Algebric numbers and transcendent numbers
The work paper starts with an introduction. In this, we have presented some information that should be taken into consideration, regarding the theory of numbers, from the crystallization of its notion as a number. At the beginning many notions were just forming on the idea of practical activities and were mixed up with the ideas of form and extent.
After that, came a long process of analysis and separation of notions. In Ancient Times, a first separation was made between the arithmetic and geometric notions.
In the XVIIth and XVIIIth centuries, the theory of numbers was already sorted out from arithmetic and from the other branches, which had as source still arithmetic, and it’s object gained configuration in the study of problems regarding the natural numbers with implications and in the multitude of rational numbers.
The theory of irrational numbers, throughout the masterworks of Dedekynd and G. Carrior (1845-1918), leads to a great analysis regarding the classes of irrational numbers and the separation of irrational numbers in algebric and transcendent numbers from the rational unit.
So, in 1844, Liouville demonstrates the existence of a whole class of transcendents. In 1873, Hermite shows the transcendence of the number “ e “, and in 1882, Linderman demonstrates the transcendence of the number “ ∏ ”.
The work paper we are analyzing “ALGEBRIC NUMBERS AND TRANSCENDENT NUMBERS” is composed out of two parts .
In the first part, we proposed four chapters, and in the second one, one chapter was proposed for debate.
In the first chapter of part one, we have studied some fundamental notions of the polygons , the definition of algebric numbers and some properties such as :
[] the Liouville theories
[] the Thue and Ruth theories
[] the discriminate of algebric numbers.
In the second chapter, we have treated some elements referring of forms and shapes of algebric numbers, bases in forms of...