In one of the earlier posts we indicated that Johann H. Lambert proved the irrationality of \exp(x) or e^{x} for non-zero rational x by means of continued fraction expansion of \tanh x. In this post we provide another proof for irrationality of e^{x} which is based on a completely different approach. I first read this proof from Carl Ludwig Siegel's wonderful book Transcendental Numbers and I was amazed by the simplicity and novelty of Siegel's argument.
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