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Nikos Kazantzakis

Nikos Kazantzakis (Νίκος Καζαντζάκης ; 2 March (OS 18 February) 1883 – 26 October 1957) was a Greek writer, journalist, politician, poet and philosopher. Widely considered a giant of modern Greek literature, he was nominated for the Nobel Prize in Literature in nine different years, and remains the most translated Greek author worldwide.

Kazantzakis's novels included Zorba the Greek (published in 1946 as Life and Times of Alexis Zorbas), Christ Recrucified (1948), Captain Michalis (1950, translated Freedom or Death), and The Last Temptation of Christ (1955). He also wrote plays, travel books, memoirs, and philosophical essays, such as The Saviors of God: Spiritual Exercises. His fame spread in the English-speaking world due to cinematic adaptations of Zorba the Greek (1964) and The Last Temptation of Christ (1988).

He also translated a number of notable works into Modern Greek, such as the Divine Comedy, Thus Spoke Zarathustra, On the Origin of Species, and Homer's Iliad and Odyssey.

Biography
Kazantzakis was born in 1883 in the town of Kandiye (now Heraklion) in Crete, with origins from the village of Myrtia. Crete had not yet joined the modern Greek state (which had been established in 1832), and was still under the rule of the Ottoman Empire. From 1902 to 1906 Kazantzakis studied law at the University of Athens; his 1906 Juris Doctor thesis title was Ο Φρειδερίκος Νίτσε εν τη φιλοσοφία του δικαίου και της πολιτείας ("Friedrich Nietzsche on the Philosophy of Law and the State"). Then he went to the Sorbonne in 1907 to study philosophy. There he fell under the influence of Henri Bergson. His 1909 doctoral dissertation at the Sorbonne was a reworked version of his 1906 dissertation under the title Friedrich Nietzsche dans la philosophie du droit et de la cité ("Friedrich Nietzsche on the Philosophy of Right and the State"). Upon his return to Greece, he began translating works of philosophy. In 1914 he met Angelos Sikelianos. Together they travelled for two years in places where Greek Orthodox Christian culture flourished, largely influenced by the enthusiastic nationalism of Sikelianos.

Kazantzakis married Galateia Alexiou in 1911; they divorced in 1926. Kazantzakis met Eleni Samiou (Helen) in 1924. They began a romantic relationship in 1928, though they were not married until 1945. Samiou helped Kazantzakis with his work, typing drafts, accompanying him on his travels, and managing his business affairs. They were married until his death in 1957. Samiou died in 2004.

Between 1922 and his death in 1957, he sojourned in Paris and Berlin (from 1922 to 1924), Italy, Russia (in 1925), Spain (in 1932), and then later in Cyprus, Aegina, Egypt, Mount Sinai, Czechoslovakia, Nice (he later bought a villa in nearby Antibes, in the Old Town section near the famed seawall), China, and Japan.

While in Berlin, where the political situation was explosive, Kazantzakis discovered communism and became an admirer of Vladimir Lenin. He never became a committed communist, but visited the Soviet Union and stayed with the Left Opposition politician and writer Victor Serge. He witnessed the rise of Joseph Stalin, and became disillusioned with Soviet-style communism. Around this time, his earlier nationalist beliefs were gradually replaced by a more universalist ideology. As a journalist in 1926 he interviewed Miguel Primo de Rivera and the Italian dictator Benito Mussolini.

During WWII he was in Athens, where he and the philologist Ioannis Kakridis translated the Iliad. In 1945, he became the leader of a small party on the non-communist left, and entered the Greek government as Minister without Portfolio. He resigned this post the following year. In 1946, Kazantzakis became the head of the UNESCO Bureau of translations, the organization which promoted translations of literary works. However, he resigned in 1947 to concentrate on writing, and indeed produced most of his literary output during the last ten years of his life.

In 1946 The Society of Greek Writers recommended that Kazantzakis and Angelos Sikelianos be awarded the Nobel Prize for Literature. In 1957, he lost the Prize to Albert Camus by a single vote. Camus later said that Kazantzakis deserved the honour "a hundred times more" than himself. In total Kazantzakis was nominated in nine different years.

Death
Late in 1957, even though suffering from leukemia, Kazantzakis set out on one last trip to China and Japan. According to one theory, while in China Kazantzakis had to be vaccinated possibly due to symptoms of smallpox and cholera. The vaccine, however, caused him gangrene and at the expense of the Chinese government he was transported first to Copenhagen and then to Freiburg. His gangrene was cured but he had contracted a severe form of Asian flu in China, which eventually led to his death.

Kazantzakis died on 26 October 1957 in Freiburg, Germany, at age 74. He is buried at the highest point of the Walls of Heraklion, the Martinengo Bastion, looking out over the mountains and sea of Crete. His epitaph reads "I hope for nothing. I fear nothing. I am free." (Δεν ελπίζω τίποτα. Δε φοβούμαι τίποτα. Είμαι λέφτερος.) Kazantzakis developed this famously pithy phrasing of the philosophical ideal of cynicism, which dates back to at least the second century CE.

The 50th anniversary of the death of Nikos Kazantzakis was selected as the main motif for a high-value euro collectors' coin; the €10 Greek Nikos Kazantzakis commemorative coin, minted in 2007. His image is on the obverse of the coin, while the reverse carries the National Emblem of Greece with his signature.

Literary work
Kazantzakis's first published work was the 1906 narrative, Serpent and Lily (Όφις και Κρίνο), which he signed with the pen name Karma Nirvami. In 1907 Kazantzakis went to Paris for his graduate studies and was deeply influenced by the philosophy of Henri Bergson, primarily the idea that a true understanding of the world comes from the combination of intuition, personal experience, and rational thought. The theme of rationalism mixed with irrationality later became central to many of Kazantzakis's later stories, characters, and personal philosophies. Later, in 1909, he wrote a one-act play titled Comedy, which was filled with existential themes, predating the post-World War II existentialist movement in Europe spearheaded by writers like Jean-Paul Sartre and Camus. After completing his studies in Paris, he wrote the tragedy, "The Master Builder" (Ο Πρωτομάστορας), based on a popular Greek folkloric myth.

Through the next several decades, from the 1910s through the 1930s, Kazantzakis traveled around Greece, much of Europe, northern Africa, and to several countries in Asia. Countries he visited include: Germany, Italy, France, The Netherlands, Romania, Egypt, Russia, Japan, and China, among others. These journeys put Kazantzakis in contact with different philosophies, ideologies, lifestyles, and people, all of which influenced him and his writings. Kazantzakis would often write about his influences in letters to friends, citing Sigmund Freud, the philosophy of Nietzsche, Buddhist theology, and communist ideology as major influences. While he continued to travel later in life, the bulk of his travel writing came from this time period.

Kazantzakis began writing The Odyssey: A Modern Sequel in 1924, and completed it in 1938 after fourteen years of writing and revision. The poem follows the hero of Homer's Odyssey, Odysseus, as he undertakes a final journey after the end of the original poem. Following the structure of Homer's Odyssey, it is divided into 24 rhapsodies and consists of 33,333 lines. While Kazantzakis felt this poem held his cumulative wisdom and experience, and that it was his greatest literary experience, critics were split, "some praised it as an unprecedented epic, [while] many simply viewed it as a hybristic act," with many scholars still being split to this day. A common criticism of The Odyssey: A Modern Sequel was aimed at Kazantzakis's over-reliance on flowery and metaphorical verse, a criticism that is also aimed at his works of fiction.

Many of Kazantzakis's most famous novels were published between 1940 and 1961, including Zorba the Greek (1946), Christ Recrucified (1948), Captain Michalis (1953), The Last Temptation of Christ (1955), and Report to Greco (1961).

Scholar Peter Bien argues that each story explores different aspects of post-World War II Greek culture such as religion, nationalism, political beliefs, the Greek Civil War, gender roles, immigration, and general cultural practices and beliefs. These works also explore what Kazantzakis believed to be the unique physical and spiritual location of Greece, a nation that belongs to neither the East nor the West, an idea he put forth in many of his letters to friends. As the scholar Peter Bien argued, "Kazantzakis viewed Greece's special mission as the reconciliation of Eastern instinct with Western reason," echoing the Bergsonian themes that balance logic against emotion found in many of Kazantzakis's novels.

Two of these works of fiction, Zorba the Greek and The Last Temptation of Christ had major motion picture adaptations in 1964 and 1988 respectively.

Language and use of Demotic Greek
During the time when Kazantzakis was writing his novels, poems, and plays, the majority of "serious" Greek artistic work was written in Katharevousa, a "pure" form of the Greek language that was created to bridge Ancient Greek with Modern, Demotic Greek, and to "purify" Demotic Greek. The use of Demotic, among writers, gradually started to gain the upper hand only in the turn of the 20th century, under the influence of the New Athenian School (or Palamian).

In his letters to friends and correspondents, Kazantzakis wrote that he chose to write in Demotic Greek to capture the spirit of the people, and to make his writing resonate with the common Greek citizen. Moreover, he wanted to prove that the common spoken language of Greek was able to produce artistic, literary works. Or, in his own words, "Why not show off all the possibilities of demotic Greek?" Furthermore, Kazantzakis felt that it was important to record the vernacular of the everyday person, including Greek peasants, and often tried to include expressions, metaphors, and idioms he would hear while traveling throughout Greece, and incorporate them into his writing for posterity. At the time of writing, some scholars and critics condemned his work because it was not written in Katharevousa, while others praised it precisely because it was written in Demotic Greek.

Several critics have argued that Kazantzakis's writing was too flowery, filled with obscure metaphors, and difficult to read, despite the fact that his works were written in Demotic Greek. Kazantzakis scholar Peter Bien argues that the metaphors and language Kazantzakis used were taken directly from the peasants he encountered when traveling Greece. Bien asserts that, since Kazantzakis was trying to preserve the language of the people, he used their local metaphors and phrases to give his narrative an air of authenticity and preserve these phrases so that they were not lost.

Socialism
Throughout his life, Kazantzakis reiterated his belief that "only socialism as the goal and democracy as the means" could provide an equitable solution to the "frightfully urgent problems of the age in which we are living." He saw the need for socialist parties throughout the world to put aside their bickering and unite so that the program of "socialist democracy" could prevail not just in Greece, but throughout the civilized world. He described socialism as a social system which "does not permit the exploitation of one person by another" and that "must guarantee every freedom."

Kazantzakis was anathema to the right-wing in Greece both before and after World War II. The right waged war against his books and called him "immoral" and a "Bolshevik troublemaker" and accused him of being a "Russian agent". He was also distrusted by the Greek and Russian Communist parties as a "bourgeois" thinker. However, upon his death in 1957, he was honored by the Chinese Communist party as a "great writer" and "devotee of peace." Following the war, he was temporarily leader of a minor Greek leftist party, while in 1945 he was, among others, a founding member of the Greek-Soviet friendship union.

Religious beliefs and relationship with the Greek Orthodox Church
While Kazantzakis was deeply spiritual, he often discussed his struggle with religious faith, specifically his Greek Orthodoxy. Baptized Greek Orthodox as a child, he was fascinated by the lives of saints from a young age. As a young man he took a month long trip to Mount Athos, a major spiritual center for Greek Orthodoxy. Most critics and scholars of Kazantzakis agree that the struggle to find truth in religion and spirituality was central to many of his works, and that some novels, like The Last Temptation of Christ and Christ Recrucified focus completely on questioning Christian morals and values. As he traveled Europe, he was influenced by various philosophers, cultures, and religions, like Buddhism, causing him to question his Christian beliefs. While never claiming to be an atheist, his public questioning and critique of the most fundamental Christian values put him at odds with some in the Greek Orthodox Church, and many of his critics. Scholars theorize that Kazantzakis's difficult relationship with many members of the clergy, and the more religiously conservative literary critics, came from his questioning. In his book Broken Hallelujah: Nikos Kazantzakis and Christian Theology, author Darren Middleton theorizes that, "Where the majority of Christian writers focus on God's immutability, Jesus' deity, and our salvation through God's grace, Kazantzakis emphasized divine mutability, Jesus' humanity, and God's own redemption through our effort," highlighting Kazantzakis's uncommon interpretation of traditional Orthodox Christian beliefs.

Many Orthodox Church clergy condemned Kazantzakis's work and a campaign was started to excommunicate him. His reply was: "You gave me a curse, Holy fathers, I give you a blessing: may your conscience be as clear as mine and may you be as moral and religious as I" ("Μου δώσατε μια κατάρα, Άγιοι πατέρες, σας δίνω κι εγώ μια ευχή: Σας εύχομαι να 'ναι η συνείδηση σας τόσο καθαρή, όσο είναι η δική μου και να 'στε τόσο ηθικοί και θρήσκοι όσο είμαι εγώ"). While the excommunication was rejected by the top leadership of the Orthodox Church, it became emblematic of the persistent disapprobation from many Christian authorities for his political and religious views.

Modern scholarship tends to dismiss the idea that Kazantzakis was being sacrilegious or blasphemous with the content of his novels and beliefs. These scholars argue that, if anything, Kazantzakis was acting in accordance to a long tradition of Christians who publicly struggled with their faith, and grew a stronger and more personal connection to God through their doubt. Moreover, scholars like Darren J. N. Middleton argue that Kazantzakis's interpretation of the Christian faith predated the more modern, personalized interpretation of Christianity that has become popular in the years after Kazantzakis's death.

Translations of The Odyssey: A Modern Sequel, in whole or in part

 * The Odyssey [Selections from], partial translation in prose by Kimon Friar, Wake 12 (1953), pp. 58–65.
 * The Odyssey, excerpt translated by Kimon Friar, Chicago Review 8, No. 2 (Spring/Summer 1954), pp. 12–18.
 * "The Return of Odysseus", partial translation by Kimon Friar, The Atlantic Monthly 195, No. 6 (June 1955), pp. 110–112.
 * The Odyssey: A Modern Sequel, a full verse-translation by Kimon Friar, New York: Simon & Schuster, 1958; London: Secker and Warburg, 1958.
 * "Death, the Ant", from The Odyssey: A Modern Sequel, Book XV, 829–63, translated by Kimon Friar, The Charioteer, No. 1 (Summer 1960), p. 39.

Travel books

 * Spain, translated by Amy Mims, New York: Simon & Schuster, 1963.
 * Japan, China, translated by George C. Pappageotes, New York: Simon & Schuster, 1963; published in the United Kingdom as Travels in China & Japan, Oxford: Bruno Cassirer, 1964; London: Faber and Faber, 1964.
 * England, translated by Amy Mims, New York: Simon & Schuster, 1965; Oxford, Bruno Cassirer, 1965.
 * Journey to the Morea, translated by F. A. Reed, New York: Simon & Schuster, 1965; published in the United Kingdom as Travels in Greece, Journey to the Morea, Oxford, Bruno Cassirer, 1966.
 * Journeying: Travels in Italy, Egypt, Sinai, Jerusalem and Cyprus, translated by Themi Vasils and Theodora Vasils, Boston and Toronto: Little, Brown and Company, 1975; San Francisco: Creative Arts Books Co., 1984.
 * Russia, translated by A. Maskaleris and M. Antonakis, Creative Arts Books Co, 1989.

Novels

 * Zorba the Greek, translated by Carl Wildman, London: John Lehmann, 1952; New York: Simon & Schuster, 1953; Oxford: Bruno Cassirer, 1959; London & Boston: Faber and Faber, 1961; New York: Ballantine Books, 1964; and Zorba the Greek: The Saint's Life of Alexis Zorba, newly translated by Peter Bien, New York: Simon & Schuster, 2014.
 * The Greek Passion, translated by Jonathan Griffin, New York, Simon & Schuster, 1954; New York, Ballantine Books, 1965; published in the United Kingdom as Christ Recrucified, Oxford: Bruno Cassirer, 1954; London: Faber and Faber, 1954.
 * Freedom or Death, translated by Jonathan Griffin, New York: Simon & Schuster, 1954; New York: Ballantine, 1965; published in the United Kingdom as Freedom or Death, Oxford: Bruno Cassirer, 1956; London: Faber and Faber, 1956.
 * The Last Temptation, translated by Peter A. Bien, New York, Simon & Schuster, 1960; New York, Bantam Books, 1961; Oxford: Bruno Cassirer, 1961; London: Faber and Faber, 1975.
 * Saint Francis, translated by Peter A. Bien, New York: Simon & Schuster, 1962; published in the United Kingdom as God's Pauper: Saint Francis of Assisi, Oxford: Bruno Cassirer, 1962, 1975; London: Faber and Faber, 1975.
 * The Rock Garden, translated from French (in which it was originally written) by Richard Howard, New York: Simon & Schuster, 1963.
 * The Fratricides, translated by Athena Gianakas Dallas, New York: Simon & Schuster, 1964; Oxford: Bruno Cassirer, 1964.
 * Toda Raba, translated from French (in which it was originally written) by Amy Mims, New York: Simon & Schuster, 1964.
 * Report to Greco — see under Memoirs, essays and letters
 * Alexander the Great. A Novel [for children], translated by Theodora Vasils, Athens (Ohio): Ohio University Press, 1982.
 * At the Palaces of Knossos. A Novel [for children], translated by Themi and Theodora Vasilis, edited by Theodora Vasilis, London: Owen, 1988. Adapted from the draft typewritten manuscript.
 * Father Yanaros [from the novel The Fratricides], translated by Theodore Sampson, in Modern Greek Short Stories, Vol. 1, edited by Kyr. Delopoulos, Athens: Kathimerini Publications, 1980.
 * Serpent and Lily, translated by Theodora Vasils, Berkeley: University of California Press, 1980.

Untranslated Novels

 * 1908: "Broken Souls" (Σπασμένες Ψυχές, Smasmenes Psyches), serialised in the O Numas magazine (1909–10)
 * 1946: "The Ascent" (Ο Ανήφορος, O Aniforos), first edition Dioptra, 2022 - ISBN 978-618-220-072-8

Short Story

 * "He Wants to Be Free – Kill Him!", translated by Athena G. Dallas, "Greek Heritage" 1, No. 1 (Winter 1963), pp. 78–82.

Plays

 * Julian the Apostate: First staged in Paris, 1948.
 * Three Plays: Melissa, Kouros, Christopher Columbus, translated by Athena Gianakas-Dallas, New York: Simon & Schuster, 1969.
 * Christopher Columbus, translated by Athena Gianakas-Dallas, Kentfield (CA): Allen Press, 1972. Edition limited to 140 copies.
 * From Odysseus, A Drama, partial translation by M. Byron Raizis, "The Literary Review" 16, No. 3 (Spring 1973), p. 352.
 * Comedy: A Tragedy in One Act, translated by Kimon Friar, "The Literary Review" 18, No. 4 (Summer 1975), pp. 417–454 {61}.
 * Sodom and Gomorrah, A Play, translated by Kimon Friar, "The Literary Review" 19, No. 2 (Winter 1976), pp. 122–256 (62).
 * Two plays: Sodom and Gomorrah and Comedy: A Tragedy in One Act, translated by Kimon Friar, Minneapolis: North Central Publishing Co., 1982.
 * Buddha, translated by Kimon Friar and Athena Dallas-Damis, San Diego (CA): Avant Books, 1983.
 * Odysseus: A Verse Tragedy, translated by Kostas Myrsiades, Boston (MA): Somerset Hall Press, 2022.

Poems

 * 1932 - 1937: "Tertsines", first edition 1960 - ISBN 978-960-794-819-9 (untranslated)
 * Christ (poetry), translated by Kimon Friar, "Journal of Hellenic Diaspora" (JHD) 10, No. 4 (Winter 1983), pp. 47–51 (60).

Memoirs, essays and letters

 * The Saviours of God: Spiritual Exercises, translated by Kimon Friar, New York: Simon & Schuster, 1960.
 * Report to Greco, translated by Peter A. Bien, New York: Simon & Schuster, 1965; Oxford: Bruno Cassirer, 1965; London: Faber and Faber, 1965; New York: Bantam Books, 1971.
 * Symposium, translated by Theodora Vasils e Themi Vasils, New York: Thomas Y. Crowell Company, 1974; New York: Minerva Press, 1974.
 * Friedrich Nietzsche on the Philosophy of Right and the State, translated by O. Makridis, New York: State University of NY Press, 2007.
 * From The Saviours of God: Spiritual Exercises, translated by Kimon Friar, "The Charioteer", No. 1 (Summer 1960), pp. 40–51; reprinted in "The Charioteer" 22 and 23 (1980/1981), pp. 116–129 {57}.
 * The Suffering God: Selected Letters to Galatea and to Papastephanou, translated by Philip Ramp and Katerina Anghelaki Rooke, New Rochelle (NY): Caratzas Brothers, 1979.
 * The Angels of Cyprus, translated by Amy Mims, in Cyprus '74: Aphrodite's Other Face, edited by Emmanuel C. Casdaglis, Athens: National Bank of Greece, 1976.
 * Burn Me to Ashes: An Excerpt, translated by Kimon Friar, "Greek Heritage" 1, No. 2 (Spring 1964), pp. 61–64.
 * Drama and Contemporary Man, An Essay, translated by Peter Bien, "The Literary Review" 19, No. 2 (Winter 1976), pp. 15–121 {62}.
 * The Homeric G.B.S., "The Shaw Review" 18, No. 3 (Sept. 1975), pp. 91–92. Greek original written for a 1946 Greek language radio broadcast by BBC Overseas Service, on the occasion of George Bernard Shaw's 90th birthday.
 * Hymn (Allegorical), translated by M. Byron Raizis, "Spirit" 37, No. 3 (Fall 1970), pp. 16–17.
 * Two Dreams, translated by Peter Mackridge, "Omphalos" 1, No. 2 (Summer 1972), p. 3.
 * Nikos Kazantzakis Pages at the Historical Museum of Crete
 * Peter Bien (ed. and tr.), The Selected Letters of Nikos Kazantzakis (Princeton, PUP, 2011) (Princeton Modern Greek Studies).

Anthologies

 * A Tiny Anthology of Kazantzakis. Remarks on the Drama, 1910–1957, compiled by Peter Bien, "The Literary Review" 18, No. 4 (Summer 1975), pp. 455–459 {61}.

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Carbonic acid

Carbonic acid is a chemical compound with the chemical formula H2CO3. The molecule rapidly converts to water and carbon dioxide in the presence of water. However, in the absence of water, it is quite stable at room temperature. The interconversion of carbon dioxide and carbonic acid is related to the breathing cycle of animals and the acidification of natural waters.

In biochemistry and physiology, the name "carbonic acid" is sometimes applied to aqueous solutions of carbon dioxide. These chemical species play an important role in the bicarbonate buffer system, used to maintain acid–base homeostasis.

Terminology in biochemical literature
In chemistry, the term "carbonic acid" strictly refers to the chemical compound with the formula. Some biochemistry literature effaces the distinction between carbonic acid and carbon dioxide dissolved in extracellular fluid.

In physiology, carbon dioxide excreted by the lungs may be called volatile acid or respiratory acid.

Anhydrous carbonic acid
At ambient temperatures, pure carbonic acid is a stable gas. There are two main methods to produce anhydrous carbonic acid: reaction of hydrogen chloride and potassium bicarbonate at 100 K in methanol and proton irradiation of pure solid carbon dioxide. Chemically, it behaves as a diprotic Brønsted acid.

Carbonic acid monomers exhibit three conformational isomers: cis–cis, cis–trans, and trans–trans.

At low temperatures and atmospheric pressure, solid carbonic acid is amorphous and lacks Bragg peaks in X-ray diffraction. But at high pressure, carbonic acid crystallizes, and modern analytical spectroscopy can measure its geometry.

According to neutron diffraction of dideuterated carbonic acid in a hybrid clamped cell (Russian alloy/copper-beryllium) at 1.85 GPa, the molecules are planar and form dimers joined by pairs of hydrogen bonds. All three C-O bonds are nearly equidistant at 1.34 Å, intermediate between typical C-O and C=O distances (respectively 1.43 and 1.23 Å). The unusual C-O bond lengths are attributed to delocalized π bonding in the molecule's center and extraordinarily strong hydrogen bonds. The same effects also induce a very short O—O separation (2.13 Å), through the 136° O-H-O angle imposed by the doubly hydrogen-bonded 8-membered rings. Longer O—O distances are observed in strong intramolecular hydrogen bonds, e.g. in oxalic acid, where the distances exceed 2.4 Å.

In aqueous solution
In even a slight presence of water, carbonic acid dehydrates to carbon dioxide and water, which then catalyzes further decomposition. For this reason, carbon dioxide can be considered the carbonic acid anhydride.

The hydration equilibrium constant at 25 °C is in pure water and ≈ 1.2×10−3 in seawater. Hence the majority of carbon dioxide at geophysical or biological air-water interfaces does not convert to carbonic acid, remaining dissolved gas. However, the uncatalyzed equilibrium is reached quite slowly: the rate constants are 0.039 s−1 for hydration and 23 s−1 for dehydration.

In biological solutions
In the presence of the enzyme carbonic anhydrase, equilibrium is instead reached rapidly, and the following reaction takes precedence:

When the created carbon dioxide exceeds its solubility, gas evolves and a third equilibrium must also be taken into consideration. The equilibrium constant for this reaction is defined by Henry's law.

The two reactions can be combined for the equilibrium in solution: When Henry's law is used to calculate the denominator care is needed with regard to units since Henry's law constant can be commonly expressed with 8 different dimensionalities.

Under high CO2 partial pressure
In the beverage industry, sparkling or "fizzy water" is usually referred to as carbonated water. It is made by dissolving carbon dioxide under a small positive pressure in water. Many soft drinks treated the same way effervesce.

Significant amounts of molecular exist in aqueous solutions subjected to pressures of multiple gigapascals (tens of thousands of atmospheres) in planetary interiors. Pressures of 0.6–1.6 GPa at 100 K, and 0.75–1.75 GPa at 300 K are attained in the cores of large icy satellites such as Ganymede, Callisto, and Titan, where water and carbon dioxide are present. Pure carbonic acid, being denser, is expected to have sunk under the ice layers and separate them from the rocky cores of these moons.

Relationship to bicarbonate and carbonate
Carbonic acid is the formal Brønsted–Lowry conjugate acid of the bicarbonate anion, stable in alkaline solution. The protonation constants have been measured to great precision, but depend on overall ionic strength $I$. The two equilibria most easily measured are as follows: where brackets indicate the concentration of specie. At 25 °C, these equilibria empirically satisfy $$\begin{alignat}{6} \log(\beta_1) =&& 0&.54&I^2 - 0&.96&I +&& 9&.93 \\ \log(\beta_2) =&& -2&.5&I^2 - 0&.043&I +&& 16&.07 \end{alignat}$$$pK$ decreases with increasing $I$, as does $pH$. In a solution absent other ions (e.g. $log(β_{1})$), these curves imply the following stepwise dissociation constants:$$\begin{alignat}{3} p\text{K}_1 &= \log(\beta_2) - \log(\beta_1) &= 6.77 \\ p\text{K}_2 &= \log(\beta_1) &= 9.93 \end{alignat}$$ Direct values for these constants in the literature include $log(β_{2})$ and $$.

To interpret these numbers, note that two chemical species in an acid equilibrium are equiconcentrated when $pK_{1} = 6.35$. In particular, the extracellular fluid (cytosol) in biological systems exhibits $pK_{2} - pK_{1} = 3.49$, so that carbonic acid will be almost 50%-dissociated at equilibrium.

Ocean acidification

The Bjerrum plot shows typical equilibrium concentrations, in solution, in seawater, of carbon dioxide and the various species derived from it, as a function of pH. As human industrialization has increased the proportion of carbon dioxide in Earth's atmosphere, the proportion of carbon dioxide dissolved in sea- and freshwater as carbonic acid is also expected to increase. This rise in dissolved acid is also expected to acidify those waters, generating a decrease in pH. It has been estimated that the increase in dissolved carbon dioxide has already caused the ocean's average surface pH to decrease by about 0.1 from pre-industrial levels.

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

title: Carbonic acid


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Integer sequence

In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers.

An integer sequence may be specified explicitly by giving a formula for its nth term, or implicitly by giving a relationship between its terms. For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... (the Fibonacci sequence) is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one: an implicit description. The sequence 0, 3, 8, 15, ... is formed according to the formula n2 − 1 for the nth term: an explicit definition.

Alternatively, an integer sequence may be defined by a property which members of the sequence possess and other integers do not possess. For example, we can determine whether a given integer is a perfect number,, even though we do not have a formula for the nth perfect number.

Computable and definable sequences
An integer sequence is a computable sequence if there exists an algorithm which, given n, calculates an, for all n > 0. The set of computable integer sequences is countable. The set of all integer sequences is uncountable (with cardinality equal to that of the continuum), and so not all integer sequences are computable.

Although some integer sequences have definitions, there is no systematic way to define what it means for an integer sequence to be definable in the universe or in any absolute (model independent) sense.

Suppose the set M is a transitive model of ZFC set theory. The transitivity of M implies that the integers and integer sequences inside M are actually integers and sequences of integers. An integer sequence is a definable sequence relative to M if there exists some formula P(x) in the language of set theory, with one free variable and no parameters, which is true in M for that integer sequence and false in M for all other integer sequences. In each such M, there are definable integer sequences that are not computable, such as sequences that encode the Turing jumps of computable sets.

For some transitive models M of ZFC, every sequence of integers in M is definable relative to M; for others, only some integer sequences are (Hamkins et al. 2013). There is no systematic way to define in M itself the set of sequences definable relative to M and that set may not even exist in some such M. Similarly, the map from the set of formulas that define integer sequences in M to the integer sequences they define is not definable in M and may not exist in M. However, in any model that does possess such a definability map, some integer sequences in the model will not be definable relative to the model (Hamkins et al. 2013).

If M contains all integer sequences, then the set of integer sequences definable in M will exist in M and be countable and countable in M.

Complete sequences
A sequence of positive integers is called a complete sequence if every positive integer can be expressed as a sum of values in the sequence, using each value at most once.

Examples
Integer sequences that have their own name include:

 * Abundant numbers
 * Baum–Sweet sequence
 * Bell numbers
 * Binomial coefficients
 * Carmichael numbers
 * Catalan numbers
 * Composite numbers
 * Deficient numbers
 * Euler numbers
 * Even and odd numbers
 * Factorial numbers
 * Fibonacci numbers
 * Fibonacci word
 * Figurate numbers
 * Golomb sequence
 * Happy numbers
 * Highly composite numbers
 * Highly totient numbers
 * Home primes
 * Hyperperfect numbers
 * Juggler sequence
 * Kolakoski sequence
 * Lucky numbers
 * Lucas numbers
 * Motzkin numbers
 * Natural numbers
 * Padovan numbers
 * Partition numbers
 * Perfect numbers
 * Practical numbers
 * Prime numbers
 * Pseudoprime numbers
 * Recamán's sequence
 * Regular paperfolding sequence
 * Rudin–Shapiro sequence
 * Semiperfect numbers
 * Semiprime numbers
 * Superperfect numbers
 * Triangular numbers
 * Thue–Morse sequence
 * Ulam numbers
 * Weird numbers
 * Wolstenholme number

title: Integer sequence


text:
Talk:Integer sequence

Merge with sequence?
I think this material should be merged with sequence; integer sequences should be reserved for sequences whose terms are integers. --AxelBoldt

 * Integer sequences ARE sequences whose terms are integers. (see the first line) — Preceding unsigned comment added by Bubba73 (talk • contribs) 19:26, 10 May 2005

Is it really necessary to include a long list of named sequences here, when we have a perfectly good category page with essentially the same information? (And btw, what is the approved method for creating internal links to category pages?) —David Eppstein 18:53, 10 September 2006 (UTC)

Links to the Definition of a Statement?
To me (a non-mathematician), the definitions of definable sequences leaves the term "statement" underdefined, so I would like to understand what a statement is, i..e. is it a finite algorithm definition, must it terminate after a finite number of steps, etc. 198.60.22.24 12:04, 29 July 2007 (UTC)

Assessment comment
Substituted at 02:14, 5 May 2016 (UTC)

Definable sequences
"The set of computable integer sequences and definable integer sequences are both countable" — Really, both? Or rather the former? "Moreover, there are countable models of ZFC in which all real numbers, all sets of real numbers, functions on the reals, etc. are definable" (quoted from Definable real number). (Related to above, and to Tarski's undefinability theorem.) (The claim about definable was written at 08:01, 16 November 2004 by 137.111.13.34; see also here: "Notably, in a system of mathematics in which everything is definable, all sets are countable.") Boris Tsirelson (talk) 08:36, 5 February 2018 (UTC)
 * Whichever definition of "definable" you choose, the formula that defines a definable object is a finite sequence of characters belonging to a finite alphabet. Thus the set of definable objects is definitively countable. I am unable to check whether the quotations are correct. The latter one is highly doubtful, as "everything is decidable" is essentially a non-sense. The former quotation seems reliably sourced. However, the paragraph refers to "countable models of set theory" and "class models". This suggests that some collections of real numbers are not sets or even that one cannot talk about them in the theory (think to Russel's paradox of the set of all sets that do not belong to themselves). This is also related to constructivist logics, where only definable objects define in the theory.
 * IMO, the section Definable real number must be removed or rewritten. I'll tagging it as too technical, as it supposes a good understanding of the logical concept of "model", which most mathematician do not know well. D.Lazard (talk) 11:23, 5 February 2018 (UTC)

 * I am not exactly sure what this thread is asking, but all of these are correct:
 * The set of computable sequences of integers is countable. This is a theorem of ZFC.
 * In any model of ZFC, the set of definable sequences of integers (definable in that model) is countable. This is a theorem of ZFC about models of ZFC.
 * There are models of ZFC in which the set of sequences of integers is countable when viewed from outside the model (these models themselves are countable). The existence of countable models of ZFC is an elementary fact from the Lowenheim-Skolem theorem. The existence of models in which each element is definable is not as elementary, but is well known, e.g.  . These models will still, internally, satisfy the sentence "the set of integer sequences is uncountable".
 * The third fact is not really relevant to this article. The real issue is that "definable" is not itself expressible from inside a model of ZFC - a model has no access to its own definability relation. This is why in item 2 we can only say that ZFC proves that the set of definable sequences in any model is countable. ZFC cannot prove simply "the set of definable integer sequences is countable" because this cannot be expressed in the language of ZFC.
 * — Carl (CBM · talk) 11:31, 5 February 2018 (UTC)
 * All above quotations refer to non-standard models of set theory. Thus this article is correct, as, unless otherwise stated, all articles of mathematics are supposed to use ZFC as logical foundation. Maybe this should be said in MOS:MATH. D.Lazard (talk) 11:47, 5 February 2018 (UTC)
 * The first two do not have anything the do with nonstandard models - at best we would say they hold for *all* models. But this article is correct, as is the one on definable real numbers, with one exception: the bare claim here that "almost all integer sequences ... cannot be defined.". The statement "almost all integer sequences cannot be defined" is not a sentence in ZFC, and as such is not provable in ZFC.  The issue then becomes more subtle, related to mathematical platonism, and certainly not resolvable within ZFC.  Hamkins et al. go into some detail on this issue in the intro of . — Carl (CBM · talk) 12:01, 5 February 2018 (UTC)
 * As far as I (not expert in logic) understand, Carl is correct, and Lazard should think again. In fact, some time ago I was trapped by this dangerous trap. But now I escaped. Indeed, the statement "almost all integer sequences cannot be defined" is not a sentence in ZFC (!). Accordingly, this article was incorrect before Carl's edit. Boris Tsirelson (talk) 18:31, 5 February 2018 (UTC)

 * Since Tarski showed that truth is undefinable, I would infer that "there exists some statement P(x) which is true for that integer sequence x and false for all other integer sequences" has no meaning unless it is relativized to some specific set-model. JRSpriggs (talk) 03:08, 6 February 2018 (UTC)

 * That's certainly the case if we take the "true" literally. It could also be a colloquial way of saying something like "ZFC proves there is a unique sequence satisfying P". For example, this happens if P says "every element of the sequence is zero". Overall this is a standard issue - definability over a model vs. definability over a theory. — Carl (CBM · talk) 16:05, 6 February 2018 (UTC)

 * Still I think you would need to require that P(x) also be absolute (perhaps provably Δ1) in order for the 'definition' to actually define anything. JRSpriggs (talk) 04:19, 7 February 2018 (UTC)

There is a flaw in CBM's argument for a 'colloquial' interpretation. You are failing to distinguish between P being a definition and it being a definition of a specific integer sequence x. I agree that we can say that P is a definition, if ZFC proves that $$\textstyle \exists x \forall y ( y = x \iff P (y) ) $$. But how do we say that x is the thing being defined? You cannot just say that P(x) because P is not a specific formula in the language of set theory, rather it is a variable over Godel numbers for formulas. So you would have to say something like P is true of x.

If you could do that, then what is to stop me from defining a function f as follows: f(k) is zero if k is not the Godel number of a definition; and f(k) = xk(k)+1 if k is the Godel number of a definition Pk which defines the function xk. Clearly this would be a contradiction since if n is the Godel number of the definition I just gave for f we get f(n) = xn(n)+1 = f(n)+1. JRSpriggs (talk) 01:48, 9 February 2018 (UTC)

 * I'm not really looking to discuss this in detail, because I think it is all well known and standard. A definitional extension only needs a proof of $$(\exists!x)\phi(x)$$. Of course the properties of the object being defined depend on φ. So φ could say "x is zero sharp if that exists, otherwise x is the constant zero sequence", and we can't prove much about the object defined by that sentence. This is the way it is for definitions over a theory, rather than definitions over a model. — Carl (CBM · talk) 02:05, 9 February 2018 (UTC)

This is not an article about foundations of mathematics, nor about model theory. The present version of the article is thus too technical, as referring to theories that are both unknown by most readers, and not needed for understanding the section. Moreover the present version does not satisfies WP:NPOV, as making too much place to theories that are outside the main stream of mathematics. If one follows what is said in this discussion, we should edit similarly every article that mention uncountable sets, as there are set theories where all sets are countable. For these reasons, I'll restore the old version, and simply add a comment that the existence of non-definable sequences may be not valid in other set theories or other underlying logics. D.Lazard (talk) 14:59, 9 February 2018 (UTC)

 * Then exclude "definable" at all (in this article). I agree that we'd better stay within ZFC. But, again: existence of non-definable sequences (of integers) is NOT a theorem of ZFC, and moreover, the notion "definable sequence" does not exist within ZFC. Within ZFC you could introduce the notion "a sequence definable under the first strongly inaccesible cardinal if it exists" or something in this spirit. But the definition of "full-blown" definability needs a map from formulas of ZFC to classes (not sets!); this is beyond ZFC. The problem is that quantifiers in ZFC are not bounded. In contrast, we non-logicians mostly use bounded quantifiers ("$$\forall x\in X$$" rather than "for all x in the class of all sets").Boris Tsirelson (talk) 15:54, 9 February 2018 (UTC)
 * I disagree: the well-founded statements of ZFC form a countable set (finite words over a finite or countable alphabet). The fact that a statement (element of a well defined set) defines an unique integer sequence is expressible in ZFC (generally not provable nor disprovable). Thus the well-founded statements that define a unique integer sequence form a subset of the well-founded statements, and thus a countable set. Thus the definable sequences form set, a thus a countable set, and this is a theorem of ZFC. However, for a given statement, there is, in general, no proof for deciding whether it defines a unique integer sequence. As defining statements are the unique way for explicitly defining a sequence, it is undecidable whether a sequence is definable. This makes possible that some sequences may be definable in some models and not in other models. Note that the argument for definable sequences is exactly the same as for computable sequences, and thus there is no need of uncountability for proving the existence of undefinable sequences. Also, in the intuitionistic logic that is used in many proof assistants, definable is exactly the same thing as computable. D.Lazard (talk) 17:32, 9 February 2018 (UTC)
 * I disagree. Yes, the formulas of ZFC form a countable set. Consider the subset of formulas with exactly one free variable (denote it $$P(x)$$); also a countable set, of course. Given such $$P(x)$$ you can form the statement $$\exists! x \, P(x)$$ on the level of syntax. The problem appears when you want to jump from syntax to semantics. Have you a formula $$T(P)$$ of ZFC that says that the statement "$$\exists! x \, P(x)$$" is true? (Here "$$P(x)$$" is treated within ZFC as a word in the appropriate alphabet, and "$$\exists! x \, P(x)$$" as another word.) No, you have no such formula. Accordingly, you cannot consider "the set of such formulas $$P(x)$$". Boris Tsirelson (talk) 17:57, 9 February 2018 (UTC)

 * If anyone checks the reference I have pointed out before, theu will find a good summary on pages 9 to 10:
 * There is no formula of ZFC so that, in every model, this formula is satisfied by exactly the definable elements.
 * In some models, the collection of definable elements is a definable class
 * In some models, the collection of definable elements is not a definable class
 * Also, on page 3, they will see that there are models inside which which every set is definable (without parameters), including every sequence of integers in the model.
 * Separately, while there are countable models of set theory, there are no models of set theory inside which every set is countable. But there are models of set theory inside which every set is definable. The reason that the argument for definable sequences is different than for computable sequences is that there is a single formula which, in every model of set theory, pulls out the computable sequences of integers. — Carl (CBM · talk) 19:36, 9 February 2018 (UTC)

Line by line explanation
Here is a specific sentence-by-sentence explanation of why the previous version of the article isn't accurate.
 * The sets of computable integer sequences and of definable integer sequences are both countable, with the computable sequences a proper subset of the definable sequences (in other words, some sequences are definable but not computable).
 * ZFC cannot prove that the set of definable integer sequences is countable. In some models of ZFC, every integer sequence is definable. But, more importantly, ZFC cannot even express the sentence "every integer sequence is definable"
 * The set of all integer sequences is uncountable (with cardinality equal to that of the continuum);
 * This is provable in ZFC
 * thus, almost all integer sequences are uncomputable and cannot be defined.
 * The uncomputable claim is provable in ZFC, but not the claim about sequences that cannot be defined
 * However, there exist non-standard set theories or non-standard underlying logics, where every integer sequence is definable (for example, if all sets are countable).
 * ZFC proves that "the set of all integer sequences is uncountable", but it is still the case that ZFC cannot prove that the set of definable sequences is countable, because there is no formula of ZFC which holds for exactly the definable sequences. Again, ZFC cannot even express the sentence "every integer sequence is definable". This is not related to nonstandard set theories or to nonclassical logics, it is just a property of ordinary ZFC.
 * — Carl (CBM · talk) 19:48, 9 February 2018 (UTC)

A set in M
To Carl: Since I am assuming "M is a transitive model of ZFC which is a set and contains all integer sequences ", I can prove that the set of all integer sequences definable relative to M is a set and indeed a countable set in M. First, we agree that it is countable. So choose a bijection between ω and the set. Then use the Cantor pairing function to combine the two arguments. This encodes the set as a single integer sequence. Thus it is in M and we can just reverse the encoding to get the bijection and the set in M. JRSpriggs (talk) 20:04, 10 February 2018 (UTC)

 * I see what you mean, but my next thought about the article was that the entire argument about "contains all integer sequences" ought to be removed, so I was not actually assuming that about M. Perhaps I only made part of the edit I had in mind. The largest challenge is that there is not really a good source for any of this, although it is all pretty elementary. There is sentence from the Hamkins, Linetsky, and Rietz paper that could be used as a source: "The surviving content of the math-tea argument seems to be the observation that in any model with access to a definability map  $$r \to \psi_r$$, the definable reals do not exhaust all the reals."  I would not mind rephrasing things to use that directly. I am more concerned about having any kind of subtle, unsourceable argument in the article, particularly given the general amount of misunderstanding around this topic. — Carl (CBM · talk) 20:10, 10 February 2018 (UTC)

 * We could just drop the whole second paragraph as too technical or unsourced, but I personally kind of like it.
 * We have to make some kind of assumption about the model as I argued above. So I think that assuming a model which is as similar as possible to the universe is desirable. That is why I made it a transitive model of ZFC and included all the things which we are talking about (the integer sequences). Making the model a set is forced on us by the fact that we cannot define truth for a proper class model.
 * If we left out some integer sequences, then it would create issues about which ones were included versus which were left out. I do not want to deal with those issues. JRSpriggs (talk) 20:25, 10 February 2018 (UTC)

 * I tried making an edit just now that is sourced, and which still has the essence of what you were talking about. I agree about transitive models - so that the "integer sequences" are actually integer sequences. The main difference is that rather than talking about "all" sequences being included, we can just be explicit: if we have the definability map, we know that not all sequences are definable.  In the end, I think the kind of argument you were making is just a different approach to the one that goes via the definability map. — Carl (CBM · talk) 20:30, 10 February 2018 (UTC)

title: Talk:Integer sequence


text:
P-adic number

In number theory, given a prime number $p$, the $p$-adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; $p$-adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number $p$ rather than ten, and extending to the left rather than to the right.

For example, comparing the expansion of the rational number $$\tfrac15$$ in base $3$ vs. the $3$-adic expansion,
 * $$\begin{alignat}{3}

\tfrac15 &{}= 0.01210121\ldots \ (\text{base } 3) &&{}= 0\cdot 3^0 + 0\cdot 3^{-1} + 1\cdot 3^{-2} + 2\cdot 3^{-3} + \cdots \\[5mu] \tfrac15 &{}= \dots 121012102 \ \ (\text{3-adic}) &&{}= \cdots + 2\cdot 3^3 + 1 \cdot 3^2 + 0\cdot3^1 + 2 \cdot 3^0. \end{alignat}$$

Formally, given a prime number $p$, a $p$-adic number can be defined as a series
 * $$s=\sum_{i=k}^\infty a_i p^i = a_k p^k + a_{k+1} p^{k+1} + a_{k+2} p^{k+2} + \cdots$$

where $p$ is an integer (possibly negative), and each $$a_i$$ is an integer such that $$0\le a_i < p.$$ A $k$-adic integer is a $p$-adic number such that $$k\ge 0.$$

In general the series that represents a $p$-adic number is not convergent in the usual sense, but it is convergent for the $p$-adic absolute value $$|s|_p=p^{-k},$$ where $p$ is the least integer $k$ such that $$a_i\ne 0$$ (if all $$a_i$$ are zero, one has the zero $i$-adic number, which has $0$ as its $p$-adic absolute value).

Every rational number can be uniquely expressed as the sum of a series as above, with respect to the $p$-adic absolute value. This allows considering rational numbers as special $p$-adic numbers, and alternatively defining the $p$-adic numbers as the completion of the rational numbers for the $p$-adic absolute value, exactly as the real numbers are the completion of the rational numbers for the usual absolute value.

$p$-adic numbers were first described by Kurt Hensel in 1897, though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using $p$-adic numbers.

Motivation
Roughly speaking, modular arithmetic modulo a positive integer $p$ consists of "approximating" every integer by the remainder of its division by $n$, called its residue modulo $n$. The main property of modular arithmetic is that the residue modulo $n$ of the result of a succession of operations on integers is the same as the result of the same succession of operations on residues modulo $n$. If one knows that the absolute value of the result is less than $n$, this allows a computation of the result which does not involve any integer larger than $n/2$.

For larger results, an old method, still in common use, consists of using several small moduli that are pairwise coprime, and applying the Chinese remainder theorem for recovering the result modulo the product of the moduli.

Another method discovered by Kurt Hensel consists of using a prime modulus $n$, and applying Hensel's lemma for recovering iteratively the result modulo $$p^2, p^3, \ldots, p^n, \ldots$$ If the process is continued infinitely, this provides eventually a result which is a $p$-adic number.

Basic lemmas
The theory of $p$-adic numbers is fundamentally based on the two following lemmas

Every nonzero rational number can be written $p^v\frac{m}{n},$ where $p$, $v$, and $m$ are integers and neither $n$ nor $m$ is divisible by $n$. The exponent $p$ is uniquely determined by the rational number and is called its $v$-adic valuation (this definition is a particular case of a more general definition, given below). The proof of the lemma results directly from the fundamental theorem of arithmetic.

Every nonzero rational number $p$ of valuation $r$ can be uniquely written $$r=ap^v+ s,$$ where $v$ is a rational number of valuation greater than $s$, and $v$ is an integer such that $$0<a<p.$$

The proof of this lemma results from modular arithmetic: By the above lemma, $r=p^v\frac{m}{n},$ where $a$ and $m$ are integers coprime with $n$. The modular inverse of $p$ is an integer $n$ such that $$nq=1+ph$$ for some integer $q$. Therefore, one has $\frac 1n=q-p\frac hn,$ and $r=p^vmq -p^{v+1}\frac{hm}{n}.$  The Euclidean division of $$mq$$ by $h$ gives $$mq= pk+a$$ where $$00$$ and
 * $$r=a_0p^v + a_1 p^{v+1} +\cdots + a_{k-1}p^{v+k-1} +p^{v+k}s_k.$$

The $p$-adic numbers are essentially obtained by continuing this infinitely to produce an infinite series.

p-adic series
The $p$-adic numbers are commonly defined by means of $p$-adic series.

A $p$-adic series is a formal power series of the form
 * $$\sum_{i=v}^\infty r_i p^{i},$$

where $$v$$ is an integer and the $$r_i$$ are rational numbers that either are zero or have a nonnegative valuation (that is, the denominator of $$r_i$$ is not divisible by $p$).

Every rational number may be viewed as a $p$-adic series with a single nonzero term, consisting of its factorization of the form $$p^k\tfrac nd,$$ with $p$ and $n$ both coprime with $d$.

Two $p$-adic series $\sum_{i=v}^\infty r_i p^{i} $ and $ \sum_{i=w}^\infty s_i p^{i} $ are equivalent if there is an integer $p$ such that, for every integer $$n>N,$$ the rational number
 * $$\sum_{i=v}^n r_i p^{i} - \sum_{i=w}^n s_i p^{i} $$

is zero or has a $N$-adic valuation greater than $p$.

A $n$-adic series $\sum_{i=v}^\infty a_i p^{i} $ is normalized if either all $$a_i$$ are integers such that $$0\le a_i 0,$$ or all $$a_i$$ are zero. In the latter case, the series is called the zero series.

Every $p$-adic series is equivalent to exactly one normalized series. This normalized series is obtained by a sequence of transformations, which are equivalences of series; see § Normalization of a $p$-adic series, below.

In other words, the equivalence of $p$-adic series is an equivalence relation, and each equivalence class contains exactly one normalized $p$-adic series.

The usual operations of series (addition, subtraction, multiplication, division) are compatible with equivalence of $p$-adic series. That is, denoting the equivalence with $~$, if $p$, $S$ and $T$ are nonzero $U$-adic series such that $$S\sim T,$$ one has
 * $$\begin{align}

S\pm U&\sim T\pm U,\\ SU&\sim TU,\\ 1/S&\sim 1/T. \end{align}$$

The $p$-adic numbers are often defined as the equivalence classes of $p$-adic series, in a similar way as the definition of the real numbers as equivalence classes of Cauchy sequences. The uniqueness property of normalization, allows uniquely representing any $p$-adic number by the corresponding normalized $p$-adic series. The compatibility of the series equivalence leads almost immediately to basic properties of $p$-adic numbers:
 * Addition, multiplication and multiplicative inverse of $p$-adic numbers are defined as for formal power series, followed by the normalization of the result.
 * With these operations, the $p$-adic numbers form a field, which is an extension field of the rational numbers.
 * The valuation of a nonzero $p$-adic number $p$, commonly denoted $$v_p(x)$$ is the exponent of $x$ in the first non zero term of the corresponding normalized series; the valuation of zero is $$v_p(0)=+\infty$$
 * The $p$-adic absolute value of a nonzero $p$-adic number $p$, is $$|x|_p=p^{-v(x)};$$ for the zero $x$-adic number, one has $$|0|_p=0.$$

Normalization of a p-adic series
Starting with the series $\sum_{i=v}^\infty r_i p^{i}, $ the first above lemma allows getting an equivalent series such that the $p$-adic valuation of $$r_v$$ is zero. For that, one considers the first nonzero $$r_i.$$ If its $p$-adic valuation is zero, it suffices to change $p$ into $v$, that is to start the summation from $i$. Otherwise, the $v$-adic valuation of $$r_i$$ is $$j>0,$$ and $$r_i= p^js_i$$ where the valuation of $$s_i$$ is zero; so, one gets an equivalent series by changing $$r_i$$ to $0$ and $$r_{i+j}$$ to $$r_{i+j} + s_i.$$ Iterating this process, one gets eventually, possibly after infinitely many steps, an equivalent series that either is the zero series or is a series such that the valuation of $$r_v$$ is zero.

Then, if the series is not normalized, consider the first nonzero $$r_i$$ that is not an integer in the interval $$[0,p-1].$$ The second above lemma allows writing it $$r_i=a_i+ps_i;$$ one gets n equivalent series by replacing $$r_i$$ with $$a_i,$$ and adding $$s_i$$ to $$r_{i+1}.$$ Iterating this process, possibly infinitely many times, provides eventually the desired normalized $p$-adic series.

Definition
There are several equivalent definitions of $p$-adic numbers. The one that is given here is relatively elementary, since it does not involve any other mathematical concepts than those introduced in the preceding sections. Other equivalent definitions use completion of a discrete valuation ring (see ), completion of a metric space (see ), or inverse limits (see ).

A $p$-adic number can be defined as a normalized $p$-adic series. Since there are other equivalent definitions that are commonly used, one says often that a normalized $p$-adic series represents a $p$-adic number, instead of saying that it is a $p$-adic number.

One can say also that any $p$-adic series represents a $p$-adic number, since every $p$-adic series is equivalent to a unique normalized $p$-adic series. This is useful for defining operations (addition, subtraction, multiplication, division) of $p$-adic numbers: the result of such an operation is obtained by normalizing the result of the corresponding operation on series. This well defines operations on $p$-adic numbers, since the series operations are compatible with equivalence of $p$-adic series.

With these operations, $p$-adic numbers form a field called the field of $p$-adic numbers and denoted $$\Q_p$$ or $$\mathbf Q_p.$$ There is a unique field homomorphism from the rational numbers into the $p$-adic numbers, which maps a rational number to its $p$-adic expansion. The image of this homomorphism is commonly identified with the field of rational numbers. This allows considering the $p$-adic numbers as an extension field of the rational numbers, and the rational numbers as a subfield of the $p$-adic numbers.

The valuation of a nonzero $p$-adic number $p$, commonly denoted $$v_p(x),$$ is the exponent of $x$ in the first nonzero term of every $p$-adic series that represents $p$. By convention, $$v_p(0)=\infty;$$ that is, the valuation of zero is $$\infty.$$ This valuation is a discrete valuation. The restriction of this valuation to the rational numbers is the $x$-adic valuation of $$\Q,$$ that is, the exponent $p$ in the factorization of a rational number as with both $v$ and $n$ coprime with $d$.

p-adic integers
The $p$-adic integers are the $p$-adic numbers with a nonnegative valuation.

A $p$-adic integer can be represented as a sequence
 * $$ x = (x_1 \operatorname{mod} p, ~ x_2 \operatorname{mod} p^2, ~ x_3 \operatorname{mod} p^3, ~ \ldots)$$

of residues $p$ mod $x_{e}$ for each integer $p^{e}$, satisfying the compatibility relations $$x_i \equiv x_j ~ (\operatorname{mod} p^i)$$ for $e$.

Every integer is a $i < j$-adic integer (including zero, since $$0<\infty$$). The rational numbers of the form $ \tfrac nd p^k$ with $p$ coprime with $d$ and $$k\ge 0$$ are also $p$-adic integers (for the reason that $p$ has an inverse mod $d$ for every $p^{e}$).

The $e$-adic integers form a commutative ring, denoted $$\Z_p$$ or $$\mathbf Z_p$$, that has the following properties.
 * It is an integral domain, since it is a subring of a field, or since the first term of the series representation of the product of two non zero $p$-adic series is the product of their first terms.
 * The units (invertible elements) of $$\Z_p$$ are the $p$-adic numbers of valuation zero.
 * It is a principal ideal domain, such that each ideal is generated by a power of $p$.
 * It is a local ring of Krull dimension one, since its only prime ideals are the zero ideal and the ideal generated by $p$, the unique maximal ideal.
 * It is a discrete valuation ring, since this results from the preceding properties.
 * It is the completion of the local ring $$\Z_{(p)} = \{\tfrac nd \mid n, d \in \Z,\, d \not\in p\Z \},$$ which is the localization of $$\Z$$ at the prime ideal $$p\Z.$$

The last property provides a definition of the $p$-adic numbers that is equivalent to the above one: the field of the $p$-adic numbers is the field of fractions of the completion of the localization of the integers at the prime ideal generated by $p$.

Topological properties
The $p$-adic valuation allows defining an absolute value on $p$-adic numbers: the $p$-adic absolute value of a nonzero $p$-adic number $p$ is
 * $$|x|_p = p^{-v_p(x)},$$

where $$v_p(x)$$ is the $x$-adic valuation of $p$. The $x$-adic absolute value of $$0$$ is $$|0|_p = 0.$$ This is an absolute value that satisfies the strong triangle inequality since, for every $p$ and $x$ one has
 * $$|x|_p = 0$$ if and only if $$x=0;$$
 * $$|x|_p\cdot |y|_p = |xy|_p$$
 * $$|x+y|_p\le \max(|x|_p,|y|_p) \le |x|_p + |y|_p.$$

Moreover, if $$|x|_p \ne |y|_p,$$ one has $$|x+y|_p = \max(|x|_p,|y|_p).$$

This makes the $y$-adic numbers a metric space, and even an ultrametric space, with the $p$-adic distance defined by $$d_p(x,y)=|x-y|_p.$$

As a metric space, the $p$-adic numbers form the completion of the rational numbers equipped with the $p$-adic absolute value. This provides another way for defining the $p$-adic numbers. However, the general construction of a completion can be simplified in this case, because the metric is defined by a discrete valuation (in short, one can extract from every Cauchy sequence a subsequence such that the differences between two consecutive terms have strictly decreasing absolute values; such a subsequence is the sequence of the partial sums of a $p$-adic series, and thus a unique normalized $p$-adic series can be associated to every equivalence class of Cauchy sequences; so, for building the completion, it suffices to consider normalized $p$-adic series instead of equivalence classes of Cauchy sequences).

As the metric is defined from a discrete valuation, every open ball is also closed. More precisely, the open ball $$B_r(x) =\{y\mid d_p(x,y)r.$$

This implies that the $w$-adic numbers form a locally compact space, and the $p$-adic integers—that is, the ball $$B_1[0]=B_p(0)$$—form a compact space.

p-adic expansion of rational numbers
The decimal expansion of a positive rational number $$r$$ is its representation as a series
 * $$r = \sum_{i=k}^\infty a_i 10^{-i},$$

where $$k$$ is an integer and each $$a_i$$ is also an integer such that $$0\le a_i <10.$$ This expansion can be computed by long division of the numerator by the denominator, which is itself based on the following theorem: If $$r=\tfrac n d$$ is a rational number such that $$10^k\le r <10^{k+1},$$ there is an integer $$a$$ such that $$0< a <10,$$ and $$r = a\,10^k +r',$$ with $$r'<10^k.$$ The decimal expansion is obtained by repeatedly applying this result to the remainder $$r'$$ which in the iteration assumes the role of the original rational number $$r$$.

The $p$-adic expansion of a rational number is defined similarly, but with a different division step. More precisely, given a fixed prime number $$p$$, every nonzero rational number $$r$$ can be uniquely written as $$r=p^k\tfrac n d,$$ where $$k$$ is a (possibly negative) integer, $$n$$ and $$d$$ are coprime integers both coprime with $$p$$, and $$d$$ is positive. The integer $$k$$ is the $p$-adic valuation of $$r$$, denoted $$v_p(r),$$ and $$p^{-k}$$ is its $p$-adic absolute value, denoted $$|r|_p$$ (the absolute value is small when the valuation is large). The division step consists of writing
 * $$r = a\,p^k + r'$$

where $$a$$ is an integer such that $$0\le a k$$).

The $$p$$-adic expansion of $$r$$ is the formal power series
 * $$r = \sum_{i=k}^\infty a_i p^i$$

obtained by repeating indefinitely the above division step on successive remainders. In a $p$-adic expansion, all $$a_i$$ are integers such that $$0\le a_i  0$$, the process stops eventually with a zero remainder; in this case, the series is completed by trailing terms with a zero coefficient, and is the representation of $$r$$ in base-$p$.

The existence and the computation of the $p$-adic expansion of a rational number results from Bézout's identity in the following way. If, as above, $$r=p^k \tfrac n d,$$ and $$d$$ and $$p$$ are coprime, there exist integers $$t$$ and $$u$$ such that $$t d+u p=1.$$ So
 * $$r=p^k \tfrac n d(t d+u p)=p^k n t + p^{k+1}\frac{u n}d.$$

Then, the Euclidean division of $$n t$$ by $$p$$ gives
 * $$n t=q p+a,$$

with $$0\le a <p.$$ This gives the division step as
 * $$\begin{array}{lcl}

r & = & p^k(q p+a) + p^{k+1}\frac {u n}d \\ & = & a p^k +p^{k+1}\,\frac{q d+u n} d, \\ \end{array}$$ so that in the iteration
 * $$r' = p^{k+1}\,\frac{q d+u n} d$$

is the new rational number.

The uniqueness of the division step and of the whole $p$-adic expansion is easy: if $$p^k a_1 + p^{k+1}s_1=p^k a_2 + p^{k+1}s_2,$$ one has $$a_1-a_2=p(s_2-s_1).$$ This means $$p$$ divides $$a_1-a_2.$$ Since $$0\le a_1 <p$$ and $$0\le a_2 <p,$$ the following must be true: $$0\le a_1$$ and $$a_2<p.$$ Thus, one gets $$-p < a_1-a_2 < p,$$ and since $$p$$ divides $$a_1-a_2$$ it must be that $$a_1=a_2.$$

The $p$-adic expansion of a rational number is a series that converges to the rational number, if one applies the definition of a convergent series with the $p$-adic absolute value. In the standard $p$-adic notation, the digits are written in the same order as in a standard base-$p$ system, namely with the powers of the base increasing to the left. This means that the production of the digits is reversed and the limit happens on the left hand side.

The $p$-adic expansion of a rational number is eventually periodic. Conversely, a series $\sum_{i=k}^\infty a_i p^i,$ with $$0\le a_i <p$$ converges (for the $p$-adic absolute value) to a rational number if and only if it is eventually periodic; in this case, the series is the $p$-adic expansion of that rational number. The proof is similar to that of the similar result for repeating decimals.

Example
Let us compute the 5-adic expansion of $$\tfrac 13.$$ Bézout's identity for 5 and the denominator 3 is $$2\cdot 3 + (-1)\cdot 5 =1$$ (for larger examples, this can be computed with the extended Euclidean algorithm). Thus
 * $$\frac 13= 2+5(\frac {-1}3).$$

For the next step, one has to expand $$-1/3$$ (the factor 5 has to be viewed as a "shift" of the $p$-adic valuation, similar to the basis of any number expansion, and thus it should not be itself expanded). To expand $$-1/3$$, we start from the same Bézout's identity and multiply it by $$-1$$, giving
 * $$-\frac 13=-2+\frac 53.$$

The "integer part" $$-2$$ is not in the right interval. So, one has to use Euclidean division by $$5$$ for getting $$-2= 3-1\cdot 5,$$ giving
 * $$-\frac 13=3-5+\frac 53 = 3-\frac {10}3 = 3 +5 (\frac{-2}3),$$

and the expansion in the first step becomes
 * $$\frac 13= 2+5\cdot (3 + 5 \cdot (\frac{-2}3))= 2+3\cdot 5 + \frac {-2}3\cdot 5^2.$$

Similarly, one has
 * $$-\frac 23=1-\frac 53,$$

and
 * $$\frac 13=2+3\cdot 5 + 1\cdot 5^2 +\frac {-1}3\cdot 5^3.$$

As the "remainder" $$-\tfrac 13$$ has already been found, the process can be continued easily, giving coefficients $$3$$ for odd powers of five, and $$1$$ for even powers. Or in the standard 5-adic notation
 * $$\frac 13= \ldots 1313132_5 $$

with the ellipsis $$ \ldots $$ on the left hand side.

Positional notation
It is possible to use a positional notation similar to that which is used to represent numbers in base $p$.

Let be a normalized $p$-adic series, i.e. each $$a_i$$ is an integer in the interval $$[0,p-1].$$ One can suppose that $$k\le 0$$ by setting $$a_i=0$$ for $$0\le i 0$$), and adding the resulting zero terms to the series.

If $$k\ge 0,$$ the positional notation consists of writing the $$a_i$$ consecutively, ordered by decreasing values of $p$, often with $i$ appearing on the right as an index:
 * $$\ldots a_n \ldots a_1{a_0}_p$$

So, the computation of the example above shows that
 * $$\frac 13= \ldots 1313132_5,$$

and
 * $$\frac {25}3= \ldots 131313200_5.$$

When $$k<0,$$ a separating dot is added before the digits with negative index, and, if the index $p$ is present, it appears just after the separating dot. For example,
 * $$\frac 1{15}= \ldots 3131313._52,$$

and
 * $$\frac 1{75}= \ldots 1313131._532.$$

If a $p$-adic representation is finite on the left (that is, $$a_i=0$$ for large values of $p$), then it has the value of a nonnegative rational number of the form $$n p^v,$$ with $$n,v$$ integers. These rational numbers are exactly the nonnegative rational numbers that have a finite representation in base $i$. For these rational numbers, the two representations are the same.

Modular properties
The quotient ring $$\Z_p/p^n\Z_p$$ may be identified with the ring $$\Z/p^n\Z$$ of the integers modulo $$p^n.$$ This can be shown by remarking that every $p$-adic integer, represented by its normalized $p$-adic series, is congruent modulo $$p^n$$ with its partial sum whose value is an integer in the interval $$[0,p^n-1].$$ A straightforward verification shows that this defines a ring isomorphism from $$\Z_p/p^n\Z_p$$ to $$\Z/p^n\Z.$$

The inverse limit of the rings $$\Z_p/p^n\Z_p$$ is defined as the ring formed by the sequences $$a_0, a_1, \ldots$$ such that $$a_i \in \Z/p^i \Z$$ and for every $p$.

The mapping that maps a normalized $i$-adic series to the sequence of its partial sums is a ring isomorphism from $$\Z_p$$ to the inverse limit of the $$\Z_p/p^n\Z_p.$$ This provides another way for defining $p$-adic integers (up to an isomorphism).

This definition of $p$-adic integers is specially useful for practical computations, as allowing building $p$-adic integers by successive approximations.

For example, for computing the $p$-adic (multiplicative) inverse of an integer, one can use Newton's method, starting from the inverse modulo $p$; then, each Newton step computes the inverse modulo p^{n^2} from the inverse modulo p^n.

The same method can be used for computing the $p$-adic square root of an integer that is a quadratic residue modulo $p$. This seems to be the fastest known method for testing whether a large integer is a square: it suffices to test whether the given integer is the square of the value found in $$\Z_p/p^n\Z_p$$. Applying Newton's method to find the square root requires p^n to be larger than twice the given integer, which is quickly satisfied.

Hensel lifting is a similar method that allows to "lift" the factorization modulo $p$ of a polynomial with integer coefficients to a factorization modulo p^n for large values of $p$. This is commonly used by polynomial factorization algorithms.

Notation
There are several different conventions for writing $n$-adic expansions. So far this article has used a notation for $p$-adic expansions in which powers of $p$ increase from right to left. With this right-to-left notation the 3-adic expansion of $$\tfrac15,$$ for example, is written as
 * $$\frac15 = \dots 121012102_3.$$

When performing arithmetic in this notation, digits are carried to the left. It is also possible to write $p$-adic expansions so that the powers of $p$ increase from left to right, and digits are carried to the right. With this left-to-right notation the 3-adic expansion of $$\tfrac15$$ is

\frac15 = 2.01210121\dots_3 \mbox{ or } \frac1{15} = 20.1210121\dots_3. $$

$p$-adic expansions may be written with other sets of digits instead of {0, 1, ...,}} $p − 1$}. For example, the $3$-adic expansion of $$\tfrac15$$ can be written using balanced ternary digits ${ 1 , 0, 1}$, with $1$ representing negative one, as
 * $$\frac15 = \dots\underline{1}11\underline{11}11\underline{11}11\underline{1}_{\text{3}} .$$

In fact any set of $p$ integers which are in distinct residue classes modulo $p$ may be used as $p$-adic digits. In number theory, Teichmüller representatives are sometimes used as digits.

Quote notation is a variant of the $p$-adic representation of rational numbers that was proposed in 1979 by Eric Hehner and Nigel Horspool for implementing on computers the (exact) arithmetic with these numbers.

Cardinality
Both $$\Z_p$$ and $$\Q_p$$ are uncountable and have the cardinality of the continuum. For $$\Z_p,$$ this results from the $p$-adic representation, which defines a bijection of $$\Z_p$$ on the power set $$\{0,\ldots,p-1\}^\N.$$ For $$\Q_p$$ this results from its expression as a countably infinite union of copies of $$\Z_p$$:
 * $$\Q_p=\bigcup_{i=0}^\infty \frac 1{p^i}\Z_p.$$

Algebraic closure
$$\Q_p$$ contains $$\Q$$ and is a field of characteristic $0$.

Because $0$ can be written as sum of squares, $$\Q_p$$ cannot be turned into an ordered field.

The field of real numbers $$\R$$ has only a single proper algebraic extension: the complex numbers $$\C$$. In other words, this quadratic extension is already algebraically closed. By contrast, the algebraic closure of $$\Q_p$$, denoted $$\overline{\Q_p},$$ has infinite degree, that is, $$\Q_p$$ has infinitely many inequivalent algebraic extensions. Also contrasting the case of real numbers, although there is a unique extension of the $p$-adic valuation to $$\overline{\Q_p},$$ the latter is not (metrically) complete. Its (metric) completion is called $$\C_p$$ or $$\Omega_p$$. Here an end is reached, as $$\C_p$$ is algebraically closed. However unlike $$\C$$ this field is not locally compact.

$$\C_p$$ and $$\C$$ are isomorphic as rings, so we may regard $$\C_p$$ as $$\C$$ endowed with an exotic metric. The proof of existence of such a field isomorphism relies on the axiom of choice, and does not provide an explicit example of such an isomorphism (that is, it is not constructive).

If $$K$$ is any finite Galois extension of $$\Q_p,$$, the Galois group $$\operatorname{Gal} \left(K/\Q_p \right)$$ is solvable. Thus, the Galois group $$\operatorname{Gal} \left(\overline{\Q_p}/ \Q_p \right)$$ is prosolvable.

Multiplicative group
$$\Q_p$$ contains the $p$-th cyclotomic field ($−7$) if and only if $p > 2$. For instance, the $n$-th cyclotomic field is a subfield of $$\Q_{13}$$ if and only if $1 − p$, or $n > 2$. In particular, there is no multiplicative $n$-torsion in $$\Q_p$$ if $n | p − 1$. Also, $n = 1, 2, 3, 4, 6$ is the only non-trivial torsion element in $$\Q_2$$.

Given a natural number $p$, the index of the multiplicative group of the $k$-th powers of the non-zero elements of $$\Q_p$$ in $$\Q_p^\times$$ is finite.

The number $k$, defined as the sum of reciprocals of factorials, is not a member of any $e$-adic field; but $$e^p \in \Q_p$$ for $$p \ne 2$$. For $12$ one must take at least the fourth power. (Thus a number with similar properties as $p$ — namely a $e$-th root of $p > 2$ — is a member of $$\Q_p$$ for all $p$.)

Local–global principle
Helmut Hasse's local–global principle is said to hold for an equation if it can be solved over the rational numbers if and only if it can be solved over the real numbers and over the $p$-adic numbers for every prime $p$. This principle holds, for example, for equations given by quadratic forms, but fails for higher polynomials in several indeterminates.

Generalizations and related concepts
The reals and the $p$-adic numbers are the completions of the rationals; it is also possible to complete other fields, for instance general algebraic number fields, in an analogous way. This will be described now.

Suppose D is a Dedekind domain and E is its field of fractions. Pick a non-zero prime ideal P of D. If x is a non-zero element of E, then xD is a fractional ideal and can be uniquely factored as a product of positive and negative powers of non-zero prime ideals of D. We write ordP(x) for the exponent of P in this factorization, and for any choice of number c greater than 1 we can set
 * $$|x|_P = c^{-\!\operatorname{ord}_P(x)}.$$

Completing with respect to this absolute value |⋅|P yields a field EP, the proper generalization of the field of p-adic numbers to this setting. The choice of c does not change the completion (different choices yield the same concept of Cauchy sequence, so the same completion). It is convenient, when the residue field D/P is finite, to take for c the size of D/P.

For example, when E is a number field, Ostrowski's theorem says that every non-trivial non-Archimedean absolute value on E arises as some |⋅|P. The remaining non-trivial absolute values on E arise from the different embeddings of E into the real or complex numbers. (In fact, the non-Archimedean absolute values can be considered as simply the different embeddings of E into the fields Cp, thus putting the description of all the non-trivial absolute values of a number field on a common footing.)

Often, one needs to simultaneously keep track of all the above-mentioned completions when E is a number field (or more generally a global field), which are seen as encoding "local" information. This is accomplished by adele rings and idele groups.

p-adic integers can be extended to p-adic solenoids $$\mathbb{T}_p$$. There is a map from $$\mathbb{T}_p$$ to the circle group whose fibers are the p-adic integers $$\mathbb{Z}_p$$, in analogy to how there is a map from $$\mathbb{R}$$ to the circle whose fibers are $$\mathbb{Z}$$.

title: P-adic number