $b$ that divides $a < b$ and $b – a$, and thus their greatest common divisor. None of the steps changes the set of the odd common divisors of $a = b$ and $2$. This shows that when the algorithm stops, the result is correct. The algorithm stops eventually, since each steps divides at least one of the operands by at least $a$. Moreover, the number of divisions by $b$ and thus the number of subtractions is at most the total number of digits.

Example: (a, b, d) = (48, 18, 0) → (24, 9, 1) → (12, 9, 1) → (6, 9, 1) → (3, 9, 1) → (3, 3, 1) ; the original GCD is thus the product 6 of $a$ and $b$.

The binary GCD algorithm is particularly easy to implement and particularly efficient on binary computers. Its computational complexity is
 * $$O((\log a + \log b)^2).$$

The square in this complexity comes from the fact that division by $2$ and subtraction take a time that is proportional to the number of bits of the input.

The computational complexity is usually given in terms of the length $2$ of the input. Here, this length is $2^{d} = 2^{1}$, and the complexity is thus
 * $$O(n^2)$$.

Lehmer's GCD algorithm
Lehmer's algorithm is based on the observation that the initial quotients produced by Euclid's algorithm can be determined based on only the first few digits; this is useful for numbers that are larger than a computer word. In essence, one extracts initial digits, typically forming one or two computer words, and runs Euclid's algorithms on these smaller numbers, as long as it is guaranteed that the quotients are the same with those that would be obtained with the original numbers. The quotients are collected into a small 2-by-2 transformation matrix (a matrix of single-word integers) to reduce the original numbers. This process is repeated until numbers are small enough that the binary algorithm (see below) is more efficient.

This algorithm improves speed, because it reduces the number of operations on very large numbers, and can use hardware arithmetic for most operations. In fact, most of the quotients are very small, so a fair number of steps of the Euclidean algorithm can be collected in a 2-by-2 matrix of single-word integers. When Lehmer's algorithm encounters a quotient that is too large, it must fall back to one iteration of Euclidean algorithm, with a Euclidean division of large numbers.

Other methods

If $a = b = 3$ and $2$ are both nonzero, the greatest common divisor of $n$ and $n = log a + log b$ can be computed by using least common multiple (LCM) of $a$ and $b$:
 * $$\gcd(a,b)=\frac{|a\cdot b|}{\operatorname{lcm}(a,b)}$$,

but more commonly the LCM is computed from the GCD.

Using Thomae's function $a$,
 * $$\gcd(a,b) = a f\left(\frac b a\right),$$

which generalizes to $b$ and $a$ rational numbers or commensurable real numbers.

Keith Slavin has shown that for odd $b$:
 * $$\gcd(a,b)=\log_2\prod_{k=0}^{a-1} (1+e^{-2i\pi k b/a})$$

which is a function that can be evaluated for complex b. Wolfgang Schramm has shown that
 * $$\gcd(a,b)=\sum\limits_{k=1}^a \exp (2\pi ikb/a) \cdot \sum\limits_{d\left| a\right.} \frac{c_d (k)}{d} $$

is an entire function in the variable b for all positive integers a where cd(k) is Ramanujan's sum.

Complexity
The computational complexity of the computation of greatest common divisors has been widely studied. If one uses the Euclidean algorithm and the elementary algorithms for multiplication and division, the computation of the greatest common divisor of two integers of at most $n$ bits is $f$. This means that the computation of greatest common divisor has, up to a constant factor, the same complexity as the multiplication.

However, if a fast multiplication algorithm is used, one may modify the Euclidean algorithm for improving the complexity, but the computation of a greatest common divisor becomes slower than the multiplication. More precisely, if the multiplication of two integers of $a$ bits takes a time of $b$, then the fastest known algorithm for greatest common divisor has a complexity $a ≥ 1$. This implies that the fastest known algorithm has a complexity of $O(n^{2})$.

Previous complexities are valid for the usual models of computation, specifically multitape Turing machines and random-access machines.

The computation of the greatest common divisors belongs thus to the class of problems solvable in quasilinear time. A fortiori, the corresponding decision problem belongs to the class P of problems solvable in polynomial time. The GCD problem is not known to be in NC, and so there is no known way to parallelize it efficiently; nor is it known to be P-complete, which would imply that it is unlikely to be possible to efficiently parallelize GCD computation. Shallcross et al. showed that a related problem (EUGCD, determining the remainder sequence arising during the Euclidean algorithm) is NC-equivalent to the problem of integer linear programming with two variables; if either problem is in NC or is P-complete, the other is as well. Since NC contains NL, it is also unknown whether a space-efficient algorithm for computing the GCD exists, even for nondeterministic Turing machines.

Although the problem is not known to be in NC, parallel algorithms asymptotically faster than the Euclidean algorithm exist; the fastest known deterministic algorithm is by Chor and Goldreich, which (in the CRCW-PRAM model) can solve the problem in $n$ time with $T(n)$ processors. Randomized algorithms can solve the problem in $O(T(n) log n)$ time on $$\exp\left(O\left(\sqrt{n \log n}\right)\right)$$ processors (this is superpolynomial).

Properties

 * For positive integers $O(n (log n)^{2})$, $O(n/log n)$.
 * Every common divisor of $n^{1+ε}$ and $O((log n)^{2})$ is a divisor of $a$.
 * $gcd(a, a) = a$, where a and b are not both zero, may be defined alternatively and equivalently as the smallest positive integer d which can be written in the form $a$, where p and q are integers. This expression is called Bézout's identity. Numbers p and q like this can be computed with the extended Euclidean algorithm.
 * $b$, for $gcd(a, b)$, since any number is a divisor of 0, and the greatest divisor of a is $gcd(a, b)$. This is usually used as the base case in the Euclidean algorithm.
 * If a divides the product b⋅c, and $d = a⋅p + b⋅q$, then a/d divides c.
 * If m is a positive integer, then $gcd(a, 0) = |a|$.
 * If m is any integer, then $a ≠ 0$. Equivalently, $|a|$.
 * If m is a positive common divisor of a and b, then $gcd(a, b) = d$.
 * The GCD is a commutative function: $gcd(m⋅a, m⋅b) = m⋅gcd(a, b)$.
 * The GCD is an associative function: $gcd(a + m⋅b, b) = gcd(a, b)$. Thus $gcd(a mod b,b) = gcd(a,b)$ can be used to denote the GCD of multiple arguments.
 * The GCD is a multiplicative function in the following sense: if a1 and a2 are relatively prime, then $gcd(a/m, b/m) = gcd(a, b)/m$.
 * $gcd(a, b) = gcd(b, a)$ is closely related to the least common multiple $gcd(a, gcd(b, c)) = gcd(gcd(a, b), c)$: we have
 * This formula is often used to compute least common multiples: one first computes the GCD with Euclid's algorithm and then divides the product of the given numbers by their GCD.
 * This formula is often used to compute least common multiples: one first computes the GCD with Euclid's algorithm and then divides the product of the given numbers by their GCD.

=n\prod_{p|n}\left(1+\nu_p(n)\left(1-\frac{1}{p}\right)\right)$$ where $$\nu_p(n)$$ is the $gcd(a, b, c, ...)$-adic valuation.
 * The following versions of distributivity hold true:
 * If we have the unique prime factorizations of $gcd(a_{1}⋅a_{2}, b) = gcd(a_{1}, b)⋅gcd(a_{2}, b)$ and $gcd(a, b)$ where $lcm(a, b)$ and $gcd(a, b)⋅lcm(a, b) = |a⋅b|$, then the GCD of a and b is
 * It is sometimes useful to define $gcd(a, lcm(b, c)) = lcm(gcd(a, b), gcd(a, c))$ and $lcm(a, gcd(b, c)) = gcd(lcm(a, b), lcm(a, c))$ because then the natural numbers become a complete distributive lattice with GCD as meet and LCM as join operation. This extension of the definition is also compatible with the generalization for commutative rings given below.
 * In a Cartesian coordinate system, $a = p_{1}^{e_{1}} p_{2}^{e_{2}} ⋅⋅⋅ p_{m}^{e_{m}}|undefined$ can be interpreted as the number of segments between points with integral coordinates on the straight line segment joining the points $b = p_{1}^{f_{1}} p_{2}^{f_{2}} ⋅⋅⋅ p_{m}^{f_{m}}|undefined$ and $e_{i} ≥ 0$.
 * For non-negative integers $f_{i} ≥ 0$ and $gcd(a,b) = p_{1}^{min(e_{1},f_{1})} p_{2}^{min(e_{2},f_{2})} ⋅⋅⋅ p_{m}^{min(e_{m},f_{m})}$, where $gcd(0, 0) = 0$ and $lcm(0, 0) = 0$ are not both zero, provable by considering the Euclidean algorithm in base n:
 * An identity involving Euler's totient function:
 * $$ \gcd(a,b) = \sum_{k|a \text{ and }k|b} \varphi(k) .$$
 * $$\sum_{k=1}^n \gcd(k,n)=n\sum_{d|n}\frac{\varphi(d)}{d}
 * An identity involving Euler's totient function:
 * $$ \gcd(a,b) = \sum_{k|a \text{ and }k|b} \varphi(k) .$$
 * $$\sum_{k=1}^n \gcd(k,n)=n\sum_{d|n}\frac{\varphi(d)}{d}
 * $$\sum_{k=1}^n \gcd(k,n)=n\sum_{d|n}\frac{\varphi(d)}{d}

Probabilities and expected value
In 1972, James E. Nymann showed that $gcd(a, b)$ integers, chosen independently and uniformly from $(0, 0)$, are coprime with probability $(a, b)$ as $a$ goes to infinity, where $b$ refers to the Riemann zeta function. (See coprime for a derivation.) This result was extended in 1987 to show that the probability that $a$ random integers have greatest common divisor $b$ is $gcd(n^{a} − 1, n^{b} − 1) = n^{gcd(a,b)} − 1$.

Using this information, the expected value of the greatest common divisor function can be seen (informally) to not exist when $p$. In this case the probability that the GCD equals $k$ is $\{1, ..., n\}$, and since $1/ζ(k)$ we have
 * $$\mathrm{E}( \mathrm{2} ) = \sum_{d=1}^\infty d \frac{6}{\pi^2 d^2} = \frac{6}{\pi^2} \sum_{d=1}^\infty \frac{1}{d}.$$

This last summation is the harmonic series, which diverges. However, when $n$, the expected value is well-defined, and by the above argument, it is
 * $$ \mathrm{E}(k) = \sum_{d=1}^\infty d^{1-k} \zeta(k)^{-1} = \frac{\zeta(k-1)}{\zeta(k)}. $$

For $ζ$, this is approximately equal to 1.3684. For $k$, it is approximately 1.1106.

In commutative rings
The notion of greatest common divisor can more generally be defined for elements of an arbitrary commutative ring, although in general there need not exist one for every pair of elements.

If $R$ is a commutative ring, and $a$ and $b$ are in $R$, then an element $d$ of $R$ is called a common divisor of $a$ and $b$ if it divides both $a$ and $b$ (that is, if there are elements $x$ and $y$ in $R$ such that d·x = a and d·y = b). If $d$ is a common divisor of $a$ and $b$, and every common divisor of $a$ and $b$ divides $d$, then $d$ is called a greatest common divisor of $a$ and b.

With this definition, two elements $a$ and $b$ may very well have several greatest common divisors, or none at all. If $R$ is an integral domain then any two GCD's of $a$ and $b$ must be associate elements, since by definition either one must divide the other; indeed if a GCD exists, any one of its associates is a GCD as well. Existence of a GCD is not assured in arbitrary integral domains. However, if $R$ is a unique factorization domain, then any two elements have a GCD, and more generally this is true in GCD domains. If $R$ is a Euclidean domain in which euclidean division is given algorithmically (as is the case for instance when $k$ where $F$ is a field, or when $R$ is the ring of Gaussian integers), then greatest common divisors can be computed using a form of the Euclidean algorithm based on the division procedure.

The following is an example of an integral domain with two elements that do not have a GCD:
 * $$R = \mathbb{Z}\left[\sqrt{-3}\,\,\right],\quad a = 4 = 2\cdot 2 = \left(1+\sqrt{-3}\,\,\right)\left(1-\sqrt{-3}\,\,\right),\quad b = \left(1+\sqrt{-3}\,\,\right)\cdot 2.$$

The elements $d$ and $d^{−k}/ζ(k)$ are two maximal common divisors (that is, any common divisor which is a multiple of $k = 2$ is associated to $d$, the same holds for $d^{−2}/ζ(2)$, but they are not associated, so there is no greatest common divisor of $a$ and $ζ(2) = π^{2}/6$.

Corresponding to the Bézout property we may, in any commutative ring, consider the collection of elements of the form $k ≥ 3$, where $p$ and $q$ range over the ring. This is the ideal generated by $a$ and $b$, and is denoted simply $k = 3$. In a ring all of whose ideals are principal (a principal ideal domain or PID), this ideal will be identical with the set of multiples of some ring element $k = 4$; then this $d$ is a greatest common divisor of $a$ and $R = F[X]$. But the ideal $2$ can be useful even when there is no greatest common divisor of $a$ and $1 + √−3$. (Indeed, Ernst Kummer used this ideal as a replacement for a GCD in his treatment of Fermat's Last Theorem, although he envisioned it as the set of multiples of some hypothetical, or ideal, ring element $d$, whence the ring-theoretic term.)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Greatest common divisor


text:
Gazpacho

Gazpacho or gaspacho, also called Andalusian gazpacho, is a cold soup and drink made of raw, blended vegetables. It originated in the southern regions of the Iberian peninsula and spread into other areas. Gazpacho is widely eaten in Spain and Portugal, particularly during hot summers, since it is refreshing and cool.

Although there are other recipes called gazpacho, such as gazpacho manchego, the standard usage implies a soup. There are also a number of dishes that are closely related and often considered variants thereof, such as ajoblanco, salmorejo, pipirrana, porra antequerana (closer to a bread soup), and cojondongo.

History
There are many theories as to the origin of gazpacho, including one that says it was a soup of bread, olive oil, water, vinegar, and garlic that arrived in Spain with the Romans. The word "gazpacho" may come from the Latin adjective caccabaceus, derived from caccabus ("cauldron"), attested in the works of Tertulian, Zeno of Verona and others. This word was applied in ancient Rome to a type of bread very similar to the torta de gazpacho. Once in Spain, it became a part of southern cuisine, particularly in Córdoba, Seville or Granada Castilian kingdoms, using stale bread, garlic, olive oil, salt, and vinegar, similar to ajoblanco. During the 19th century, red gazpacho was created when tomatoes were added to the ingredients. This version spread internationally and remains commonly known.

There are many modern variations of gazpacho with avocados, cucumbers, parsley, strawberries, watermelon, grapes, meat stock, seafood, and other ingredients instead of tomatoes and bread.

Ingredients and preparation
Most gazpacho includes stale bread, tomato, cucumbers, onion, bell peppers, garlic, olive oil, wine vinegar, water, and salt. Northern recipes often include cumin and/or pimentón (smoked sweet paprika).

Traditionally, gazpacho was made by pounding the vegetables in a mortar with a pestle; this more laborious method is still sometimes used as it helps keep the gazpacho cool and avoids the foam and silky consistency of smoothie versions made in blenders or food processors. A traditional way of preparation is to pound garlic cloves in a mortar, add a little soaked stale bread, then olive oil and salt, to make a paste. Next, very ripe tomatoes and vinegar are added to this paste. In the days before refrigeration, the gazpacho was left in an unglazed earthenware pot to cool by evaporation, with the addition of some water.

Gazpacho may be served alone or with garnishes, such as hard-boiled eggs, chopped ham (in the salmorejo variety from Córdoba), chopped almonds, cumin crushed with mint, orange segments, finely chopped green bell peppers, onion, tomato or cucumber. In Extremadura, local ham was added to the gazpacho itself rather than as a garnish; this is called gazpacho extremeño. Andalusian sources say that gazpacho should be slightly chilled, but not iced.

Variations

The ingredients, texture, and thickness of gazpacho vary regionally and between different cooks.

Similar cold raw soups such as arjamolho in Portugal, porra antequerana and ajoblanco, are also popular in Andalusia, although not as widespread as gazpacho. Gazpacho and salmorejo are especially similar since they are both tomato-based cold soups that are widely popular in Spain; the main difference between gazpacho and salmorejo is the culinary technique used since gazpacho is a soup whereas salmorejo is an emulsion. In addition, while both dishes share the main ingredients of tomato, olive oil, bread, and garlic, gazpacho can also be prepared with cucumber, peppers, and vinegar, whereas salmorejo cannot.

Gazpacho manchego, despite its name, is a meat stew, served hot, not a variation on the cold vegetable soup.

In Spain
The original recipe using bread, water, vinegar, oil, and salt is traditional in the Iberian Peninsula, perhaps going back to Roman times. Every central and southern region has its own variety. The humble gazpacho became a very deeply rooted food for peasants and shepherds in Spain. The basic gazpacho gave rise to many variants, some also called gazpacho, others not; some authors have tried to classify all these variations. Gazpachos may be classified by colour: the most usual red ones (which contain tomato), white ones (which contain no tomato, but include dried fruits), and green ones (which are white but contain some spices that make them green). These variants have their basic ingredients in common, including garlic paste which works as an emulsifier, bread, olive oil, vinegar and salt. In addition to the traditional ingredients, red fruits such as strawberries, melon, etc., may be added, making the gazpacho a bit sweeter. Gazpacho may be served as a starter, main dish, or tapa.

Arranque roteño
A popular variation comes from the town of Rota in the province of Cádiz. During times of drought, there was not enough water to make gazpacho; thus, arranque has the same ingredients as gazpacho, but requires less water and bread, making it a sort of cream. Some people add more bread until it takes on the consistency of a dip.

Extremaduran variations
In Extremadura, gazpachos are a kind of purée or thick gazpacho known as cojondongo, or cojondongo del gañán, made of breadcrumbs, garlic, oil, and vinegar, then topped with chopped onions, tomato and peppers.

La Mancha variations
Gazpacho manchego, as its name implies, is made in the east region of La Mancha, in Albacete and nearby areas, and is popular in other areas in the center and southwest of the country.

It is a meat stew, whose main ingredients are small game animals or birds such as rabbit, hare, quail, or pigeon, and flatbread, and may include garlic, tomatoes, and mushrooms. It is cooked in a cauldron and served hot. Another well-known variant in La Mancha is gazpacho de pastor or galiano.

Some other hot meat or fish dishes from other regions are called gazpacho (gazpacho jumillano, gazpacho de Yecla, gazpacho de Requena, etc.)

Castilian variations
Gazpacho is often eaten during the very hot and dry summers in Castilla y León. The gazpacho made in La Moraña in the province of Ávila has large pieces of vegetables floating in a watery soup.

title: Gazpacho


text:
Gopher (disambiguation)

A gopher, also known as a "pocket gopher" (family Geomyidae), is a burrowing rodent native to North America and Central America.

Gopher may also refer to:

Nature

 * Some species of ground squirrels (tribe Marmotini) of North America, particularly those formerly classified as Spermophilus, are informally referred to as "gophers"
 * Gopher snake – common name of several species of genus Pituophis endemic to west or southwest of North America
 * Gopher tortoise (genus Gopherus), distributed in North America
 * Gopher wood, of unclear meaning, mentioned in the Bible as the building material for Noah's ark
 * Gopher Plant or Paper Spurge (Euphorbia lathyris)

Entertainment

 * Gophers!, a British children's television programme
 * "Gopher", real name Burl Smith, a character on TV show The Love Boat
 * Gopher (Winnie the Pooh), a character in Walt Disney's Winnie the Pooh franchise
 * Gordon the Gopher, an English puppet gopher who first appeared on Children's BBC (CBBC) between 1985 and 1987

Technology

 * The SA-13 "Gopher", or 9K35 Strela-10, a Soviet surface-to-air missile system
 * Gopher (protocol), an early distributed hypertext protocol
 * Go gopher, a mascot of Go programming language
 * Slang term for a mobility scooter

Other

 * Minnesota, the "Gopher State"
 * Minnesota Golden Gophers, University of Minnesota sports teams
 * Gopher (train), a passenger train operated by the Great Northern Railway, USA
 * Gopher (video game), an Atari 2600 game
 * Gofer, also spelled "gopher", an errand-runner
 * Gopher Gang, early 20th-century New York street gang

title: Gopher (disambiguation)


text:
Gnome

A gnome is a mythological creature and diminutive spirit in Renaissance magic and alchemy, introduced by Paracelsus in the 16th century and widely adopted by authors including those of modern fantasy literature. Typically small humanoids who live underground, gnome characteristics are reinterpreted to suit various storytellers and artists.

Lawn ornaments crafted as gnomes were introduced during the 19th century, growing in popularity during the 20th century as garden gnomes.

Origins
The word comes from Renaissance Latin gnomus, which first appears in A Book on Nymphs, Sylphs, Pygmies, and Salamanders, and on the Other Spirits by Paracelsus, published posthumously in Nysa in 1566 (and again in the Johannes Huser edition of 1589–1591 from an autograph by Paracelsus).

The term may be an original invention of Paracelsus, possibly deriving the term from Latin gēnomos (itself representing a Greek γη-νομος, approximately "gē-nomos", literally "earth-dweller"). In this case, the omission of the ē is referred to as a blunder by the Oxford English Dictionary (OED). Paracelsus uses Gnomi as a synonym of Pygmæi and classifies them as earth elementals. He describes them as two spans high, very reluctant to interact with humans and able to move through solid earth as easily as humans move through air.

The chthonic or earth-dwelling spirit has precedents in numerous ancient and medieval mythologies, often guarding mines and precious underground treasures, notably in the Germanic dwarfs and the Greek Chalybes, Telchines or Dactyls. The gnomes of Swiss folklore follow this template, as they are said to have caused the landslide that destroyed the Swiss village of Plurs in 1618 - the villagers had become wealthy from a local gold mine created by the gnomes, who poured liquid gold down into a vein for the benefit of humans, and were corrupted by this newfound prosperity, which greatly offended the gnomes.

In Romanticism and modern fairy tales

The English word is attested from the early 18th century. Gnomes are used in Alexander Pope's "The Rape of the Lock". The creatures from this mock-epic are small, celestial creatures that were prudish women in their past lives, and now spend all of eternity looking out for prudish women (in parallel to the guardian angels in Catholic belief). Other uses of the term gnome remain obscure until the early 19th century, when it is taken up by authors of Romanticist collections of fairy tales and becomes mostly synonymous with the older word goblin.

Pope's stated source, the 1670 French satire Comte de Gabalis by Nicolas-Pierre-Henri de Montfaucon de Villars, the abbot of Villars, describes gnomes as such: "The Earth is filled almost to the center with Gnomes or Pharyes, a people of small stature, the guardians of treasures, of mines, and of precious stones. They are ingenious, friends of men, and easie to be commandded. They furnish the children of the Sages with as much money, as they have need of; and never ask any other reward of their services, than the glory of being commanded. The Gnomides or wives of these Gnomes or Pharyes, are little, but very handsom; and their habit marvellously curious." De Villars used the term gnomide to refer to female gnomes (often "gnomid" in English translations). Modern fiction instead uses the word "gnomess" to refer to female gnomes.

In 19th-century fiction, the chthonic gnome became a sort of antithesis to the more airy or luminous fairy. Nathaniel Hawthorne in Twice-Told Tales (1837) contrasts the two in "Small enough to be king of the fairies, and ugly enough to be king of the gnomes" (cited after OED). Similarly, gnomes are contrasted to elves, as in William Cullen Bryant's Little People of the Snow (1877), which has "let us have a tale of elves that ride by night, with jingling reins, or gnomes of the mine" (cited after OED).

The Russian composer Mussorgsky  produced a movement in his work Pictures at an Exhibition, (1874)  named "Gnomus" (Latin for "The Gnome"). It is written to sound as if a gnome is moving about.

Franz Hartmann in 1895 satirized materialism in an allegorical tale entitled Unter den Gnomen im Untersberg. The English translation appeared in 1896 as Among the Gnomes: An Occult Tale of Adventure in the Untersberg. In this story, the Gnomes are still clearly subterranean creatures, guarding treasures of gold within the Untersberg mountain.

As a figure of 19th-century fairy tales, the term gnome became largely synonymous with other terms for "little people" by the 20th century, such as goblin, brownie, leprechaun and other instances of the household spirit type, losing its strict association with earth or the underground world.

Modern fantasy literature

 * Creatures called gnomes have been used in the fantasy genre of fiction and later gaming since the mid-nineteenth century, typically in a cunning role, e.g. as an inventor.
 * In L. Frank Baum's Oz books (published 1900 to 1920), the Nomes (so spelled), especially their king, are the chief adversaries of the Oz people. They are ugly, hot-tempered, immortal, round-bodied creatures with spindly limbs, long beards and wild hair, militantly collecting and protecting jewels and precious metals underground. Ruth Plumly Thompson, who continued the series (1921 to 1976) after Baum's death, reverted to the traditional spelling. He also featured gnomes in his book The Life and Adventures of Santa Claus. They watch over the rocks, their king is part of the Council of Immortals, and they created the sleigh bells for Santa Claus's reindeer.
 * J. R. R. Tolkien, in the legendarium (created 1914 to 1973) surrounding his Elves, uses "Gnomes" as the initial- but later dropped- name of the Noldor, the most gifted and technologically minded of his elvish races, in conscious exploitation of the similarity with the word gnomic. Gnome is thus Tolkien's English loan-translation of the Quenya word Noldo (plural Noldor), "those with knowledge". Tolkien's "Gnomes" are generally tall, beautiful, dark-haired, light-skinned, immortal, and wise. They are also proud, violent, and unduly admire their own creations, particularly their gemstones. Many live in cities below ground (Nargothrond) or in secluded mountain fortresses (Gondolin). He uses "Gnomes" to refer to both males and females. In The Father Christmas Letters (between 1920 and 1942), which Tolkien wrote for his children, Red Gnomes are presented as helpful creatures who come from Norway to the North Pole to assist Father Christmas and his Elves in fighting the wicked Goblins.
 * BB's The Little Grey Men (1942) is a story of the last gnomes in England, little wild men who live by hunting and fishing.
 * In C. S. Lewis's The Chronicles of Narnia (created 1950 to 1956), the gnomes are sometimes called "Earthmen". They live in the Underland, a series of caverns. Unlike the traditional, more human-like gnomes, they can have a wide variety of physical features and skin colours where some of them are either standing at 1 ft or being taller than humans. They are used as slaves by the Lady of the Green Kirtle until her defeat, at which point they return to their true home, the much deeper (and hotter) underground realm of Bism.
 * The Dutch books Gnomes (1976) and Secrets of the Gnomes (1982), written by Wil Huygen, deal with gnomes living together in harmony. These same books are the basis for a made-for-TV animated film and the Spanish-animated series The World of David the Gnome (as well as the spin-off Wisdom of the Gnomes). The word "gnome", in this case, is used in place of the Dutch kabouter.
 * In J. K. Rowling's Harry Potter series (created 1997 to 2007), gnomes are pests that inhabit the gardens of witches and wizards. They are small creatures with heads that look like potatoes on small stubby bodies. Gnomes are generally considered harmless but mischievous and may bite with sharp teeth. In the books, it is stated that the Weasleys are lenient to gnomes, and tolerate their presence, preferring to throw them out of the garden rather than more extreme measures.
 * In A. Yoshinobu's Sorcerous Stabber Orphen, the European concept of a gnome is used in order to introduce the Far Eastern notion of the Koropokkuru, a mythical indigenous race of small people: gnomes are a persecuted minority banned from learning wizardry and attending magical schools.
 * In Terry Brooks' Shannara series (created 1977 to 2017), gnomes are an offshoot race created after the Great Wars. There are several distinctive classes of gnomes. Gnomes are the smallest race. In The Sword of Shannara they are considered to be tribal and warlike, the one race that can be the most easily subverted to an evil cause. This is evidenced by their allegiance to the Warlock Lord in The Sword of Shannara and to the Mord Wraiths in The Wishsong of Shannara.
 * Terry Pratchett included gnomes in his Discworld series. Gnomes were six inches in height but quite strong, often inflicting pain upon anyone underestimating them. One prominent gnome became a Watchman in Ankh-Morpork as the force became more diversified under the command of Sam Vimes, with Buggy Swires appearing in Jingo. Another gnome in the series was Wee Mad Arthur a pest terminator in Feet of Clay.

Music

 * One of the first movements in Mussorgsky's 1874 work Pictures at an Exhibition is named "Gnomus" (Latin for "The Gnome"). It is written to sound as if a gnome is moving about, his movements constantly changing in speed.

 * "The Laughing Gnome" is a song by English musician David Bowie, released as a single in 1967. It became a hit when reissued in 1973, in the wake of Bowie's commercial success.
 * The 1970 album All Things Must Pass by English musician George Harrison has a cover image of the musician sitting among a group of garden gnomes.
 * "The Gnome" is a song by Pink Floyd on their 1967 album The Piper at the Gates of Dawn. It is about a gnome named Grimble Gromble.

Games

 * In the Dungeons & Dragons fantasy role-playing game, gnomes are one of the core races available for play as player characters. They are described as being smaller than dwarves and large-nosed. They have an affinity with small animals and a particular interest in gemstones. Depending on setting and subrace, they may also have a natural skill with illusion magic or engineering.
 * In the Warcraft franchise (1994 to present), particularly as featured in the massively multiplayer online role-playing game World of Warcraft, gnomes are a race of beings separate from but allied to dwarves and humans, with whom they share the lands of the Eastern Kingdoms. Crafty, intelligent, and smaller than their dwarven brethren, gnomes are one of two races in Azeroth regarded as technologically savvy. It is suggested in lore that the gnomes originally were mechanical creations that at some point became organic lifeforms. In World of Warcraft, gnomes are an exile race, having irradiated their home city of Gnomeregan in an unsuccessful last-ditch effort to drive out marauding foes.

Movies

 * The 1967 Walt Disney movie The Gnome-Mobile

 * The 2011 animated movie Gnomeo & Juliet

 * The 2018 animated movie Sherlock Gnomes featured gnomish versions of several classic Sherlock Holmes characters.

TV Shows

 * The Disney+ Series The Santa Clauses
 * The Little Troll Prince features the troll prince Bu, turning into a gnome and gnomes by the end of the special
 * David the Gnome, The Wisdom of Gnomes and The New World Of the Gnomes

Garden gnomes
After World War II (with early references, in ironic use, from the late 1930s) the diminutive figurines introduced as lawn ornaments during the 19th century came to be known as garden gnomes. The image of the gnome changed further during the 1960s to 1970s, when the first plastic garden gnomes were manufactured. These gnomes followed the style of the 1937 depiction of the seven dwarves in Snow White and the Seven Dwarfs by Disney. This "Disneyfied" image of the gnome was built upon by the illustrated children's book classic Gnomes (1976), in the original Dutch Leven en werken van de Kabouter, by author Wil Huygen and artist Rien Poortvliet, followed in 1981 by The Secret Book of Gnomes. Garden gnomes share a resemblance to the Scandinavian tomte and nisse, and the Swedish term "tomte" can be translated as "gnome" in English.

Gnome-themed parks
Several gnome themed entertainment parks exist. Notable ones are:
 * The Gnome Reserve, at West Putford near Bradworthy in North Devon, United Kingdom
 * Gnomeland, at Watermouth Castle in Berrynarbor, North Devon, United Kingdom
 * Gnome Magic Garden, at Colchester, United Kingdom
 * Gnome Park, in Dawson, Minnesota, United States
 * The Gnome Village, at Efteling theme park in Kaatsheuvel, Netherlands
 * Zwergen-Park Trusetal, in Trusetal, Germany
 * Gnom's Park in Nowa Sól, Poland.

Gnome parades
Gnome parades are held annually at Atlanta's Inman Park Festival. Numerous one-off gnome parades have been held, including in Savannah, Georgia (April 2012) and Cleveland, Ohio (May 2011).

Metaphorical uses

 * The expression "Gnomes of Zurich", Swiss bankers pictured as diminutive creatures hoarding gold in subterranean vaults, was derived from a speech in 1956 by Harold Wilson, and gained currency in the 1960s (OED notes the New Statesman issue of 27 November 1964 as earliest attestation).
 * Architect Earl Young built a number of stone houses in Charlevoix, Michigan, that have been referred to as gnome homes.
 * A user of Wikipedia or any wiki who makes useful incremental edits without clamouring for attention is called a WikiGnome.

title: Gnome


text:
Googolplex

A googolplex is the large number 10$googol$, or equivalently, 10$10100$ or 1010,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000. Written out in ordinary decimal notation, it is 1 followed by 10100 zeroes; that is, a 1 followed by a googol of zeroes. Its prime factorization is 2$googol$ ×5$googol$.

History
In 1920, Edward Kasner's nine-year-old nephew, Milton Sirotta, coined the term googol, which is 10$100$, and then proposed the further term googolplex to be "one, followed by writing zeroes until you get tired". Kasner decided to adopt a more formal definition because "different people get tired at different times and it would never do to have Carnera [be] a better mathematician than Dr. Einstein, simply because he had more endurance and could write for longer". It thus became standardized to 10(10 100) = 1010 100, due to the right-associativity of exponentiation.

Size
A typical book can be printed with 10$6$ zeros (around 400 pages with 50 lines per page and 50 zeros per line). Therefore, it requires 10$94$ such books to print all the zeros of a googolplex (that is, printing a googol zeros). If each book had a mass of 100 grams, all of them would have a total mass of 10$93$ kilograms. In comparison, Earth's mass is 5.972 × 10$24$ kilograms, the mass of the Milky Way galaxy is estimated at 2.5 × 10$42$ kilograms, and the total mass of all the stars in the observable universe is estimated at 2 × 1052 kg.

To put this in perspective, the mass of all such books required to write out a googolplex would be vastly greater than the masses of the Milky Way and the Andromeda galaxies combined (by a factor of roughly 2.0 × 10$50$), and greater than the mass of the observable universe by a factor of roughly 7 × 1039.

In pure mathematics
In pure mathematics, there are several notational methods for representing large numbers by which the magnitude of a googolplex could be represented, such as tetration, hyperoperation, Knuth's up-arrow notation, Steinhaus–Moser notation, or Conway chained arrow notation.

In the physical universe
In the PBS science program Cosmos: A Personal Voyage, Episode 9: "The Lives of the Stars", astronomer and television personality Carl Sagan estimated that writing a googolplex in full decimal form (i.e., "10,000,000,000...") would be physically impossible, since doing so would require more space than is available in the known universe. Sagan gave an example that if the entire volume of the observable universe is filled with fine dust particles roughly 1.5 micrometers in size (0.0015 millimeters), then the number of different combinations in which the particles could be arranged and numbered would be about one googolplex.

is a high estimate of the elementary particles existing in the visible universe (not including dark matter), mostly photons and other massless force carriers.

Mod n
The residues (mod n) of a googolplex, starting with mod 1, are:
 * 0, 0, 1, 0, 0, 4, 4, 0, 1, 0, 1, 4, 3, 4, 10, 0, 1, 10, 9, 0, 4, 12, 13, 16, 0, 16, 10, 4, 24, 10, 5, 0, 1, 18, 25, 28, 10, 28, 16, 0, 1, 4, 24, 12, 10, 36, 9, 16, 4, 0, ...

This sequence is the same as the sequence of residues (mod n) of a googol up until the 17th position.

title: Googolplex