Transcendence of e and π
نویسندگان
چکیده
When proving it is impossible to ‘square’ the circle by a ruler–and–compass construction we have to appeal to the theorem that π is transcendental. It is our goal to prove this theorem. Since the algebraic numbers are the roots of integer polynomials, they are countably many. Cantor’s proof in 1874 of the uncountability of the real numbers guaranteed the existence of (uncountably many) transcendental numbers. Thirty years earlier Liouville had actually constructed the transcendental number +∞ ∑
منابع مشابه
On the possible exceptions for the transcendence of the log-gamma function at rational values and its consequences for the transcendence of log π and π e
In a recent work published in this journal [JNT 129, 2154 (2009)], it has been argued that the numbers log Γ(x) + log Γ(1− x), x being a rational number between 0 and 1, are transcendental with at most one possible exception, but the proof presented there is incorrect. Here in this paper, I point out the mistake committed in that proof and I present a theorem that establishes the transcendence ...
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