Algebra Euclid discovered important properties of numbers through a study of geometry Diophantus another famous early Greek mathematician has been called the father of algebra He treated algebra from a purely numerical point of view He made a special study of certain types of equations that are today called Diophantine equations Our modern word algebra comes from the Arabic al Jabr which appeared in the title of an algebra text written in about 825 CE by the Arab astronomer and mathematician al Khwarizmi The word algorithm is derived from his name MODERN ALGEBRA The study of the algebra of real numbers and the recent recognition of the fundamental importance of the basic principles have led to the development of what is now called mod ern algebra or abstract algebra The concepts of modern algebra have been found to be extremely useful in other branches of mathematics as well as in the physical and social sciences A chemist may use modern algebra in a study of the structure 12
WHAT IS ALGEBRA DIOPHANTUS Diophantus of Alexandria C 250 CE was a Greek mathematician famous for his work in algebra One mathematical work he is believed to have written is a large and extremely influential treatise called Arithmetica continued on the next page Pierre de Fermat extended the work of Diophantus by marking up a copy of the earlier mathematician s treatise Arithmetica Courtesy of De Agostini Picture Library Getty Images 13 R a costs
Algebra continued from the previous page Its historical importance is twofold it is the first known work to employ algebra in a modern style and it inspired the rebirth of number theory Although he had limited algebraic tools at his disposal Diophantus managed to solve a great variety of problems and Arithmetica inspired Arabic mathematicians such as al Karaji c 980 1030 to apply his methods The most famous extension of Diophantus's work was by Pierre de Fermat 1601 65 the founder of modern number theory In the margins of his copy of Arithmetica Fermat wrote various remarks proposing new solutions corrections and generalizations of Diophantus's methods as well as some conjectures including what became known as Fermat's last theorem which occupied mathematicians for gen erations to come Indeterminate equations restricted to integral solutions have come to be known though inap propriately as Diophantine equations of crystals An engineer may use modern alge bra in designing a computer A mathematician may use modern algebra in a study of logic ementar algebra students study the properties of addition and multiplication of real numbers that follow as a consequence of certain basic principles Students of modern algebra may work with any set of objects not 14
WHAT IS ALGEBRA only with real numbers Modern algebra con siders certain operations on these objects that are not restricted to addition and subtraction as with elementary algebra In modern alge bra it is agreed that certain basic principles are satisfied by these operations then vari ous properties are derived from assumed basic principles Modern or abstract algebra will not be discussed further in this book 15 R a costs
