It is very sound. You can think of i as being a variable 'x', and therefore, you can see Complex numbers are polynomials over real numbers (denoted as R[x]). Bu

It is very sound. You can think of i as being a variable 'x', and therefore, you can see Complex numbers are polynomials over real numbers (denoted as R[x]). But you can simplify the polynomials by using the fact that x2 = 1 for this construction (since x = i on this situation). This is the exact construction of Complex numbers as an algebraic extension over the field of Real numbers[1] What is more amazing: there is no such way to do the same on the complex numbers. This is because the Complex numbers form an Algebraically closed field[2] [1] http://en.wikipedia.org/wiki/Field_extension#Examples http://en.wikipedia.org/wiki/Field_extension#Examples [2] http://en.wikipedia.org/wiki/Algebraically_closed_field http://en.wikipedia.org/wiki/Algebraically_closed_field