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Diophantus ' work created a foundation for work on algebra and in fact much of advanced mathematics is based on algebra.
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Diophantus and work
The Arithmetica is the major work of Diophantus and the most prominent work on algebra in Greek mathematics.
Diophantus himself refers to a work which consists of a collection of lemmas called The Porisms ( or Porismata ), but this book is entirely lost.
Diophantus ' work has had a large influence in history.
It is addressed to Diophantus and conveys a moral, that one should work and not dream, illustrated by the story of an old fisherman who dreams that he has caught a fish of gold and narrates his vision to his mate.
Diophantus and created
Around AD 250, Diophantus created a different form of the Pell equation
Diophantus and for
Diophantus was the first Greek mathematician who recognized fractions as numbers ; thus he allowed positive rational numbers for the coefficients and solutions.
However, it seems that many of the methods for solving linear and quadratic equations used by Diophantus go back to Babylonian mathematics.
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 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.
“ 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 .”
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.
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.
Diophantus also studies the equations of some non-rational curves, for which no rational parametrisation is possible.
Some consider him to be merely reworking the ideas of others ( he was influenced by Diophantus ) but most regard him as more original, in particular for the beginnings of freeing algebra from geometry.
Diophantus and on
Diophantus is also known to have written on polygonal numbers, a topic of great interest to Pythagoras and Pythagoreans.
One may say that Diophantus was studying rational points — i. e., points whose coordinates are rational — on curves and algebraic varieties ; however, unlike the Greeks of the Classical period, who did what we would now call basic algebra in geometrical terms, Diophantus did what we would now call basic algebraic geometry in purely algebraic terms.
While Diophantus is concerned largely with rational solutions, he assumes some results on integer numbers ; in particular, he seems to assume that every integer is the sum of four squares, though he never states as much explicitly.
Alkarkhi worked on similar problems to Diophantus.
* Diophantus writes Arithmetica, the first systematic treatise on algebra.
As Leonidas of Tarentum wrote epigrams on fishermen, and one of them is a dedication of his tackle to Poseidon by Diophantus, the fisher, it is likely that the author of this poem was an imitator of Leonidas.
* PDF files of many of Heath's works, including those on Diophantus, Apollonius, etc.
Arithmetica is an ancient Greek text on mathematics written by the mathematician Diophantus in the 3rd century AD.
Diophantus and algebra
The word Diophantine refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who made a study of such equations and was one of the first mathematicians to introduce symbolism into 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.
For this, and other, reasons mathematical historian Kurt Vogel writes: “ Diophantus was not, as he has often been called, the father of algebra.
He is thus considered to be the father of algebra by some, although the Greek mathematician Diophantus has also been given this title.
Diophantus of Alexandria, in turn, was the " father of algebra ".
Like the algebra of Diophantus, the algebra of Brahmagupta was syncopated.
Diophantus and much
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.
Diophantus and mathematics
Diophantus and his works have also influenced Arab mathematics and were of great fame among Arab mathematicians.
Diophantus and is
The mathematical study of Diophantine problems Diophantus initiated is now called " Diophantine analysis ".
Little is known about the life of Diophantus.
Much of our knowledge of the life of Diophantus is derived from a 5th century Greek anthology of number games and strategy puzzles.
Although the original copy in which Fermat wrote this is lost today, Fermat's son edited the next edition of Diophantus, published in 1670.
Although The Porisms is lost, we know three lemmas contained there, since Diophantus refers to them in the Arithmetica.
It has been studied recently by Wilbur Knorr, who suggested that the attribution to Hero is incorrect, and that the true author is Diophantus.
There is no evidence that suggests Diophantus even realized that there could be two solutions to a quadratic equation.
Very little is known about Diophantus of Alexandria ; he probably lived in the third century CE, that is, about five hundred years after Euclid.
In modern language, what Diophantus does is to find rational parametrisations of many varieties ; in other words, he shows how to obtain infinitely many rational numbers satisfying a system of equations by giving a procedure that can be made into an algebraic expression
Diophantus and .
Title page of the 1621 edition of Diophantus ' Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac.
concluded that a certain equation considered by Diophantus had no solutions, and noted without elaboration that he had found " a truly marvelous proof of this proposition ," now referred to as Fermat's Last Theorem.
Diophantus also made advances in mathematical notation.
:' Here lies Diophantus ,' the wonder behold.
This puzzle implies that Diophantus lived to be 84 years old.
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.
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.
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.
: “ 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.
Fermat was not the first mathematician so moved to write in his own marginal notes to Diophantus ; the Byzantine scholar John Chortasmenos ( 14th / 15th C .) had written " Thy soul, Diophantus, be with Satan because of the difficulty of your theorems " next to the same problem.
Diophantus wrote several other books besides Arithmetica, but very few of them have survived.
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