Thursday, 22 June 2017

Irrational ^ Irrational

I came upon this very clever but simple proof recently ...

It is not known which irrational numbers x and y would make xy rational.  But it can be easily proved that such rational xy exists.  We know that 2 is irrational.  Take, for instance, the number

which must either be rational or irrational.  If it is rational, then we're done.  If it is not, then the number

must be rational.  So either case proves the existence of rational xy where both x and y are irrational.

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