>>13845889Five solutions exist.
First, you can prove this geometrically. If you plot the function int he complex plane, there are five zeros. No ifs, ands, or buts. Just because you can't find a way to express them doesn't prove that they aren't there when you can see it on the graph.
Secondly, there are real but non-rational numbers. Wildberger objects to this, claiming that you can't just describe a process to get to the answer, you have to have the answer. By doing so, he is disqualifying a lot of numbers with infinite digits, but that's where the solution to his problem is.
Additionally, if you can prove that you can get arbitrarily close to a solution, then that's the same as proving that the solution exists. 0.999 repeating = 1. This restriction just isn't mathematically sound.
