Math Department (Thesis/2008-09)

Algebraic and Transcendental Numbers

Under the direction of: Susan Niefield

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A real (or complex) number is called algebraic if it is a root of a polynomial equation of the form $a_{n}x^{n}+a_{n-1}x^{n-1}+ \ldots+a_{1}x+a_{0}$, where $a_{0}, a_{1}, \ldots a_{n}$ are integers. It is called transcendental if it not algebraic. Examples of algebraic numbers include all rational numbers as well as many irrational real numbers (e.g., $\sqrt 2$, $1+\sqrt 2$, $\root 3 \of 5$, $2 - \root 3 \of 5$) and complex numbers (e.g., $i$, $2+i$, $3-i\sqrt 2$). It can be shown that $\pi$, $e$, and $2^{\sqrt 2}$ are transcendental, as well any number of the form $\alpha^{\beta}$, where $\alpha$ and $\beta$ are algebraic and $\beta$ is irrational. There are many curiosities in this area. For example, $e^{\pi}$ is known to be transcendental, but the fate of ${\pi}^e$ is still at large. Also, $${1\over 10^{5}}+ {1\over 10^{8}}+{1\over 10^{11}}+{1\over 10^{14}}+\ldots$$ is algebraic, since it can be shown to be a root of $99900x-1$, but the following similar looking number is not. $${1\over 10}+ {1\over 10^{2}}+ {1\over 10^{6}}+{1\over 10^{24}}+\ldots+{1\over 10^{n!}}+\ldots$$ This topic would be a study of the history and mathematics of algebraic and transcendental numbers with the history/mathematics balance depending upon your interests and background.

Math Department web pages Created: 15 Apr 2002 Last modified: 25 Apr 2008 15:38:38 Comments to: math@union.edu

Thesis Topics - S. Niefield Thesis Topics for 2008-2009 Thesis Topics - S. Niefield