Why are Exp[3] and 2 treated differently within Complex?

Question

Why doesn't the last command below split the complex number into its real and imaginary parts?

Complex[2,2]
(* 2 + 2 I *)
Complex[N[Exp[3]], N[Exp[3]]]
(* 20.0855 + 20.0855 I *)
Complex[Exp[3],Exp[3])
(* Complex[E^3,E^3] *)

Answer 1

I understand why this can be very confusing, but essentially

Complex[Exp[3],Exp[3]]

is invalid syntax. Do not do this.

Complex is an atomic type (AtomQ). It is not a compound expression with head Complex and two arguments. It is a fundamental type that is indivisible.

What happens when you evaluate Complex[1,2]? What you type here is indeed a compound expression with head Complex and two integer arguments. However, it evaluates to an atomic integer-based complex immediately.

AtomQ[Complex[1, 2]]
(* True *)

AtomQ[Unevaluated@Complex[1, 2]]
(* False *)

The fundamental complex type can be either integer-based or real-based, and that's it. It cannot contain arbitrary symbolic expressions.

If you put such symbolic expressions into Complex[...], it simply won't evaluate. What you get is a compound expression that is not usable as a complex number. It is not of the atomic Complex type. Im, Re, Abs, ComplexExpand, etc. won't handle it.

z = Complex[1, Sqrt[2]];

AtomQ[z]
(* False *)

{Re[z], Im[z], Abs[z]}
(* {Re[Complex[1, Sqrt[2]]], Im[Complex[1, Sqrt[2]]], Abs[Complex[1, Sqrt[2]]]} *)

The correct way to represent such number is Exp[3] + I*Exp[3] instead. The structure will then be like this: