Gytax said:
So why exactly in this case imaginary numbers aren't allowed and elsewhere they are?
Every number has
two square roots. For real numbers, we can define
the square root (or 1/2 power) to be the positive root. However, there is no order for the complex numbers that makes it an ordered field. In particular, that means we cannot distinguish between "positive" and "negative" roots for complex numbers. Because of that, your error was writing "take
the square root" and treating it as if it were a uniquely defined number.
That is also why "defining" i to be "\sqrt{-1}" or "
the number whose square is -1" are technically wrong. They do not distinguish between the two possible roots. A more valid approach is to define the complex numbers as
pairs of real numbers, (a, b), and then define addition by (a, b)+ (c, d)= (a+ c, b+ d) and define multplication by (a, b)(c, d)= (ac- bd, ad+ bc). Then we can identify every real number a with the complex number (a, 0) and define i to be (0, 1). Then, (0, 1)(0, 1)= (0*0- 1*1, 0*1+ 1*0)= (-1, 0) so that i^2= -1. Of course, (-i)^2= (0, -1)^2= (-1, 0) also but now we can distinguish between i= (0, 1) and -i= (0, -1).
