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{{for|the general|Diophantus  (general)}}
[[Image:Diophantus-cover.jpg|right|thumb|200px|Title page of the 1621 edition of Diophantus' ''Arithmetica'', translated into [[Latin]] by [[Claude Gaspard Bachet de Méziriac]].]]
 
'''Diophantus of Alexandria''' ({{lang-grc|'''Διόφαντος ὁ Ἀλεξανδρεύς'''}}. b. between AD 201 and 215, d. between 285 and 299 at age 84), sometimes called "the father of [[algebra]]", was an [[Alexandria]]n [[Greek mathematics|Greek mathematician]]<ref name="Dictionary of Scientific Biography">{{cite book | title=The Hutchinson dictionary of scientific biography | author = Research Machines plc. | location = Abingdon, Oxon | publisher = Helicon Publishing | year = 2004 | page = 312 | quote = '''Diophantus (lived ''c.'' A.D. 270-280)''' Greek mathematician who, in solving linear mathematical problems, developed an early form of algebra.}}</ref><ref>{{cite book|first=Carl B.|last=Boyer|authorlink=Carl Benjamin Boyer|title=A History of Mathematics|edition=Second|publisher=John Wiley & Sons, Inc.|year=1991|chapter=Revival and Decline of Greek Mathematics|page=178|isbn=0-471-54397-7|quote=At the beginning of this period, also known as the Later [[Alexandrian Age]], we find the leading Greek algebraist, Diophantus of Alexandria, and toward its close there appeared the last significant Greek geometer, Pappus of Alexandria.}}</ref><ref>{{cite book|first=Roger|last=Cooke|authorlink=Roger Cooke|title=The History of Mathematics: A Brief Course|publisher=Wiley-Interscience|year=1997|chapter=The Nature of Mathematics|page=7|isbn=0-471-18082-3|quote=Some enlargement in the sphere in which symbols were used occurred in the writings of the third-century Greek mathematician Diophantus of Alexandria, but the same defect was present as in the case of Akkadians.}}</ref><ref name="Katz 184">Victor J. Katz (1998). ''A History of Mathematics: An Introduction'', p. 184. Addison Wesley, ISBN 0-321-01618-1.
{{quote|"But what we really want to know is to what extent the Alexandrian mathematicians of the period from the first to the fifth centuries C.E. were Greek. Certainly, all of them wrote in Greek and were part of the Greek intellectual community of Alexandria. And most modern studies conclude that the Greek community coexisted [...] So should we assume that [[Ptolemy]] and Diophantus, [[Pappus of Alexandria|Pappus]] and [[Hypatia]] were ethnically Greek, that their ancestors had come from Greece at some point in the past but had remained effectively isolated from the Egyptians? It is, of course, impossible to answer this question definitively. But research in papyri dating from the early centuries of the common era demonstrates that a significant amount of intermarriage took place between the Greek and Egyptian communities [...] And it is known that Greek marriage contracts increasingly came to resemble Egyptian ones. In addition, even from the founding of Alexandria, small numbers of Egyptians were admitted to the privaleged classes in the city to fulfill numerous civic roles. Of course, it was essential in such cases for the Egyptians to become "Hellenized," to adopt Greek habits and the Greek language. Given that the Alexandrian mathematicians mentioned here were active several hundred years after the founding of the city, it would seem at least equally possible that they were ethnically Egyptian as that they remained ethnically Greek. In any case, it is unreasonable to portray them with purely European features when no physical descriptions exist."}}</ref> and the author of a series of books called ''[[Arithmetica]]'',  many of which are now lost. These texts deal with solving [[algebraic equation]]s. While reading  [[Claude Gaspard Bachet de Méziriac]]'s edition of Diophantus' ''Arithmetica,'' [[Pierre de Fermat]]
concluded that a certain equation considered by Diophantus had no solutions, and noted in the margin without elaboration that he had found  "a truly marvelous proof of this proposition," now referred to as [[Fermat's Last Theorem]]. This led to tremendous advances in [[number theory]], and the study of [[Diophantine equation]]s ("Diophantine geometry") and of [[Diophantine approximation]]s remain important areas of mathematical research. Diophantus coined the term παρισὀτης to refer to an approximate equality.<ref>{{citation
| last1 = Katz | first1 = Mikhail G.
| author1-link = Mikhail Katz
| last2 = Schaps | first2 = David
| last3 = Shnider | first3 = Steve
| author3-link = Steve Shnider
| arxiv = 1210.7750
| doi =
| issue = 3
| journal = [[Perspectives on Science]]
| pages =
| title = Almost Equal: The Method of [[Adequality]] from Diophantus to Fermat and Beyond
| volume = 21
| year = 2013}}</ref>  This term was rendered as ''adaequalitat'' in Latin, and became the technique of [[adequality]] developed by [[Pierre de Fermat]] to find maxima for functions and tangent lines to curves.  Diophantus was the first [[Hellenistic civilization|Greek]] mathematician who recognized fractions as numbers; thus he allowed [[positive number|positive]] [[rational number]]s for the coefficients and solutions. In modern use, Diophantine equations are usually algebraic equations with [[integer]] coefficients, for which integer solutions are sought. Diophantus also made advances in mathematical notation.
 
==Biography==
 
Little is known about the life of Diophantus. He lived in [[Alexandria]], [[Egypt]], probably from between AD 200 and 214 to 284 or 298. Much of our knowledge of the life of Diophantus is derived from a 5th-century [[Greek language|Greek]] anthology of number games and puzzles created by [[Metrodorus (grammarian)|Metrodorus]]. One of the problems (sometimes called his epitaph) states:
 
:'Here lies Diophantus,' the wonder behold.
:Through art algebraic, the stone tells how old:
:'God gave him his boyhood one-sixth of his life,
:One twelfth more as youth while whiskers grew rife;
:And then yet one-seventh ere marriage begun;
:In five years there came a bouncing new son.
:Alas, the dear child of master and sage
:After attaining half the measure of his father's life chill fate took him. After consoling his fate by the science of numbers for four years, he ended his life.'
 
This puzzle implies that Diophantus' age <math> x </math> can be expressed as
 
:<math>x = \frac{x}{6} + \frac{x}{12} + \frac{x}{7} + 5 + \frac{x}{2} + 4 </math>
 
which gives <math>x</math> a value of 84 years. However, the accuracy of the information cannot be independently confirmed.
 
In popular culture, this puzzle was the Puzzle No.142 in ''[[Professor Layton and Pandora's Box]]'' as one of the hardest solving puzzles in the game, which needed to be unlocked by solving other puzzles first.
 
==Arithmetica==
{{see also|Arithmetica}}
 
The Arithmetica is the major work of Diophantus and the most prominent work on algebra in Greek mathematics. It is a collection of problems giving numerical solutions of both determinate and indeterminate [[equation]]s. Of the original thirteen books of which Arithmetica consisted only six have survived, though there are some who believe that four  Arab books discovered in 1968 are also by Diophantus.<ref name="Books IV-VII of Diophantus' Arithmetica">{{cite book | title=Books IV to VII of Diophantus' ''Arithmetica'' in the Arabic Translation Attributed to Qusta ibn Luqa | author = J. Sesiano | location = New York/Heidelberg/Berlin | publisher = Springer-Verlag | year = 1982 | page = 502}}</ref> Some Diophantine problems from Arithmetica have been found in Arabic sources.
 
It should be mentioned here that Diophantus never used general methods in his solutions. [[Hermann Hankel]], renowned German mathematician made the following remark regarding Diophantus.
 
“Our author (Diophantos) not the slightest trace of a general, comprehensive method is discernible; each problem calls for some special method which refuses to work even for the most closely related problems.  For this reason it is difficult for the modern scholar to solve the 101st problem even after having studied 100 of Diophantos’s solutions” <ref>Hankel H., “Geschichte der mathematic im altertum und mittelalter, Leipzig, 1874. (translated to English by Ulrich Lirecht in Chinese Mathematics in the thirteenth century, Dover publications, New York, 1973.</ref>{{Dubious|date=February 2011}}
 
===History===
 
Like many other Greek mathematical treatises, Diophantus was forgotten in Western Europe during the so-called [[Dark Ages (historiography)|Dark Ages]], since the study of ancient Greek had greatly declined.  The portion of the Greek ''Arithmetica'' that survived, however, was, like all ancient Greek texts transmitted to the early modern world, copied by, and thus known to, medieval Byzantine scholars.  In addition, some portion of the ''Arithmetica'' probably survived in the Arab tradition (see above). In 1463 German mathematician [[Regiomontanus]] wrote:
 
: “No one has yet translated from the Greek into Latin the thirteen books of Diophantus, in which the very flower of the whole of arithmetic lies hidden . . . .”
 
''Arithmetica'' was first translated from Greek into [[Latin]] by [[Rafael Bombelli|Bombelli]] in 1570, but the translation was never published. However, Bombelli borrowed many of the problems for his own book ''Algebra''. The ''[[editio princeps]]'' of ''Arithmetica'' was published in 1575 by [[Guilielmus Xylander|Xylander]]. The best known Latin translation of ''Arithmetica'' was made by [[Bachet]] in 1621 and became the first Latin edition that was widely available. [[Pierre de Fermat]] owned a copy, studied it, and made notes in the margins.
 
===Margin writing by Fermat and Chortasmenos===
[[Image:Diophantus-II-8-Fermat.jpg|right|thumb|200px|Problem II.8 in the ''Arithmetica'' (edition of 1670), annotated with Fermat's comment which became [[Fermat's Last Theorem]].]]
 
The 1621 edition of Arithmetica by [[Bachet]] gained fame after [[Pierre de Fermat]] wrote his famous "[[Fermat's Last Theorem|Last Theorem]]" in the margins of his copy:
 
:“If an integer ''n'' is greater than 2, then <math>a^n + b^n = c^n</math> has no solutions in non-zero integers ''a'', ''b'', and ''c''. I have a truly marvelous proof of this proposition which this margin is too narrow to contain.”
 
Fermat's proof was never found, and the problem of finding a proof for the theorem went unsolved for centuries. A proof was finally found in 1994 by [[Andrew Wiles]] after working on it for seven years.  It is believed that Fermat did not actually have the proof he claimed to have. Although the original copy in which Fermat wrote this is lost today, Fermat's son edited the next edition of Diophantus, published in 1670. Even though the text is otherwise inferior to the 1621 edition, Fermat's annotations—including the "Last Theorem"—were printed in this version.
 
Fermat was not the first mathematician so moved to write in his own marginal notes to Diophantus; the Byzantine scholar [[John Chortasmenos]] (1370–1437) had written "Thy soul, Diophantus, be with Satan because of the difficulty of your theorems" next to the same problem.
 
==Other works==
 
Diophantus wrote several other books besides ''Arithmetica'', but very few of them have survived.
 
===The ''Porisms''===
Diophantus himself refers to a work which consists of a collection of [[Lemma (mathematics)|lemmas]] called ''The Porisms'' (or ''Porismata''), but this book is entirely lost.
 
Although ''The Porisms'' is lost, we know three lemmas contained there, since Diophantus refers to them in the ''Arithmetica''. One lemma states that the difference of the cubes of two rational numbers is equal to the sum of the cubes of two other rational numbers, i.e. given any ''a'' and ''b'', with ''a'' > ''b'', there exist ''c'' and ''d'', all positive and rational, such that
 
:<math>a^3 - b^3 = c^3 + d^3.\ </math>
 
===Polygonal numbers and geometric elements===
Diophantus is also known to have written on [[polygonal number]]s, a topic of great interest to [[Pythagoras]] and [[Pythagoreans]]. Fragments of a book dealing with polygonal numbers are extant.
 
A book called ''Preliminaries to the Geometric Elements'' has been traditionally attributed to [[Hero of Alexandria]]. It has been studied recently by [[Wilbur Knorr]], who suggested that the attribution to Hero is incorrect, and that the true author is Diophantus.<ref>Knorr, Wilbur: Arithmêtike stoicheiôsis: On Diophantus and Hero of Alexandria, in: Historia Matematica, New York, 1993, Vol.20, No.2, 180-192</ref>
 
==Influence==
Diophantus' work has had a large influence in history. Editions of  Arithmetica exerted a profound influence on the development of algebra in Europe in the late sixteenth and through the 17th and 18th centuries.  Diophantus and his works have also influenced [[Arab mathematics]] and were of great fame among Arab mathematicians. Diophantus' work created a foundation for work on algebra and in fact much of advanced mathematics is based on algebra. As far as we know Diophantus did not affect the lands of the Orient much and how much he affected India is a matter of debate.
 
===The father of algebra?===
Diophantus is often called “the father of [[algebra]]" because he contributed greatly to number theory, mathematical notation, and because Arithmetica contains the earliest known use of syncopated notation.<ref>Carl B. Boyer, A History of Mathematics, Second Edition (Wiley, 1991), page 228</ref> However, it seems that many of the methods for solving linear and quadratic equations used by Diophantus go back to [[Babylonian mathematics]]. For this, and other, reasons mathematical historian [[Kurt Vogel]] writes: “Diophantus was not, as he has often been called, the father of algebra. Nevertheless, his remarkable, if unsystematic, collection of indeterminate problems is a singular achievement that was not fully appreciated and further developed until much later.”<ref>Harald Kittel, ''Übersetzung: ein internationales Handbuch zur Übersetzungsforschung'', Volume 2 p. 1123, 1124</ref>
 
==Diophantine analysis==
{{see also|Diophantine equation}}
 
Today Diophantine analysis is the area of study where integer (whole number) solutions are sought for equations, and Diophantine equations are polynomial equations with integer coefficients to which only integer solutions are sought. It is usually rather difficult to tell whether a given Diophantine equation is solvable. Most of the problems in Arithmetica lead to quadratic equations. Diophantus looked at 3 different types of quadratic equations: <math>ax^2 + bx = c</math>, <math>ax^2 = bx + c</math>, and <math>ax^2 + c = bx</math>. The reason why there were three cases to Diophantus, while today we have only one case, is that he did not have any notion for zero and he avoided negative coefficients by considering the given numbers <math>a, b, c</math> to all be positive in each of the three cases above.  Diophantus was always satisfied with a rational solution and did not require a whole number which means he accepted fractions as solutions to his problems. Diophantus considered negative or irrational square root solutions "useless", "meaningless", and even "absurd". To give one specific example, he calls the equation <math>4 = 4x + 20</math> 'absurd' because it would lead to a negative value for ''x''. One solution was all he looked for in a quadratic equation. There is no evidence that suggests Diophantus even realized that there could be two solutions to a quadratic equation. He also considered simultaneous quadratic equations.
 
==Mathematical notation==
Diophantus made important advances in mathematical notation. He was the first person to use algebraic notation and symbolism. Before him everyone wrote out equations completely. Diophantus introduced an algebraic symbolism that used an abridged notation for frequently occurring operations, and an abbreviation for the unknown and for the powers of the unknown. Mathematical historian Kurt Vogel states:
 
“The symbolism that Diophantus introduced for the first time, and undoubtedly devised himself, provided a short and readily comprehensible means of expressing an equation... Since an abbreviation is also employed for the word ‘equals’, Diophantus took a fundamental step from verbal algebra towards symbolic algebra.”{{Citation needed|date=February 2007}}
 
Although Diophantus made important advances in symbolism, he still lacked the necessary notation to express more general methods. This caused his work to be more concerned with particular problems rather than general situations. Some of the limitations of Diophantus' notation are that he only had notation for one unknown and, when problems involved more than a single unknown, Diophantus was reduced to expressing "first unknown", "second unknown", etc. in words. He also lacked a symbol for a general number n. Where we would write <math>(12 + 6n)/(n^2 -3)</math>, Diophantus has to resort to constructions like :  ... a sixfold number increased by twelve, which is divided by the difference by which the square of the number exceeds three.
 
Algebra still had a long way to go before very general problems could be written down and solved succinctly.
 
==See also==
* [[Erdős–Diophantine graph]]
* [[Diophantus II.VIII]]
* [[Polynomial Diophantine equation]]
 
==Notes==
{{reflist|colwidth=60em}}
 
==References==
* Allard, A. "Les scolies aux arithmétiques de Diophante d'Alexandrie dans le Matritensis Bibl.Nat.4678 et les Vatican Gr.191 et 304" ''Byzantion'' 53. Brussels, 1983: 682-710.
*Christianidis, J. "Maxime Planude sur le sens du terme diophantien "plasmatikon"", ''Historia Scientiarum'', 6 (1996)37-41.
*Christianidis, J. "Une interpretation byzantine de Diophante", ''Historia Mathematica'', 25 (1998) 22-28.
*[[T. L. Heath|Heath, Sir Thomas]], ''Diophantos of Alexandria: A Study in the History of Greek Algebra'', Cambridge: Cambridge University Press, 1885, 1910.
*Robinson, D. C. and Luke Hodgkin. ''History of Mathematics'', [[King's College London]], 2003.
*Sesiano, Jacques. ''Books IV to VII of Diophantus’ Arithmetica in the Arabic translation attributed to Qusṭā ibn Lūqā'',  Heidelberg: Springer-Verlag, 1982. ISBN 0-387-90690-8.
*Tannery, P. L. ''Diophanti Alexandrini Opera omnia: cum Graecis commentariis'', Lipsiae: In aedibus B.G. Teubneri, 1893-1895.
*Ver Eecke, P. ''Diophante d’Alexandrie: Les Six Livres Arithmétiques et le Livre des Nombres Polygones'', Bruges: Desclée, De Brouwer, 1921.
 
==Further reading==
*{{cite book|last=Heath|first=Sir Thomas|title=A history of Greek mathematics|url=http://www.archive.org/details/ahistorygreekma00heatgoog|year=1981|publisher=Cambridge |location=Cambridge University Press|volume=2}}
*{{cite encyclopedia|last=Vogel|first=Kurt|title=Diophantus of Alexandria|encyclopedia=Dictionary of Scientific Biography|volume=4|publisher=Scribner|location=New York|year=1970}}
 
==External links==
* {{MacTutor Biography|id=Diophantus}}
* [http://mathworld.wolfram.com/DiophantussRiddle.html Diophantus's Riddle] Diophantus' epitaph, by E. Weisstein
* Norbert Schappacher (2005). [http://www-irma.u-strasbg.fr/~schappa/NSch/Publications_files/Dioph.pdf Diophantus of Alexandria : a Text and its History].
*[http://www.wilbourhall.org Tannery's edition of the Works of Diophantus, now in the public domain] (Classical Greek)
* [http://www.math.uu.nl/people/hogend/sesiano.html Review of Sesiano's Diophantus] Review of J. Sesiano, Books IV to VII of Diophantus' Arithmetica, by Jan P. Hogendijk
* [http://echo.mpiwg-berlin.mpg.de/ECHOdocuViewfull?mode=imagepath&url=/mpiwg/online/permanent/library/W770Y3H9/pageimg&viewMode=images Latin translation from 1575 by [[Wilhelm Xylander]]]
 
{{Greek mathematics}}
 
{{Authority control|VIAF=2604182}}
{{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
| NAME              = Diophantus
| ALTERNATIVE NAMES =
| SHORT DESCRIPTION = Ancient Greek mathematician
| DATE OF BIRTH    =
| PLACE OF BIRTH    =
| DATE OF DEATH    =
| PLACE OF DEATH    =
}}
[[Category:Number theorists|Diophantus of Alexandria]]
[[Category:Ancient Alexandrians]]
[[Category:Ancient Greek mathematicians|Diophantus of Alexandria]]
[[Category:3rd-century births]]
[[Category:3rd-century deaths]]

Revision as of 20:09, 24 January 2014

28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance.

Title page of the 1621 edition of Diophantus' Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac.

Diophantus of Alexandria (Template:Lang-grc. b. between AD 201 and 215, d. between 285 and 299 at age 84), sometimes called "the father of algebra", was an Alexandrian Greek mathematician[1][2][3][4] and the author of a series of books called Arithmetica, many of which are now lost. These texts deal with solving algebraic equations. While reading Claude Gaspard Bachet de Méziriac's edition of Diophantus' Arithmetica, Pierre de Fermat concluded that a certain equation considered by Diophantus had no solutions, and noted in the margin without elaboration that he had found "a truly marvelous proof of this proposition," now referred to as Fermat's Last Theorem. This led to tremendous advances in number theory, and the study of Diophantine equations ("Diophantine geometry") and of Diophantine approximations remain important areas of mathematical research. Diophantus coined the term παρισὀτης to refer to an approximate equality.[5] This term was rendered as adaequalitat in Latin, and became the technique of adequality developed by Pierre de Fermat to find maxima for functions and tangent lines to curves. Diophantus was the first Greek mathematician who recognized fractions as numbers; thus he allowed positive rational numbers for the coefficients and solutions. In modern use, Diophantine equations are usually algebraic equations with integer coefficients, for which integer solutions are sought. Diophantus also made advances in mathematical notation.

Biography

Little is known about the life of Diophantus. He lived in Alexandria, Egypt, probably from between AD 200 and 214 to 284 or 298. Much of our knowledge of the life of Diophantus is derived from a 5th-century Greek anthology of number games and puzzles created by Metrodorus. One of the problems (sometimes called his epitaph) states:

'Here lies Diophantus,' the wonder behold.
Through art algebraic, the stone tells how old:
'God gave him his boyhood one-sixth of his life,
One twelfth more as youth while whiskers grew rife;
And then yet one-seventh ere marriage begun;
In five years there came a bouncing new son.
Alas, the dear child of master and sage
After attaining half the measure of his father's life chill fate took him. After consoling his fate by the science of numbers for four years, he ended his life.'

This puzzle implies that Diophantus' age can be expressed as

which gives a value of 84 years. However, the accuracy of the information cannot be independently confirmed.

In popular culture, this puzzle was the Puzzle No.142 in Professor Layton and Pandora's Box as one of the hardest solving puzzles in the game, which needed to be unlocked by solving other puzzles first.

Arithmetica

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The Arithmetica is the major work of Diophantus and the most prominent work on algebra in Greek mathematics. It is a collection of problems giving numerical solutions of both determinate and indeterminate equations. Of the original thirteen books of which Arithmetica consisted only six have survived, though there are some who believe that four Arab books discovered in 1968 are also by Diophantus.[6] Some Diophantine problems from Arithmetica have been found in Arabic sources.

It should be mentioned here that Diophantus never used general methods in his solutions. Hermann Hankel, renowned German mathematician made the following remark regarding Diophantus.

“Our author (Diophantos) not the slightest trace of a general, comprehensive method is discernible; each problem calls for some special method which refuses to work even for the most closely related problems. For this reason it is difficult for the modern scholar to solve the 101st problem even after having studied 100 of Diophantos’s solutions” [7]To succeed in selling a home, it is advisable be competent in real estate advertising and marketing, authorized, monetary, operational aspects, and other information and skills. This is essential as a result of you want to negotiate with more and more sophisticated buyers. You could outperform rivals, use latest technologies, and stay ahead of the fast altering market.

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History

Like many other Greek mathematical treatises, Diophantus was forgotten in Western Europe during the so-called Dark Ages, since the study of ancient Greek had greatly declined. The portion of the Greek Arithmetica that survived, however, was, like all ancient Greek texts transmitted to the early modern world, copied by, and thus known to, medieval Byzantine scholars. In addition, some portion of the Arithmetica probably survived in the Arab tradition (see above). In 1463 German mathematician Regiomontanus wrote:

“No one has yet translated from the Greek into Latin the thirteen books of Diophantus, in which the very flower of the whole of arithmetic lies hidden . . . .”

Arithmetica was first translated from Greek into Latin by Bombelli in 1570, but the translation was never published. However, Bombelli borrowed many of the problems for his own book Algebra. The editio princeps of Arithmetica was published in 1575 by Xylander. The best known Latin translation of Arithmetica was made by Bachet in 1621 and became the first Latin edition that was widely available. Pierre de Fermat owned a copy, studied it, and made notes in the margins.

Margin writing by Fermat and Chortasmenos

Problem II.8 in the Arithmetica (edition of 1670), annotated with Fermat's comment which became Fermat's Last Theorem.

The 1621 edition of Arithmetica by Bachet gained fame after Pierre de Fermat wrote his famous "Last Theorem" in the margins of his copy:

“If an integer n is greater than 2, then has no solutions in non-zero integers a, b, and c. I have a truly marvelous proof of this proposition which this margin is too narrow to contain.”

Fermat's proof was never found, and the problem of finding a proof for the theorem went unsolved for centuries. A proof was finally found in 1994 by Andrew Wiles after working on it for seven years. It is believed that Fermat did not actually have the proof he claimed to have. Although the original copy in which Fermat wrote this is lost today, Fermat's son edited the next edition of Diophantus, published in 1670. Even though the text is otherwise inferior to the 1621 edition, Fermat's annotations—including the "Last Theorem"—were printed in this version.

Fermat was not the first mathematician so moved to write in his own marginal notes to Diophantus; the Byzantine scholar John Chortasmenos (1370–1437) had written "Thy soul, Diophantus, be with Satan because of the difficulty of your theorems" next to the same problem.

Other works

Diophantus wrote several other books besides Arithmetica, but very few of them have survived.

The Porisms

Diophantus himself refers to a work which consists of a collection of lemmas called The Porisms (or Porismata), but this book is entirely lost.

Although The Porisms is lost, we know three lemmas contained there, since Diophantus refers to them in the Arithmetica. One lemma states that the difference of the cubes of two rational numbers is equal to the sum of the cubes of two other rational numbers, i.e. given any a and b, with a > b, there exist c and d, all positive and rational, such that

Polygonal numbers and geometric elements

Diophantus is also known to have written on polygonal numbers, a topic of great interest to Pythagoras and Pythagoreans. Fragments of a book dealing with polygonal numbers are extant.

A book called Preliminaries to the Geometric Elements has been traditionally attributed to Hero of Alexandria. It has been studied recently by Wilbur Knorr, who suggested that the attribution to Hero is incorrect, and that the true author is Diophantus.[8]

Influence

Diophantus' work has had a large influence in history. Editions of Arithmetica exerted a profound influence on the development of algebra in Europe in the late sixteenth and through the 17th and 18th centuries. Diophantus and his works have also influenced Arab mathematics and were of great fame among Arab mathematicians. Diophantus' work created a foundation for work on algebra and in fact much of advanced mathematics is based on algebra. As far as we know Diophantus did not affect the lands of the Orient much and how much he affected India is a matter of debate.

The father of algebra?

Diophantus is often called “the father of algebra" because he contributed greatly to number theory, mathematical notation, and because Arithmetica contains the earliest known use of syncopated notation.[9] However, it seems that many of the methods for solving linear and quadratic equations used by Diophantus go back to Babylonian mathematics. For this, and other, reasons mathematical historian Kurt Vogel writes: “Diophantus was not, as he has often been called, the father of algebra. Nevertheless, his remarkable, if unsystematic, collection of indeterminate problems is a singular achievement that was not fully appreciated and further developed until much later.”[10]

Diophantine analysis

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Today Diophantine analysis is the area of study where integer (whole number) solutions are sought for equations, and Diophantine equations are polynomial equations with integer coefficients to which only integer solutions are sought. It is usually rather difficult to tell whether a given Diophantine equation is solvable. Most of the problems in Arithmetica lead to quadratic equations. Diophantus looked at 3 different types of quadratic equations: , , and . The reason why there were three cases to Diophantus, while today we have only one case, is that he did not have any notion for zero and he avoided negative coefficients by considering the given numbers to all be positive in each of the three cases above. Diophantus was always satisfied with a rational solution and did not require a whole number which means he accepted fractions as solutions to his problems. Diophantus considered negative or irrational square root solutions "useless", "meaningless", and even "absurd". To give one specific example, he calls the equation 'absurd' because it would lead to a negative value for x. One solution was all he looked for in a quadratic equation. There is no evidence that suggests Diophantus even realized that there could be two solutions to a quadratic equation. He also considered simultaneous quadratic equations.

Mathematical notation

Diophantus made important advances in mathematical notation. He was the first person to use algebraic notation and symbolism. Before him everyone wrote out equations completely. Diophantus introduced an algebraic symbolism that used an abridged notation for frequently occurring operations, and an abbreviation for the unknown and for the powers of the unknown. Mathematical historian Kurt Vogel states:

“The symbolism that Diophantus introduced for the first time, and undoubtedly devised himself, provided a short and readily comprehensible means of expressing an equation... Since an abbreviation is also employed for the word ‘equals’, Diophantus took a fundamental step from verbal algebra towards symbolic algebra.”Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

Although Diophantus made important advances in symbolism, he still lacked the necessary notation to express more general methods. This caused his work to be more concerned with particular problems rather than general situations. Some of the limitations of Diophantus' notation are that he only had notation for one unknown and, when problems involved more than a single unknown, Diophantus was reduced to expressing "first unknown", "second unknown", etc. in words. He also lacked a symbol for a general number n. Where we would write , Diophantus has to resort to constructions like : ... a sixfold number increased by twelve, which is divided by the difference by which the square of the number exceeds three.

Algebra still had a long way to go before very general problems could be written down and solved succinctly.

See also

Notes

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References

  • Allard, A. "Les scolies aux arithmétiques de Diophante d'Alexandrie dans le Matritensis Bibl.Nat.4678 et les Vatican Gr.191 et 304" Byzantion 53. Brussels, 1983: 682-710.
  • Christianidis, J. "Maxime Planude sur le sens du terme diophantien "plasmatikon"", Historia Scientiarum, 6 (1996)37-41.
  • Christianidis, J. "Une interpretation byzantine de Diophante", Historia Mathematica, 25 (1998) 22-28.
  • Heath, Sir Thomas, Diophantos of Alexandria: A Study in the History of Greek Algebra, Cambridge: Cambridge University Press, 1885, 1910.
  • Robinson, D. C. and Luke Hodgkin. History of Mathematics, King's College London, 2003.
  • Sesiano, Jacques. Books IV to VII of Diophantus’ Arithmetica in the Arabic translation attributed to Qusṭā ibn Lūqā, Heidelberg: Springer-Verlag, 1982. ISBN 0-387-90690-8.
  • Tannery, P. L. Diophanti Alexandrini Opera omnia: cum Graecis commentariis, Lipsiae: In aedibus B.G. Teubneri, 1893-1895.
  • Ver Eecke, P. Diophante d’Alexandrie: Les Six Livres Arithmétiques et le Livre des Nombres Polygones, Bruges: Desclée, De Brouwer, 1921.

Further reading

External links

Template:Greek mathematics

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  1. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  2. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  3. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  4. ↑ Victor J. Katz (1998). A History of Mathematics: An Introduction, p. 184. Addison Wesley, ISBN 0-321-01618-1. 31 year-old Systems Analyst Bud from Deep River, spends time with pursuits for instance r/c cars, property developers new condo in singapore singapore and books. Last month just traveled to Orkhon Valley Cultural Landscape.
  5. ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

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  6. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  7. ↑ Hankel H., “Geschichte der mathematic im altertum und mittelalter, Leipzig, 1874. (translated to English by Ulrich Lirecht in Chinese Mathematics in the thirteenth century, Dover publications, New York, 1973.
  8. ↑ Knorr, Wilbur: Arithmêtike stoicheiôsis: On Diophantus and Hero of Alexandria, in: Historia Matematica, New York, 1993, Vol.20, No.2, 180-192
  9. ↑ Carl B. Boyer, A History of Mathematics, Second Edition (Wiley, 1991), page 228
  10. ↑ Harald Kittel, Übersetzung: ein internationales Handbuch zur Übersetzungsforschung, Volume 2 p. 1123, 1124
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