In general, determining if two arbitrary reals are the same is impossible per the halting problem. People claim to measure 'real' numbers. This is a lie. People can only measure rational numbers. A real number is either a rational or the supremum of some arbitrary set of rationals (perhaps an infinite one). A set is described by whether or not a number is in it. To be able to determine what number is in your set you need to have some sort of decision procedure (a program). However, more real numbers exist than there are possible written programs. Thus, the full set of reals is inexpressible

On the other hand, it's very easy to see and measure rational complex numbers with a protractor.

Dummit and Foote is the classic abstract Algebra textbook to learn about how to precisely define these. Its treatment of ring theory is very well motivated and easy to grasp