Biographies of Mathematicians - Diophantus

Diophantus's Life

{dy-oh-fan'-tuhs}

Born: about 200 A.D. in Alexandria , Egypt

Died: about 284 A.D. in Alexandria , Egypt

Diophantus worked during the middle of the third century and is best known for his Arithmetica, a work on the theory of numbers.

Little is known of Diophantus's life. The most details we have (and these may not be accurate) say that he married at the age of 33 and had a son who died at the age of 42, four years before Diophantus himself died at approximately 84.

Diophantus's epitaph

The most details we have found of Diophantus's life have come from Greek Anthology epigrams. Which is a collection of number games and strategy puzzles. Because of the speculation of his age the only significant evidence of his lifespan is through his collection of puzzles.

"This tomb hold Diophantus. Ah, what a marvel! And the tomb tells scientifically the measure of his life. God vouchsafed that he should be a boy for the sixth part of his life; when a twelfth was added, his cheeks acquired a beard; He kindled for him the light of marriage after a seventh, and in the fifth year after his marriage He granted him a son. Alas! late-begotten and miserable child, when he had reached the measure of half his father's life, the chill grave took him. After consoling his grief by this science of numbers for four years, he reached the end of his life."

J R Newman (ed.) The World of Mathematics (New York 1956).

The Arithmetica

The Arithmetica , Diophantus's book , is a collection of 130 problems giving numerical solutions of determinate equations (those with a unique solution), and indeterminate equations. The equations in the book are called Diophantine equations. The method for solving these equations is known as Diophantine analysis. Only 6 of the original 13 books survive and the others must have been lost quite soon after they were written. There are many Arabic translations but only material from these six books appeared. Most of the Arithmetica problems lead to quadratic equations.

Diophantus refers to other works in The Porisms, a collection of lemmas. The book has been lost but three of the lemmas are known since Diophantus refers to them in Arithmetica. One Lemma is the difference of the cubes of two rational number is equal to the sum of the cubes of two other rational numbers, i.e. given any numbers a, b then there exists numbers c,d such that a³-b³=c³+d³.

Diophantus' Book
http://www-groups.dcs.st-and.ac.uk/~history/Bookpages/Diophantus.html

Diophantus' Work

Diophantine equations

A Diophantine equation is a polynomial equation with integral coefficients to which only integral solutions are sought. The study of Diophantine equations is one of the central areas of number theory. In general, it is difficult to tell whether a given Diophantine equation is solvable. For example, the Diophantine equation x² - 94y² = 1 is solvable, although the smallest solution is x = 2,143,295 and y = 221,064. The Diophantine equation x² - 94y² = -1, however, has no solutions.

Diophantus was always satisfied with a rational solution and did not require a whole number. He did not deal in negative solutions. He considered negative or irrational square root solutions "useless", "meaningless", and even "absurd". One solution was all he required to a quadratic equation. There is no evidence that suggests Diophantus even realized that there could be two solutions to a quadratic equation.

Linear Diophantine Equations

A Linear Diophantine Equation is an equation of the form: D(x1,...,x_m)=0 Where D is a polynomial with integer coefficiants. A more general equation would be: D_L(x1,...,x_m=D_R(x1,...,x_m) Where D_L and D_R are again polynomials with integer coefficients.

Other Facts on Diophantus

Although Diophantus made important advances in symbolism, he still lacked the necessary notation to express more general methods. For instance 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 (12 + 6n)/(n2 -3), Diophantus has to write in words:

... a sixfold number increased by twelve, which is divided by the difference by which the square of the number exceeds three.

Diophantus is often regarded as the 'father of algebra' but there is no doubt that many of the methods for solving linear and quadratic equations go back to Babylonian mathematics. For this reason Vogel writes:

Diophantus has numerous works in the mathematical field including Plus and Minus Signs, Geodesic Diophantine boxes, and Four Squares from Three Numbers

... 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.