Diophantus

Lore & Background

The exact details of Diophantus' life are obscure. He probably flourished in the third century CE, but may have lived anywhere between 170 BCE and 350 CE. The only definitive information about his life comes from a puzzle in the Greek Anthology, attributed to Metrodorus, which describes his age as 84 years through an algebraic riddle, though its accuracy cannot be confirmed.

Diophantus' major work, the *Arithmetica*, is a collection of 290 algebraic problems that solve determinate and indeterminate equations. It is the earliest extant work to solve arithmetic problems by algebra, though algebra existed before him. The book considers integer solutions to equations with integer coefficients, now called Diophantine equations.

Diophantus introduced an algebraic symbolism with abbreviations for the unknown and powers of the unknown. His method of solution matches medieval Arabic algebra in concepts and procedure, using three stages: naming an unknown and setting up an equation, simplifying to a standard form, and solving. Unlike modern notation, his lacked symbols for operations and relations.

Reader's Guide

Diophantus' significance lies in his *Arithmetica*, which became the standard within Neoplatonic schools of Late antiquity and influenced later algebra through its translation into Arabic in the 9th century. Joseph-Louis Lagrange called him 'the inventor of algebra.' The 1621 edition by Bachet gained fame after Pierre de Fermat wrote his 'Last Theorem' in its margins. In modern use, Diophantine equations are algebraic equations with integer coefficients for which integer solutions are sought. Diophantine geometry and Diophantine approximations are subareas of number theory named after him. Some problems from the *Arithmetica* have inspired modern work in abstract algebra and number theory. His notation, though limited, was a step toward symbolic algebra. The *Arithmetica* originally had thirteen books; six survive in Greek and four in Arabic, discovered in 1968. Most problems lead to quadratic equations. Diophantus did not classify equations into six types like Al-Khwarizmi, but he mentioned solving three-term equations later, suggesting that part is lost.

Did You Know?

• Diophantus' age is given as 84 years by a puzzle in the Greek Anthology, but its accuracy cannot be confirmed.
• Only six of the thirteen books of Arithmetica survive in Greek; four more were discovered in Arabic in 1968.
• Pierre de Fermat wrote his famous 'Last Theorem' in the margins of the 1621 edition of Arithmetica by Bachet.
• Diophantus introduced an algebraic symbolism with abbreviations for the unknown and powers of the unknown.

Notable Quotes (Wikiquote)

"The 16th century Italian mathematician Rafael Bombelli, L’Algebra, published in 1572." — Diophantus "Florian Cajori, A History of Mathematics (1893)." — Diophantus "In this work [Arithmetica] is introduced the idea of an algebraic equation expressed in algebraic symbols. His treatment is purely analytical and completely divorced from geometrical methods." — Diophantus "Florian Cajori, A History of Mathematics (1893)." — Diophantus