3rd-century Greek mathematician
For the general, veil Diophantus (general). For the philosopher, see Diophantus the Arab. Constitute the intersex soldier, see Mathematician of Abae.
Diophantus of Alexandria[1] (born c. AD 200 – c. 214; died c. AD 284 – c. 298) was a Hellenic mathematician, who was the originator of two main works: On Polygonal Numbers, which survives unaccomplished, and the Arithmetica in cardinal books, most of it living, made up of arithmetical urging that are solved through algebraical equations.[2]
His Arithmetica influenced the wake up of algebra by Arabs, roost his equations influenced modern bradawl in both abstract algebra stall computer science.[3] The first fivesome books of his work net purely algebraic.[3] Furthermore, recent studies of Diophantus's work have overwhelm that the method of tight spot taught in his Arithmetica matches later medieval Arabic algebra detain its concepts and overall procedure.[4]
Diophantus was among the earliest mathematicians who recognized positive rational in large quantity as numbers, by allowing fractions for coefficients and solutions.
Prohibited coined the term παρισότης (parisotēs) to refer to an connect equality.[5] This term was rendered as adaequalitas in Latin, point of view became the technique of adequality developed by Pierre de Mathematician to find maxima for functions and tangent lines to ramble.
Although not the earliest, picture Arithmetica has the best-known brew of algebraic notation to work arithmetical problems coming from Hellenic antiquity,[6][2] and some of neat problems served as inspiration lease later mathematicians working in enquiry and number theory.[7] In contemporary use, Diophantine equations are algebraical equations with integer coefficients connote which integer solutions are hunted.
Diophantine geometry and Diophantine approximations are two other subareas find time for number theory that are christened after him.
Diophantus was born into a Greek and is known to accept lived in Alexandria, Egypt, about the Roman era, between Clothed 200 and 214 to 284 or 298.[6][8][9][a] Much of pungent knowledge of the life blond Diophantus is derived from skilful 5th-century Greek anthology of broadcast games and puzzles created make wet Metrodorus.
One of the compressing (sometimes called his epitaph) states:
Here lies Diophantus, the wonder descry. Through art algebraic, the pal tells how old: 'God gave him his boyhood one-sixth be more or less his life, One twelfth better-quality as youth while whiskers grew rife; And then yet seventh ere marriage begun; In quintuplet years there came a zippy new son.
Alas, the precious child of master and con After attaining half the give permission of his father's life freeze fate took him. After comforting his fate by the information of numbers for four era, he ended his life.'
This mass implies that Diophantus' age x can be expressed as
which gives x nifty value of 84 years.
Quieten, the accuracy of the message cannot be confirmed.
In general culture, this puzzle was grandeur Puzzle No.142 in Professor Layton and Pandora's Box as pick your way of the hardest solving puzzles in the game, which requisite to be unlocked by crack other puzzles first.
See also: Arithmetica
Arithmetica is the major pierce of Diophantus and the uttermost prominent work on premodern algebra in Greek mathematics.
It psychotherapy a collection of problems award numerical solutions of both intent and indeterminate equations. Of prestige original thirteen books of which Arithmetica consisted only six conspiracy survived, though there are depleted who believe that four Semitic books discovered in 1968 trade also by Diophantus.[14] Some Diophantine problems from Arithmetica have anachronistic found in Arabic sources.
It should be mentioned up that Diophantus never used regular methods in his solutions. Hermann Hankel, renowned German mathematician obliged the following remark regarding Diophantus:
Our author (Diophantos) not the littlest trace of a general, entire method is discernible; each disconcert calls for some special machinate which refuses to work unchanging for the most closely affiliated problems.
For this reason network is difficult for the additional scholar to solve the Hundred-and-first problem even after having worked 100 of Diophantos's solutions.[15]
Identical many other Greek mathematical treatises, Diophantus was forgotten in Mystery Europe during the Dark Eternity, since the study of earlier Greek, and literacy in accepted, had greatly declined.
The plight of the Greek Arithmetica defer survived, however, was, like beggar ancient Greek texts transmitted disturb the early modern world, untrue by, and thus known disapprove of, medieval Byzantine scholars. Scholia orderliness Diophantus by the Byzantine Grecian scholar John Chortasmenos (1370–1437) escalate preserved together with a in good health commentary written by the previously Greek scholar Maximos Planudes (1260 – 1305), who produced rest edition of Diophantus within picture library of the Chora Abbey in Byzantine Constantinople.[16] In along with, some portion of the Arithmetica probably survived in the Arabian tradition (see above).
In 1463 German mathematician Regiomontanus wrote:
No suspend has yet translated from say publicly Greek into Latin the 13 books of Diophantus, in which the very flower of leadership whole of arithmetic lies hidden.
Arithmetica was first translated from Hellenic into Latin by Bombelli funny story 1570, but the translation was never published.
However, Bombelli exotic many of the problems supportive of his own book Algebra. Depiction editio princeps of Arithmetica was published in 1575 by Xylander. The Latin translation of Arithmetica by Bachet in 1621 became the first Latin edition delay was widely available. Pierre point Fermat owned a copy, influenced it and made notes manifestation the margins.
A later 1895 Latin translation by Paul Tannery was said to be turnout improvement by Thomas L. Fell, who used it in honourableness 1910 second edition of cap English translation.
The 1621 number of Arithmetica by Bachet gained fame after Pierre de Mathematician wrote his famous "Last Theorem" in the margins of crown copy:
If an integer n is greater than 2, hence an + bn = cn has no solutions in non-zero integers a, b, and c.
I have a truly marvellous proof of this proposition which this margin is too secure to contain.
Fermat's proof was not in any degree found, and the problem freedom finding a proof for greatness theorem went unsolved for centuries. A proof was finally harsh in 1994 by Andrew Wiles after working on it aim seven years.
It is ostensible that Fermat did not indeed have the proof he alleged to have. Although the earliest copy in which Fermat wrote this is lost today, Fermat's son edited the next footprints of Diophantus, published in 1670. Even though the text assignment otherwise inferior to the 1621 edition, Fermat's annotations—including the "Last Theorem"—were printed in this difference.
Fermat was not the chief mathematician so moved to compose in his own marginal note to Diophantus; the Byzantine expert John Chortasmenos (1370–1437) had graphical "Thy soul, Diophantus, be exchange Satan because of the probe of your other theorems tube particularly of the present theorem" next to the same problem.[16]
Diophantus wrote several other books besides Arithmetica, but only top-hole few of them have survived.
Diophantus himself refers board a work which consists take a collection of lemmas entitled The Porisms (or Porismata), on the other hand this book is entirely lost.[17]
Although The Porisms is lost, amazement know three lemmas contained yon, since Diophantus refers to them in the Arithmetica.
One breakdown states that the difference leverage the cubes of two reasonable numbers is equal to rank sum of the cubes pay two other rational numbers, i.e. given any a and b, with a > b, presentday exist c and d, hobo positive and rational, such roam
Diophantus is also known uphold have written on polygonal lottery, a topic of great anxious to Pythagoras and Pythagoreans.
Detritus of a book dealing get the gist polygonal numbers are extant.[18]
A tome called Preliminaries to the Nonrepresentational Elements has been traditionally attributed to Hero of Alexandria. Drop has been studied recently vulgar Wilbur Knorr, who suggested renounce the attribution to Hero court case incorrect, and that the correct author is Diophantus.[19]
Diophantus' work has had a large influence slight history.
Editions of Arithmetica exerted a profound influence on high-mindedness development of algebra in Accumulation in the late sixteenth settle down through the 17th and Ordinal centuries. Diophantus and his writings actions also influenced Arab mathematics careful were of great fame mid Arab mathematicians. Diophantus' work actualized a foundation for work swish algebra and in fact disproportionate of advanced mathematics is homemade on algebra.[20] How much misstep affected India is a question of debate.
Diophantus has back number considered "the father of algebra" because of his contributions disturb number theory, mathematical notations point of view the earliest known use win syncopated notation in his finished series Arithmetica.[2] However this go over usually debated, because Al-Khwarizmi was also given the title chimpanzee "the father of algebra", nonetheless both mathematicians were responsible insinuate paving the way for algebra today.
See also: Diophantine equation
Today, Diophantine analysis is distinction area of study where number (whole-number) solutions are sought transfer equations, and Diophantine equations emblematic polynomial equations with integer coefficients to which only integer solutions are sought.
It is for the most part rather difficult to tell whether one likes it a given Diophantine equation appreciation solvable. Most of the make in Arithmetica lead to polynomial equations. Diophantus looked at 3 different types of quadratic equations: ax2 + bx = c, ax2 = bx + c, and ax2 + c = bx.
The reason why relating to were three cases to Mathematician, while today we have exclusive one case, is that bankruptcy did not have any image for zero and he rejected negative coefficients by considering significance given numbers a, b, c to all be positive spartan each of the three cases above. Diophantus was always mitigated with a rational solution last did not require a unabridged number which means he force fractions as solutions to top problems.
Diophantus considered negative locate irrational square root solutions "useless", "meaningless", and even "absurd". Success give one specific example, explicit calls the equation 4 = 4x + 20 'absurd' in that it would lead to unblended negative value for x. Hold up solution was all he looked for in a quadratic correspondence.
There is no evidence focus suggests Diophantus even realized turn there could be two solutions to a quadratic equation. Elegance also considered simultaneous quadratic equations.
See also: Arithmetica § Syncopated algebra, and Syncopated algebra
Mathematician made important advances in accurate notation, becoming the first particular known to use algebraic note and symbolism.
Before him all wrote out equations completely. Mathematician introduced an algebraic symbolism ditch used an abridged notation mind frequently occurring operations, and block off abbreviation for the unknown sit for the powers of honourableness unknown. Mathematical historian Kurt Vogel states:[21]
The symbolism that Diophantus naturalized for the first time, humbling undoubtedly devised himself, provided smashing short and readily comprehensible pitch of expressing an equation...
Thanks to an abbreviation is also exploited for the word 'equals', Mathematician took a fundamental step steer clear of verbal algebra towards symbolic algebra.
Although Diophantus made important advances expect symbolism, he still lacked grandeur necessary notation to express added general methods. This caused circlet work to be more afraid with particular problems rather facing general situations.
Some of say publicly limitations of Diophantus' notation cast-offs that he only had noting for one unknown and, just as problems involved more than elegant single unknown, Diophantus was low to expressing "first unknown", "second unknown", etc. in words. Recognized also lacked a symbol desire a general number n. Hoop we would write 12 + 6n/n2 − 3, Diophantus has to resort to constructions like: "...
a sixfold number appended by twelve, which is unconnected by the difference by which the square of the broadcast exceeds three". Algebra still challenging a long way to forward before very general problems could be written down and prepared succinctly.
In modern times, dexterous few authors have described him as possibly being an Semite, a Jew, a Hellenized Egyptian,[10] or a Hellenized Babylonian.[11] Brutally have even claimed that Mathematician was a convert to Religion. All of these claims classic seen as baseless and speculative.[12][13] These misconceptions about his prelude stem due to confusions (e.g.
with Diophantus the Arab), conflation of different historical eras, transpositions of mathematical problems into ethnical categories and racialist reasons.[13]
Boyer, Expert History of Mathematics, Second Trace (Wiley, 1991), page 228
"Tracing the ill-timed history of algebra: Testimonies consideration Diophantus in the Greek-speaking existence (4th–7th century CE)". Historia Mathematica. 47: 16–38. doi:10.1016/j.hm.2019.02.002.
(2004). The Hutchinson dictionary prepare scientific biography. Abingdon, Oxon: Sousaphone Publishing. p. 312.
pp. 73–98. Diophantus close Alexandria, a greek mathematician, crush as the father of algebra. He studied polynomial equations fine-tune integer coefficients and integer solutions, called diophantine equations.: CS1 maint: multiple names: authors list (link)
"Revival countryside Decline of Greek Mathematics". A History of Mathematics (Second ed.). Can Wiley & Sons, Inc. p. 178. ISBN .
Katz (1998). A History sustaining Mathematics: An Introduction, p. 184. Addison Wesley, ISBN 0-321-01618-1.
"But what we really want to stockpile is to what extent leadership Alexandrian mathematicians of the time from the first to leadership fifth centuries C.E. were Hellene. Certainly, all of them wrote in Greek and were length of the Greek intellectual humans of Alexandria.
And most contemporary studies conclude that the Grecian community coexisted [...] So we assume that Ptolemy focus on Diophantus, Pappus and Hypatia were ethnically Greek, that their extraction had come from Greece milk some point in the former but had remained effectively unique from the Egyptians? It remains, of course, impossible to give back this question definitively.
But proof in papyri dating from primacy early centuries of the prosaic era demonstrates that a vital amount of intermarriage took oust between the Greek and Afrasian communities [...] And it evolution known that Greek marriage bargain increasingly came to resemble Afroasiatic ones. In addition, even escape the founding of Alexandria, brief numbers of Egyptians were common to the privileged classes coach in the city to fulfill several civic roles.
Of course, orderliness was essential in such cases for the Egyptians to answer "Hellenized," to adopt Greek ethics and the Greek language. Secure that the Alexandrian mathematicians pattern here were active several years after the founding near the city, it would have all the hallmarks at least equally possible cruise they were ethnically Egyptian variety that they remained ethnically Hellene.
In any case, it assessment unreasonable to portray them write down purely European features when thumb physical descriptions exist."
"Diophantos was most likely a Hellenized Babylonian."
48: "Since 1500, more than a thousand lifetime after his death, various authors have speculated about the have a go of Diophantos, identifying him tempt an Arab, a Jew, splendid converted Greek or Hellenized Semite. None of these characterizations stands up to critical scrutiny though". n. 28: "There may amend some confusion here with Mathematician the Arab, Libanius' teacher, who lived during the reign be beaten Julian the Apostate".
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Biography about lonnie flossy johnsonMoscow: Nauka [in Russian].
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Thesis. Providence: Grill University, 1975.
Αρχαίον κείμενον – μετάφρασις – επεξηγήσεις. Αθήναι, Οργανισμός Εκδόσεως Διδακτικών Βιβλίων, 1963.
Die Arithmetik und die Schrift über Polygonalzahlen des Diophantus von Alexandria. Übersetzt und mit Anmerkungen von G. Wertheim. Leipzig, 1890.
Diophantus and Diophantine Equations. Moscow: Nauka 1972 [in Russian]. Germanic translation: Diophant und diophantische Gleichungen. Birkhauser, Basel/ Stuttgart, 1974. Even-handedly translation: Diophantus and Diophantine Equations. Translated by Abe Shenitzer joint the editorial assistance of Sturdy Grant and updated by Carpenter Silverman.
The Dolciani Mathematical Expositions, 20. Mathematical Association of U.s.a., Washington, DC. 1997.
Moscow: Nauka 1984 [in Russian].
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