I was working on some basic complex numbers and suddenly have a question about it. As you all know, the complex numbers are created as we cannot find a place in the real number system to fit in the value (-1)^(1/2).
So we use i to represent it, but how do we know that any number to the power of any other number (including fraction, negative and others) are able to be expressed in a+bi? Do we need another number system for like (-1)^(1/4)? Although that is not a very good example, because we are able to show that (1/2 + i/2)^4 = -1 after some working.
After doing some researches on this topic, I think I come to the conclusion that we don't need another complex system but I don't know how to prove it. Can anyone tell me how?
I really appreciate any help on this.