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Diophant von Alexandrien We know almost nothing about Diophantus, except that he lived, in Alexandria. We know that he quotes Hypsicles (150 BC) and he is quoted by Theon of Alexandria (whose date is fixed by the solar eclipse of June 16, 364 AD ). 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). From his epitaph, which can be written as a linear equation x = x/6 + x/12 + x/7 + 5+ x/2 + 4 , (where x is the number of years of Diophantus lived ) we find a solution x = 84. We know therefore that he married at the age of 33, had a son who died when he was 42, 4 years before Diophantus himself died. Assumimg that he was born around 200 AD then Diophantus lived in the period (200 - 284 ) AD . Diophantus began the systematic study of equations with integer coefficients. He was a pioneer in solving certain indeterminate algebraic equations, i.e., an equation in which the variables can take on integer values (Diophantine equations) and has an infinite but denumerable set of solutions: e.g., x+2y=3. He published his results in the book Arithmetica. Other work: On polygonal numbers only fragments survived, the book Porismata was lost. For the Diophantine equations of the form ax + by = c Euclid's algorithm provides an answer if a solution exists (integer x, y). If d is the greatest common divisor of a,b and c can be divided by d then such a solution exist. Take the equation 6x+15y = 12. Then the greatest common divisor of 6 and 15 is 3 and 12 can by divided by 3. A solution exists and if we try we will find (x = -3 and y =2 which gives -18 + 30 = 12). Did Diophantus know negative numbers? He says in Greek: Leiyis epileiyin pollaplasiasqeisa poiei uparcin, leiyis de epi uparcin poiei leiyin. (Book I, 20). Translated this means: minus times minus is plus and minus times plus is minus. The second chapter comments that Diophantus extended the notion of number to include negatives and rationals, describes his symbols for exponents from -6 to 6, and notes that he moved beyond Greek traditions in permitting addition of non-homogeneous magnitudes. (Review by D. Graves of a book of I. G. Bashmakova about the Arithmetica)
Diophantus used for negative exponents the symbol X. For example ςX for 1/x = x -1.
KY αςβ Almost 1500 years later | ||||||||||||||||
was used for the minus sign (a symbol derived from Lambda, I and leiyis), 
was used for the constant term.