I have a set of arithmetic functions from $D\subset\mathbb C$ to $\mathbb C$ (addition, division, trigonometric functions, ...). Each of those functions can also be restricted from $E\subset \mathbb R$ to $\mathbb R$.
By using integer constant, I can define many different reals using my set of functions, for example $\pi=4\tan^{-1}(1)$.
Can I define some reals using the complex that I can't define using only the real definition ? For example, if I have only exponential function, I can define $\Re(e^{i.a})=\cos(a)$ But can I define $\cos(a)$ without trigonometric function and without complex ? I don't think so. Are they other examples where I do not restrict any use of usual functions ?
Are they some sets of functions "complete" (and which ones)? I mean that using such a set of functions, if I restrict my functions to the reals, I can define the same reals that If I can use the more general complex notations. (Regarding the previous question, the set $\{\Re,\exp\}$ would not be complete for example).
Is the set $\{+,-,\times,\div,\exp,\ln,\cos,\sin,\tan^{-1},\Re\}$ complete ?