Interesting Example from Kac and Ulam’s ‘Mathematics and Logic’

On pages 6 and 7 of that fascinating book, the irrationality of the square root of 2 is rephrased in a way that, as the authors express it, ‘is so surprising as to be paradoxical.’ They prove that [square root of 2]/2 will never be covered by an open covering consisting of open intervals constructed from all rational numbers a/b plus and minus [1/4 * b ^ 2], despite the fact that the rational numbers are dense in the reals. I have had a lot of math up through measure theory, functional analysis, topology, and the like, and I had never run across this clever and quite elementary illustration. Any similar examples, or texts/articles discussing the phenomenon more generally?

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Nevin Manimala

Nevin Manimala is interested in blogging and finding new blogs https://nevinmanimala.com

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