Their equivalence can be given by a Turing machine, though, so the non-computability comes directly from the Halting Problem. This is worth mentioning because

Their equivalence can be given by a Turing machine, though, so the non-computability comes directly from the Halting Problem. This is worth mentioning because some folks get the incorrect impression that it's not possible to write a program that, say, outputs whether two computable reals are equal to each other. It is possible! But the program might not halt. I shouldn't trivialize the difficulty. It is a PITA. I remember my first foray into this; I wrote a program which computed sqrt(8), which worked fine, and then another program which computed sqrt(8)^2, which didn't halt. The edges are tricky.