Childhood memory: I learned that i is formally defined to be the square root of -1. Well, I thought, that worked well, what about the square root of i? Surely that must be yet a new kind of number. I just had to check that (a+bi)^2 can never be i. But whoa, you can solve that! (sqrt(2)/2) + (sqrt(2)/2)i does the trick. I was kind of bowled over by this. i, you sneaky bastard — you anticipated my next move and got ahead of me! I had no idea what “algebraically closed” meant, or anything like that. But it was one of my first experience of the incredible power of the right definition. Once the definition is right, you can just do everything.