Math
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Fred Richman Few mathematical structures have undergone as many revisions or have been presented in as many guises as the real numbers. Every generation reexamines the reals in the light of its values and mathematical objectives. Arguing whether 0.999... is equal to 1 is a popular sport on the newsgroup sci.math---a thread that will not die. It seems to me that people are often too quick to dismiss the idea that these two numbers might be different. The issues here are closely related to Zeno's paradox, and to the notion of potential infinity versus actual infinity.
Is 0.999... = 1?
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A few days ago, Endre Szemerédi was awarded the 2012 Abel prize “for his fundamental contributions to discrete mathematics and theoretical computer science, and in recognition of the profound and lasting impact of these contributions on additive number theory and ergodic theory.” The full citation for the prize may be found here , and the written notes for a talk given by Tim Gowers on Endre’s work at the announcement may be found here (and video of the talk can be found here ). As I was on the Abel prize committee this year, I won’t comment further on the prize, but will instead focus on what is arguably Endre’s most well known result, namely Szemerédi’s theorem on arithmetic progressions : Theorem 1 (Szemerédi’s theorem) Let be a set of integers of positive upper density, thus , where . Then contains an arithmetic progression of length for any