What little is known of Diophantus of Alexandria’s life is inferred from a letter of the eleventh-century Byzantine scholar Michael Psellus. The only other information, which is uncertain, comes from the solution of an arithmetic epigram, which says that Diophantus married at 33 and had a son who died at age 42, four years before Diophantus’s death at 84.

 

Mathematics—Algebra, Number Theory

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The Arithmetica, the treatise on which Diophantus’s fame rests, purports to be in thirteen books, but none of the surviving Greek manuscripts consists of more than six, though one has the same text in seven. The missing books were apparently lost early, for there is no evidence that the Arabs who translated or commented on Diophantus ever had access to more than is now extant. The Arithmetica is not a work of theoretical arithmetic in the sense understood by the Pythagoreans. It deals, rather, with the computational arithmetic used in the solution of some 150 practical problems. Many of the procedures used by Diophantus can be found in pre-existing Greek, Babylonian, and Chinese texts. Consequently, he certainly was not, as he has often been called, the father of algebra. Nevertheless, his remarkable, if unsystematic, collection of indeterminate problems, with its new symbolism, is a singular achievement that was not fully appreciated and further developed until much later.

 

The word “indeterminate” refers to a system of polynomial equations that has fewer equations than variables and “determinate” if the number of equations equals or exceeds the number of variables. Thus

 

                                    x + y = a,  a constant,

 

                                    xy = 1,

 

is a determinate system since there are an equal number of equations (2) and variables x and y make two as well. On the other hand:

 

                                    x² + y² = z²

 

is indeterminate since there are three variables (x, y, and z) and only one equation. Note that polynomial equations in one variable, such as quadratic or cubic equations, are determinate.

 

The term “indeterminate” is most commonly applied to systems with fewer equations than variables for which integer or rational solutions are sought. Such systems, also called “Diophantine,” have been particularly influential in the development of algebra.(1) The fact that these indeterminate systems are also called “Diophantine” demonstrates the significance of Diophantus’s Arithmetica.

 

Before Diophantus, all algebra, including the problem, operations and logic, and solution, was expressed without symbolism. He was the first to introduce symbolism into Greek algebra. For an unknown quantity, he used only one symbol (called arithmos), which characterized an undefined number of units. To avoid confusion in problems having more than one unknown term, Diophantus expressed all unknowns in terms of one of them whenever possible. The only other algebraical symbol that he used was for subtraction, also used by Hero of Alexandria (first century C.E.). The Arithmetica is valuable also for the propositions in the theory of numbers. “His number-theory propositions were taken up by mathematicians of the seventeenth century, generalized, and proved, thereby creating modern number theory.”(2)

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Footnotes:

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(1) Isabella Bashmakova and Galina Smirnova, translated from the Russian by Abe Shenitzer, The Beginnings & Evolution of Algebra (Washington, D.C., 2000), p. xvi.

(2) C.C. Gillispie (Editor), Concise Dictionary of Scientific Biography (Chicago, 1980), p. 205.

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