This paper is devoted to the development of transcendental numbers and to proofs that e and Pi are transcendental. Chapter I presents a brief account of the historical development of transcendental numbers. Chapter II is composed of the existence proof of transcendental numbers which is based on a proof due to G. Canto, and also a method of construction for transcendental numbers. Chapter III is devoted to the proof of the transcendency of e and chapter IV, to the proof of the transcendency of Pi. / Typescript. / "August, 1955." / "A Paper." / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Arts." / Advisor: Dwight B. Goodner, Professor Directing Paper. / Includes bibliographical references.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1969. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliography.

A remarkable inequality among integer numbers is given. Easily deduced from Fermat's Last Theorem, it would be nevertheless very difficult to establish through other means.

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We study bounds on the distances of roots of integer polynomials and applications of such results. The separation of complex roots for reducible monic integer polynomials of fourth degree is thoroughly explained. Lemmas on roots of polynomials in the p-adic setting are proved. Explicit families of polynomials of general degree as well as families in some classes of quadratic and cubic polynomials with very good separation of roots in the same setting are exhibited. The second part of the thesis is concerned with results on p-adic versions of Mahler's and Koksma's functions wn and w*n and the related classifications of transcendental numbers in Cp. The main result is a construction of numbers such that the two functions wn and w*n differ on them for every n and later on expanding the interval of possible values for wn-w*n. The inequalities linking values of Koksma's functions for algebraically dependent numbers are proved.

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This report explores the effect of interactive visualizations on the advancement of mathematics understanding. Not only do interactive visualizations aid mathematicians to expand the body of knowledge of mathematics but it also allows students an efficient way to process the information taught in schools. There are many concepts in mathematics that utilize interactive visualizations and examples of such concepts are illustrated within this report. / text

We study bounds on the distances of roots of integer polynomials and applications of such results. The separation of complex roots for reducible monic integer polynomials of fourth degree is thoroughly explained. Lemmas on roots of polynomials in the p-adic setting are proved. Explicit families of polynomials of general degree as well as families in some classes of quadratic and cubic polynomials with very good separation of roots in the same setting are exhibited. The second part of the thesis is concerned with results on p-adic versions of Mahler's and Koksma's functions wn and w*n and the related classifications of transcendental numbers in Cp. The main result is a construction of numbers such that the two functions wn and w*n differ on them for every n and later on expanding the interval of possible values for wn-w*n. The inequalities linking values of Koksma's functions for algebraically dependent numbers are proved.

Nous étudions les bornes sur les distances des racines des polynômes entiers et les applications de ces résultats. La séparation des racines complexes pour les polynômes réductibles normalisés de quatrième degré à coefficients entiers est examinée plus à fond. Différents lemmes sur les racines des polynômes en nombres p-adiques sont prouvés. Sont fournies les familles explicites de polynômes de degré général, ainsi que les familles dans certaines classes de polynômes quadratiques et cubiques avec une très bon separation des racins dans le cadre p-adique. Le reste de la thèse est dédié aux résultats liés aux versions p-adiques des fonctions de Mahler et de Koksma wn et w*n , ainsi qu'aux classifications correspondantes des nombres transcendants dans Cp. Le résultat principal est une construction des nombres pour lesquelles les deux fonctions wn et w*n sont différentes pour tous les n et puis l'intervalle de valeurs possibles pour wn-w*n est élargi. Les inégalités reliant les valeurs des fonctions de Koksma en nombres algébriquement dépendants sont prouvées. / We study bounds on the distances of roots of integer polynomials and applications of such results. The separation of complex roots for reducible monic integer polynomials of fourth degree is thoroughly explained. Lemmas on roots of polynomials in the p-adic setting are proved. Explicit families of polynomials of general degree as well as families in some classes of quadratic and cubic polynomials with very good separation of roots in the same setting are exhibited. The second part of the thesis is concerned with results on p-adic versions of Mahler's and Koksma's functions wn and w*n and the related classifications of transcendental numbers in Cp. The main result is a construction of numbers such that the two functions wn and w*n differ on them for every n and later on expanding the interval of possible values for wn-w*n. The inequalities linking values of Koksma's functions for algebraically dependent numbers are proved.