Of tremendous importance to the develpment of algebra and of great influence
on later European number theorists was Diophantus of Alexandria. Diophantus
was another mathematician, like Heron, of uncertain date and nationality.
Although there is some tenuous evidence that he may have beem a contemporary,
contemporary, of Heron, most historians tend to place him in the
third century of our era. Beyond the fact that he flourished at
Alexandria,nothing certain is known about him,
although there is an epigram in the Greek
Anthology that purports to give some details of his life.
Diophantus wrote three works: Arithmetica, his most important one, of
which the first six of thirteen books are extant; On Polygonal Numbers, of
which only a fragment is extant;and Porisms, which is lost.
The Arithmentica is an analytic treatment of algebraic number theory and
marks the author as a genius in this field. The extant portion of the work is
debeted to the solution of about 130problems, of considerable wariety, leading
to equations of the first and sevond degree. One very special cubic is solved.
The first book concerns itself with determinate equations in one unknown, and
the remaining books with indeterminate equations of the second, and sometimes
higher, degree in two and three unknowns. Striking is the lack of general
methods and the repeated application of ingenious devices designed for the
needs of each individual problem. Diophantus recognized only positive rationa
answers and was, in most cases, satisfied with only one answer to a problem.
Indeterminate algebraic problems in which one must find only the rational
solutions have beacome known as Diophantine problems. In fact. modern usage
of the terminology often implies the restriction of the solutions to integers.
diophantus did not originate problems of this sort, however. Also he was not,
as is sometimes stated, the first to work with indeterminate equations or the
first to solve quadratic equations nongeometrically. He may have been, hoqever,
the first to take steps towards an algebraic notation. These steps were in
the nature of stenographic abbreviations.
Diophantus had abbreviations for the unknown, powers of the unknown up
through the sixth, subtraction, equality, and reciprocals.
During his life in Greece, Geometry was mostly studied
and arithmetic and algebra wasn't separated.
As Diophantus introduced abbreviated letter (or character),
algebra was obviously separated from arithmetic.
The biggest difference between two is whether character is used or not.
Another one is whether negative numbers are recognized as numbers.
A question is written on his epitaph, which
represents his contributions and passon