Posts

MODERN ALGEBRA

Image
  Modern history Italian mathematician  Girolamo Cardano  published the solutions to the  cubic  and  quartic equations  in his 1545 book  Ars magna . François Viète 's work on  new algebra  at the close of the 16th century was an important step towards modern algebra. In 1637,  René Descartes  published  La Géométrie , inventing  analytic geometry  and introducing modern algebraic notation. Another key event in the further development of algebra was the general algebraic solution of the cubic and quartic equations, developed in the mid-16th century. The idea of a  determinant  was developed by  Japanese mathematician   Seki Kōwa  in the 17th century, followed independently by  Gottfried Leibniz  ten years later, for the purpose of solving systems of simultaneous linear equations using  matrices .  Gabriel Cramer  also did some work on matrices and determinants in...
Post a Comment
Read more

Algebraic structure

Image
  Basic concepts [ edit ] Main article:  Algebraic structure By abstracting away various amounts of detail, mathematicians have defined various algebraic structures that are used in many areas of mathematics. For instance, almost all systems studied are  sets , to which the theorems of  set theory  apply. Those sets that have a certain binary operation defined on them form  magmas , to which the concepts concerning magmas, as well those concerning sets, apply. We can add additional constraints on the algebraic structure, such as associativity (to form  semigroups ); identity, and inverses (to form  groups ); and other more complex structures. With additional structure, more theorems could be proved, but the generality is reduced. The "hierarchy" of algebraic objects (in terms of generality) creates a hierarchy of the corresponding theories: for instance, the theorems of  group theory  may be used when studying  rings  (algebrai...
Post a Comment
Read more