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Essay / Research Paper Abstract
A 3 page overview of the book Abel's Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability by Pesic. Bibliography lists 1 source.
Page Count:
3 pages (~225 words per page)
File: MM12_PGabel.rtf
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Unformatted sample text from the term paper:
Pesics book tells about Abels problem and his proof, which has to do with solving algebraic equations. Pesic begins the story with Pythagoras and the brotherhood, credited with beginning
mathematics and whose major contribution is reputed to be the concept of mathematical proof. Pesic discusses the many notions of incommensurability and irrationality in Greek thought, with examples of Platos
use of mathematical ideas in some of his works and Euclids exploration into the concept of irrationality. This discussion is not only interesting, it sets the foundation for Abels problem.
For instance, Pesics discussion of Vi?tes belief that all polynomial equations should be able to be solved using the same methods as used for lower degrees. Pesic suggests that Netons
demonstration of the impossibility of expressing the area within an oval using a finite algebraic formula actually provided a very early warning that the quintic would be unsolvable.
Solutions to the quadratic equation can be expressed in radicals, such as a square root. Other equations at a higher level have been demonstrated to be solvable, such as third
degree or cubic equations and fourth degree or quartic equations. It is not until Chapter 6 that Pesic presents Abels problem, that of the quintic or fifth degree equation.
There are some quintic equations that have solutions that can be expressed in radicals but there are so many coefficient values that the exact value of x cannot be determined.
The problem is expressing the equation in a finite formula that incorporates only powers and roots. The same is true for all types of equations beyond quintic, higher than five,
like the sixth degree. None of these can be solved in terms of radicals. Algebraic equations are supposed to have solutions so what is it at the higher degrees
...