Why you need a special theory for complex numbers.

  There is a big flaw in teaching the complex number theory. The first lecture or book's first paragraph usually starts with a boring description of what is complex number and its representation in different forms followed by the introduction of the theory. It does not answer the question - Why do we need a special theory and what is wrong with the real numbers algebra when applied to complex numbers?

   I believe that after defining a complex number it must be stressed that it is not possible to introduce ordering ( see Order_theory ) in the complex number space while you have a natural ordering for real numbers ( for irrational numbers introduced via nested intervals with rational borders the order is easily introduced ). This is the difference that does not allow to map a complex number space on a real number space and use a familiar real number algebra. So you need a new theory.

   It looks like that this is a common belief that this idea of lack of ordering in complex space is too complicated for first-year or second-year students and it is not possible to prove it at the level of introductory course.