# Definition of a real without using complex

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 $\pi=4\tan^{-1}(1)$.

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 ?

2. 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).

3. Is the set $\{+,-,\times,\div,\exp,\ln,\cos,\sin,\tan^{-1},\Re\}$ complete ?

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en.wikipedia.org/wiki/… –  xavierm02 Aug 18 '12 at 14:58
And how do you get $\sqrt 2$ with your functions? You need to add at least $exp^{-1}$ so that you can use $\sqrt 2 = 2^{\frac{1}{2}}$ –  xavierm02 Aug 18 '12 at 15:02
I add $\ln$ to the list –  Xoff Aug 18 '12 at 15:07
I'm not sure I understand #2 and #3. What do you mean by "define"? For any real $y$ there is a real $x$ such that $\ln x = y$, so would you say that $\{\ln\}$ is complete? –  Antonio Vargas Aug 18 '12 at 16:55
My only constants are integers, so I can define any fraction using the division, then I can define any power (like $\sqrt{2}$) using $\exp$ and $\ln$, and so one... But I think some algebric numbers can not be define (written) this way. –  Xoff Aug 18 '12 at 17:58