A) I understand that complex numbers come in the form z= a+ib where a and b are real numbers. In the special case that b = 0 you get pure real numbers which are a subset of complex numbers. I read that both real and imaginary numbers are complex numbers so I am a little confused with notations. What then is the purpose of saying something is a member of the reals or a member of complex if real numbers are also part of the complex numbers. Also how do you determine what is a larger number when comparing say 4i with 3. Do I take the modulus of 4i (which is 2) and compare that? B) Now going to the complex plane, we use something like vector addition to map a complex number using some amount of real and some amount of imaginary. The real and imaginary axis are orthogonal to each other so I am wondering if real and imaginary numbers are orthogonal? For example if I have two real orthogonal vectors u=(1,1) and v=(1,-1), the dot product gives 0. But two complex numbers that are "orthogonal" say s=1+i and t=1-i give a "dot product" of 2 (using i*-i=1). Also since a complex number uses real and imaginary components, why can I not do a dot product (or vector multiplication) and say s (dot) t =|s||t|cosα and since α= 90, then s (dot) t=0? I'm not sure if I am asking the right questions here or if I even understand what I'm asking but can someone try to clear this up?