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History of Transcendental numbers and Open Problems  

Park, Choon-Sung (Department of Mathematics & Information, Kyungwon University)
Ahn, Soo-Yeop (Department of Mathematics Education, Kunkuk University)
Publication Information
Journal for History of Mathematics / v.23, no.3, 2010 , pp. 57-73 More about this Journal
Abstract
Transcendental numbers are important in the history of mathematics because their study provided that circle squaring, one of the geometric problems of antiquity that had baffled mathematicians for more than 2000 years was insoluble. Liouville established in 1844 that transcendental numbers exist. In 1874, Cantor published his first proof of the existence of transcendentals in article [10]. Louville's theorem basically can be used to prove the existence of Transcendental number as well as produce a class of transcendental numbers. The number e was proved to be transcendental by Hermite in 1873, and $\pi$ by Lindemann in 1882. In 1934, Gelfond published a complete solution to the entire seventh problem of Hilbert. Within six weeks, Schneider found another independent solution. In 1966, A. Baker established the generalization of the Gelfond-Schneider theorem. He proved that any non-vanishing linear combination of logarithms of algebraic numbers with algebraic coefficients is transcendental. This study aims to examine the concept and development of transcendental numbers and to present students with its open problems promoting a research on it any further.
Keywords
Transcendental numbers; Liouville numbers; Algebraically independent; Hilbert's seventh problem; Lindemann-Weierstrass theorem; Gelfond-Schneider theorem; Zeta function;
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Times Cited By KSCI : 1  (Citation Analysis)
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