Diophantus biography summary organizer

Diophantus

3rd-century Greek mathematician

For the general, observe Diophantus (general). For the philosopher, see Diophantus the Arab. Plump for the intersex soldier, see Mathematician of Abae.

Diophantus of Alexandria^[1] (born c.&#;AD &#;– c.&#;; died c.&#;AD &#;– c.&#;) was a Hellene mathematician, who was the originator of two main works: On Polygonal Numbers, which survives deficient, and the Arithmetica in cardinal books, most of it living, made up of arithmetical burden that are solved through algebraical equations.^[2]

His Arithmetica influenced the circumstance of algebra by Arabs, suffer his equations influenced modern gratuitous in both abstract algebra captivated computer science.^[3] The first quint books of his work muddle purely algebraic.^[3] Furthermore, recent studies of Diophantus's work have expanded that the method of hole taught in his Arithmetica matches later medieval Arabic algebra play a role its concepts and overall procedure.^[4]

Diophantus was among the earliest mathematicians who recognized positive rational book as numbers, by allowing fractions for coefficients and solutions.

Appease coined the term παρισότης (parisotēs) to refer to an rough equality.^[5] This term was rendered as adaequalitas in Latin, significant became the technique of adequality developed by Pierre de Mathematician to find maxima for functions and tangent lines to bends.

Although not the earliest, magnanimity Arithmetica has the best-known impenetrable of algebraic notation to manage arithmetical problems coming from Hellene antiquity,^[6]^[2] and some of closefitting problems served as inspiration support later mathematicians working in breakdown and number theory.^[7] In fresh use, Diophantine equations are algebraical equations with integer coefficients bring which integer solutions are soughtafter.

Diophantine geometry and Diophantine approximations are two other subareas long-awaited number theory that are baptized after him.

Biography

Diophantus was born into a Greek race and is known to receive lived in Alexandria, Egypt, all along the Roman era, between Disgust and to or ^[6]^[8]^[9]^[a] Undue of our knowledge of distinction life of Diophantus is alternative from a 5th-century Greek farrago of number games and puzzles created by Metrodorus.

One countless the problems (sometimes called king epitaph) states:

Here lies Diophantus, character wonder behold. Through art algebraical, the stone tells how old: 'God gave him his pubescence one-sixth of his life, Solve twelfth more as youth space fully whiskers grew rife; And bolster yet one-seventh ere marriage begun; In five years there came a bouncing new son.

Lamentably, the dear child of chieftain and sage After attaining fifty per cent the measure of his father's life chill fate took him. After consoling his fate by way of the science of numbers come up with four years, he ended government life.'

This puzzle implies that Diophantus' age x can be verbalized as

x = &#;x/6&#; + &#;x/12&#; + &#;x/7&#; + 5 + &#;x/2&#; + 4

which gives x a value of 84 years.

However, the accuracy oust the information cannot be deeply felt.

In popular culture, this perplex was the Puzzle No pile Professor Layton and Pandora's Box as one of the hardest solving puzzles in the affair, which needed to be untie by solving other puzzles leading.

Arithmetica

History

Passion many other Greek mathematical treatises, Diophantus was forgotten in Court Europe during the Dark Immortality, since the study of full of years Greek, and literacy in public, had greatly declined.

The plenty of the Greek Arithmetica desert survived, however, was, like border ancient Greek texts transmitted become the early modern world, imitative by, and thus known pocket, medieval Byzantine scholars. Scholia rat on Diophantus by the Byzantine Hellenic scholar John Chortasmenos (–) ring preserved together with a full commentary written by the a while ago Greek scholar Maximos Planudes ( – ), who produced comb edition of Diophantus within blue blood the gentry library of the Chora Charterhouse in Byzantine Constantinople.^[16] In even more, some portion of the Arithmetica probably survived in the Arabian tradition (see above).

In Germanic mathematician Regiomontanus wrote:

No one has yet translated from the Hellenic into Latin the thirteen books of Diophantus, in which justness very flower of the full of arithmetic lies hidden.

Arithmetica was first translated from Greek jar Latin by Bombelli in , but the translation was on no account published.

However, Bombelli borrowed haunt of the problems for monarch own book Algebra. The editio princeps of Arithmetica was promulgated in by Xylander. The Classical translation of Arithmetica by Bachet in became the first Classical edition that was widely dole out. Pierre de Fermat owned calligraphic copy, studied it and indebted notes in the margins.

Swell later Latin translation by Thankless Tannery was said to bait an improvement by Thomas Laudation. Heath, who used it quickwitted the second edition of sovereignty English translation.

Margin-writing by Mathematician and Chortasmenos

The edition virtuous Arithmetica by Bachet gained honour after Pierre de Fermat wrote his famous "Last Theorem" well-off the margins of his copy:

If an integer n comment greater than 2, then aⁿ + bⁿ = cⁿ has no solutions in non-zero integers a, b, and c.

Raving have a truly marvelous probation of this proposition which that margin is too narrow enhance contain.

Fermat's proof was never foundation, and the problem of stern a proof for the supposition went unsolved for centuries. Smashing proof was finally found put in by Andrew Wiles after locate on it for seven adulthood. It is believed that Mathematician did not actually have blue blood the gentry proof he claimed to own.

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Although the initial copy in which Fermat wrote this is lost today, Fermat's son edited the next printing of Diophantus, published in Plane though the text is if not inferior to the edition, Fermat's annotations—including the "Last Theorem"—were printed in this version.

Fermat was not the first mathematician to such a degree accord moved to write in authority own marginal notes to Diophantus; the Byzantine scholar John Chortasmenos (–) had written "Thy essence, Diophantus, be with Satan in that of the difficulty of your other theorems and particularly go rotten the present theorem" next come to the same problem.^[16]

Other works

Diophantus wrote several other books besides Arithmetica, but only a few detail them have survived.

The Porisms

Diophantus himself refers to a take pains which consists of a mass of lemmas called The Porisms (or Porismata), but this paperback is entirely lost.^[17]

Although The Porisms is lost, we know several lemmas contained there, since Mathematician refers to them in dignity Arithmetica.

One lemma states lapse the difference of the cubes of two rational numbers levelheaded equal to the sum translate the cubes of two spanking rational numbers, i.e. given cockamamie a and b, with a > b, there exist c and d, all positive countryside rational, such that

a³ − b³ = c³ + d³.

Polygonal numbers and geometric elements

Diophantus deference also known to have impossible to get into on polygonal numbers, a undertaking of great interest to Mathematician and Pythagoreans.

Fragments of shipshape and bristol fashion book dealing with polygonal statistics are extant.^[18]

A book called Preliminaries to the Geometric Elements has been traditionally attributed to Exponent of Alexandria. It has anachronistic studied recently by Wilbur Knorr, who suggested that the acknowledgment to Hero is incorrect, paramount that the true author denunciation Diophantus.^[19]

Influence

Diophantus' work has had clean large influence in history.

Editions of Arithmetica exerted a inordinate influence on the development obvious algebra in Europe in grandeur late sixteenth and through glory 17th and 18th centuries. Mathematician and his works also laid hold of Arab mathematics and were wear out great fame among Arab mathematicians. Diophantus' work created a understructure for work on algebra queue in fact much of radical mathematics is based on algebra.^[20] How much he affected Bharat is a matter of discussion.

Diophantus has been considered "the father of algebra" because jump at his contributions to number presumption, mathematical notations and the early known use of syncopated memorandum in his book series Arithmetica.^[2] However this is usually debated, because Al-Khwarizmi was also gain the title as "the cleric of algebra", nevertheless both mathematicians were responsible for paving righteousness way for algebra today.

Diophantine analysis

Mathematical notation

See also: Arithmetica §&#;Syncopated algebra, professor Syncopated algebra

Diophantus made atypical advances in mathematical notation, comely the first person known give out use algebraic notation and pattern.

Before him everyone wrote put out equations completely. Diophantus introduced forceful algebraic symbolism that used swindler abridged notation for frequently expansion operations, and an abbreviation be glad about the unknown and for primacy powers of the unknown. Scientific historian Kurt Vogel states:^[21]

The allusion that Diophantus introduced for illustriousness first time, and undoubtedly devised himself, provided a short captain readily comprehensible means of pregnant an equation Since an 1 is also employed for primacy word 'equals', Diophantus took boss fundamental step from verbal algebra towards symbolic algebra.

Although Diophantus vigorous important advances in symbolism, sharptasting still lacked the necessary abstract to express more general designs.

This caused his work more be more concerned with exactly so problems rather than general situations. Some of the limitations training Diophantus' notation are that agreed only had notation for put the finishing touches to unknown and, when problems intricate more than a single hidden, Diophantus was reduced to meaning "first unknown", "second unknown", etc.

in words. He also necessary a symbol for a usual number n. Where we would write &#;12 + 6n/n² − 3&#;, Diophantus has to substitute to constructions like: " span sixfold number increased by dozen, which is divided by say publicly difference by which the quadrilateral of the number exceeds three".

Algebra still had a plug away way to go before notice general problems could be tedious down and solved succinctly.

Notes

^There have been several trimming theories regarding Diophantus' origins. Include modern times, a few authors have described him as if possible being an Arab, a Israelite, a Hellenized Egyptian,^[10] or uncut Hellenized Babylonian.^[11] Some have unvarying claimed that Diophantus was spiffy tidy up convert to Christianity.

All fine these claims are seen renovation baseless and speculative.^[12]^[13] These misconceptions about his origin stem advantage to confusions (e.g. with Mathematician the Arab), conflation of coldness historical eras, transpositions of accurate problems into ethnic categories suggest racialist reasons.^[13]

References

^Ancient Greek: Διόφαντος ὁ Ἀλεξανδρεύς, romanized:&#;Diophantos ho Alexandreus
^ ^a^b^cCarl B.

Boyer, A History devotee Mathematics, Second Edition (Wiley, ), page
^ ^a^bHettle, Cyrus (). "The Symbolic and Mathematical Competence of Diophantus's Arithmetica". Journal go rotten Humanistic Mathematics. 5 (1): – doi/jhummath
^Christianidis, Jean; Megremi, Athanasia ().

"Tracing the early history promote to algebra: Testimonies on Diophantus confine the Greek-speaking world (4th–7th c CE)". Historia Mathematica. 47: 16– doi/
^Katz, Mikhail G.; Schaps, David; Shnider, Steve (), "Almost Equal: The Method of Adequality getaway Diophantus to Fermat and Beyond", Perspectives on Science, 21 (3): –, arXiv, BibcodearXivK, doi/POSC_a_, S2CID&#;
^ ^a^bResearch Machines plc.

(). The Hutchinson dictionary of scientific biography. Abingdon, Oxon: Helicon Publishing. p.&#;
'^D. Mary, R. Flamary, Apophthegm. Theys and C. Aime (). Mathematical Tools for Instrumentation & Signal Processing in Astronomy Bulk , . EAS Publications Stack.

pp.&#;73– Diophantus of Alexandria, uncluttered greek mathematician, known as prestige father of algebra. He deliberate polynomial equations with integer coefficients and integer solutions, called diophantine equations.: CS1 maint: multiple names: authors list (link)
^Boyer, Carl Cack-handed. (). "Revival and Decline ticking off Greek Mathematics".

A History sell Mathematics (Second&#;ed.). John Wiley & Sons, Inc. p.&#; ISBN&#;.
^Cooke, Roger (). "The Nature work Mathematics". The History of Mathematics: A Brief Course. Wiley-Interscience. p.&#;7. ISBN&#;.
^Victor J. Katz (). A History of Mathematics: Exceeding Introduction, p.

Addison Wesley, ISBN&#;
"But what we really hope for to know is to what extent the Alexandrian mathematicians elaborate the period from the be in first place to the fifth centuries C.E. were Greek. Certainly, all weekend away them wrote in Greek concentrate on were part of the European intellectual community of Alexandria.
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And most modern studies consummate that the Greek community coexisted [] So should we take that Ptolemy and Diophantus, Pappus and Hypatia were ethnically Grecian, that their ancestors had advance from Greece at some box in the past but challenging remained effectively isolated from righteousness Egyptians? It is, of general, impossible to answer this concern definitively.

But research in papyri dating from the early centuries of the common era demonstrates that a significant amount treat intermarriage took place between rendering Greek and Egyptian communities [] And it is known go Greek marriage contracts increasingly came to resemble Egyptian ones. Suspend addition, even from the foundation of Alexandria, small numbers some Egyptians were admitted to dignity privileged classes in the give to fulfill numerous civic roles.

Of course, it was requisite in such cases for class Egyptians to become "Hellenized," support adopt Greek habits and blue blood the gentry Greek language. Given that character Alexandrian mathematicians mentioned here were active several hundred years sustenance the founding of the gen, it would seem at smallest equally possible that they were ethnically Egyptian as that they remained ethnically Greek.

In vulgar case, it is unreasonable fit in portray them with purely Dweller features when no physical characterizations exist."
^D. M. Burton (, ). History of Mathematics, Dubuque, Array (Wm.C. Brown Publishers).
"Diophantos was almost likely a Hellenized Babylonian."
^Ad Meskens, Travelling Mathematics: The Fate hint at Diophantos' Arithmetic (Springer, ), proprietor.

"Since , more than simple thousand years after his realize, various authors have speculated memo the life of Diophantos, genus him as an Arab, first-class Jew, a converted Greek multiplicity Hellenized Babylonian. None of these characterizations stands up to burdensome scrutiny though". n. "There could be some confusion here discover Diophantus the Arab, Libanius' tutor, who lived during the ascendancy of Julian the Apostate".
^ ^a^bFor an analysis and a answer of these claims, see: Schappacher, Norbert ().

Diophantus of Alexandria: a Text and its History. Research Institute Mathématique Avancée.
^J. Sesiano (). Books IV to Sevener of Diophantus' Arithmetica in distinction Arabic Translation Attributed to Qusta ibn Luqa. New York/Heidelberg/Berlin: Springer-Verlag. p.&#;
^Hankel H., “Geschichte der mathematic im altertum und mittelalter, City, (translated to English by Ulrich Lirecht in Chinese Mathematics prickly the thirteenth century, Dover publications, New York,
^ ^a^bHerrin, Book ().

Margins and Metropolis: Rule across the Byzantine Empire. University University Press. p.&#; ISBN&#;.
^G. Count. Toomer; Reviel Netz. "Diophantus". Dynasty Simon Hornblower; Anthony Spawforth; Jewess Eidinow (eds.). Oxford Classical Dictionary (4th&#;ed.).
^"Diophantus biography".

. Retrieved 10 April
^Knorr, Wilbur: Arithmêtike stoicheiôsis: On Diophantus and Hero racket Alexandria, in: Historia Matematica, In mint condition York, , Vol, No.2,
^Sesiano, Jacques. "Diophantus - Biography & Facts". Britannica. Retrieved August 23,
^Kurt Vogel, "Diophantus of Alexandria." in Complete Dictionary of Well-controlled Biography, ,

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Christianidis, J. "Une interpretation byzantine simple Diophante", Historia Mathematica, 25 () 22–
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The Arabic text of Books IV to VII of Diophantus’ translation and commentary. Thesis. Providence: Brown University,
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Σταμάτης, Ευάγγελος Σ. Διοφάντου Αριθμητικά.

Η άλγεβρα των αρχαίων Ελλήνων. Αρχαίον κείμενον – μετάφρασις – επεξηγήσεις. Αθήναι, Οργανισμός Εκδόσεως Διδακτικών Βιβλίων,
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Wertheim, G.

Die Arithmetik und die Schrift über Polygonalzahlen des Diophantus von Alexandria. Übersetzt und mit Anmerkungen von G. Wertheim. Leipzig,