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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.1 The Axiom of Choice

Consider the following problem: it is easy to find two irrational numbers a and b such that a + b is rational, or such that ab is rational (in both cases one could take a = Imageand b = - Image), but is it possible for ab to be rational? Here is an elegant proof that the answer is yes. Let x = ImageImage. If x is rational then we have our example. ...

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