Home on Jonathan Bergknoff · Nov 8, 2014

A Peculiar, Non-Constructive Proof

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A friend of mine loves to trot out this oddity, and I always manage to forget it afterwards. Now that I have it in mind, I will take a minute to write it here for posterity.
Theorem: There exist a,b∈Ra,b∈R, irrational, such that ab∈Qab∈Q.
In other words, the theorem asserts that an irrational raised to an irrational may be rational. This isn’t especially interesting and, while…

A friend of mine loves to trot out this oddity, and I always manage to forget it afterwards. Now that I have it in mind, I will take a minute to write it here for posterity. Theorem: There exist a,b∈Ra,b∈R, irrational, such that ab∈Qab∈Q. In other words, the theorem asserts that an irrational raised to an irrational may be rational. This isn’t especially interesting and, while surprising, I don’t claim to have any sort of intuition violated here.

Read on /journal/peculiar-non-constructive-proof/ ↗

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