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By Vitushkin A. G.

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For this is what mathematicians do: When they cannot solve a problem, they re-state it and turn it around and try to find other ways to look at it. This is nothing to be ashamed of. As the great classic work on problem-solving—P´olya’s How to Solve It [POL]— will tell you, one of the mathematician’s most powerful tools is to restate a problem. We shall encounter this technique repeatedly in the present book. But, unfortunately, this method of re-statement did not help the Greeks. Like all people in all civilizations, they had an interlocking system of beliefs to which their reasoning was wedded.

14103 . 3 Archimedes 37 This new approximation of π is accurate to nearly three decimal places. Archimedes himself considered regular polygons with nearly 500 sides. His method did not yield an approximation as accurate as ours. But, historically, it was one of the first estimations of the size of π. Exercises √ 1. Verify that the number 17 is irrational. √ 2. The number α = 5 9 is that unique positive real number that satisfies α5 = 9. Verify that this α is irrational. 3. , a natural √ number).

After we state Zeno’s paradox, we shall be able to analyze it quickly and easily. ). Our main source of information concerning this influential thinker is Plato’s dialogue Parmenides. Although Plato gives a positive account of Zeno’s teachings, he does not necessarily believe all the paradoxes that we usually attribute to Zeno. The philosopher Diogenes Laertius also wrote of Zeno’s life, but his reports are today deemed to be unreliable. Zeno was certainly a philosopher, and was the son of Teleutagoras.

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Analytic capacity and rational approximations by Vitushkin A. G.

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