$(a/b)$ is a transposition of a and b.

@RyanBudney I am quite sure about that. I am sure, that every mental task doable by humans will be sooner or later doable by a machine. Once, somebody will prove a theorem, which will be the last interesting theorem proven by a human. And Math will be done by machines afterwards :)

There could be one more application of such system. We can use it to prove theorems which humanity wasn't able to prove yet. Even though we will work with undecidable theories, it does not mean that our system can't eventually prove something useful. Actually, that's what we people do during last 3000 years. No mathematician was sure he will prove a formula, before he started looking for a proof. And using some smart heuristics and enormous mechanical power, I think machines may eventually beat human mathematicians.

Thank you! I took a look at Taylor series as definition of exponentiation by complex numbers (Taylor series of $y=e^x$ around 0), I put $ix$ instead of x and I got Euler's formula! :)