Diophantus of Alexandria (
GreekGreek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...
: . b. between 200 and 214 CE, d. between 284 and 298 CE), sometimes called "the father of
algebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
", was an
AlexandriaAlexandria is the second-largest city of Egypt, with a population of 4.1 million, extending about along the coast of the Mediterranean Sea in the north central part of the country; it is also the largest city lying directly on the Mediterranean coast. It is Egypt's largest seaport, serving...
n
Greek mathematicianGreek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...
and the author of a series of books called
ArithmeticaArithmetica is an ancient Greek text on mathematics written by the mathematician Diophantus in the 3rd century AD. It is a collection of 130 algebraic problems giving numerical solutions of determinate equations and indeterminate equations.Equations in the book are called Diophantine equations...
. These texts deal with solving
algebraic equations, many of which are now lost. In studying
Arithmetica, Pierre de FermatPierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus, including his adequality...
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 TheoremIn number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
. This led to tremendous advances in
number theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, and the study of
Diophantine equationIn mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...
s ("Diophantine geometry") and of
Diophantine approximationIn number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers....
s remain important areas of mathematical research. Diophantus was the first
GreekHellenistic civilization represents the zenith of Greek influence in the ancient world from 323 BCE to about 146 BCE...
mathematician who recognized fractions as numbers; thus he allowed positive
rational numberIn mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s for the coefficients and solutions. In modern use, Diophantine equations are usually algebraic equations with
integerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
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
AlexandriaAlexandria is the second-largest city of Egypt, with a population of 4.1 million, extending about along the coast of the Mediterranean Sea in the north central part of the country; it is also the largest city lying directly on the Mediterranean coast. It is Egypt's largest seaport, serving...
,
EgyptEgypt , officially the Arab Republic of Egypt, Arabic: , is a country mainly in North Africa, with the Sinai Peninsula forming a land bridge in Southwest Asia. Egypt is thus a transcontinental country, and a major power in Africa, the Mediterranean Basin, the Middle East and the Muslim world...
, probably from between 200 and 214 to 284 or 298 AD. Much of our knowledge of the life of Diophantus is derived from a 5th century
GreekGreek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...
anthology of number games and strategy puzzles. 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 lived to be 84 years old. 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, known in Australia and Europe as Professor Layton and Pandora's Box, is the second game in the Professor Layton series by Level-5. It was followed by a third game, Professor Layton and the Unwound Future...
as one of the hardest solving puzzles in the game, which needed to be unlocked by solving other puzzles first.
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
equationAn equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...
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. 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 HankelHermann Hankel was a German mathematician who was born in Halle, Germany and died in Schramberg , Imperial Germany....
, 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”
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
RegiomontanusJohannes Müller von Königsberg , today best known by his Latin toponym Regiomontanus, was a German mathematician, astronomer, astrologer, translator and instrument maker....
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
LatinLatin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
by
BombelliRafael Bombelli was an Italian mathematician.Born in Bologna, he is the author of a treatise on algebra and is a central figure in the understanding of imaginary numbers....
in 1570, but the translation was never published. However, Bombelli borrowed many of the problems for his own book
Algebra. The
editio princepsIn classical scholarship, editio princeps is a term of art. It means, roughly, the first printed edition of a work that previously had existed only in manuscripts, which could be circulated only after being copied by hand....
of
Arithmetica was published in 1575 by
XylanderWilhelm Xylander was a German classical scholar and humanist....
. 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 FermatPierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus, including his adequality...
owned a copy, studied it, and made notes in the margins.
Margin writing by Fermat and Chortasmenos
The 1621 edition of Arithmetica by Bachet gained fame after
Pierre de FermatPierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus, including his adequality...
wrote his famous "
Last TheoremIn number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
" 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 WilesSir Andrew John Wiles KBE FRS is a British mathematician and a Royal Society Research Professor at Oxford University, specializing in number theory...
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 (14th/15th C.) 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
lemmasIn mathematics, a lemma is a proven proposition which is used as a stepping stone to a larger result rather than as a statement in-and-of itself...
called
The Porisms (or
Porismata), but this book is entirely lost. Some scholars think that
The porisms may have actually been a section of
Arithmetica that is now 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 numberIn mathematics, a polygonal number is a number represented as dots or pebbles arranged in the shape of a regular polygon. The dots were thought of as alphas . These are one type of 2-dimensional figurate numbers.- Definition and examples :...
s, a topic of great interest to
PythagorasPythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...
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 AlexandriaHero of Alexandria was an ancient Greek mathematician and engineerEnc. Britannica 2007, "Heron of Alexandria" who was active in his native city of Alexandria, Roman Egypt...
. It has been studied recently by
Wilbur KnorrWilbur Richard Knorr was an American historian of mathematics and a professor in the departments of philosophy and classics at Stanford University. He has been called "one of the most profound and certainly the most provocative historian of Greek mathematics" of the 20th century.-Biography:Knorr...
, who suggested that the attribution to Hero is incorrect, and that the true author is Diophantus.
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 seventeenth and eighteenth 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
algebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
" because he contributed greatly to number theory, mathematical notation, and because Arithmetica contains the earliest known use of syncopated notation. However, it seems that many of the methods for solving linear and quadratic equations used by Diophantus go back to
Babylonian mathematicsBabylonian mathematics refers to any mathematics of the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited...
. 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.”
Diophantine analysis
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.”
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
- Erdős–Diophantine graph
- Diophantus II.VIII
The eighth problem of the second book of Diophantus's Arithmetica is to divide a square into a sum of two squares.-The solution given by Diophantus:Diophantus takes the square to be 16 and solves the problem as follows:...
- Polynomial Diophantine equation
In mathematics, a polynomial Diophantine equation is an indeterminate polynomial equation whose solutions are restricted to be polynomials in the indeterminate. A Diophantine equation, in general, is one where the solutions are restricted to some algebraic system, typically integers...
External links