Interesting observations.<p>I'd like to write this up - I might later - I don't have time now - here's some material from Feynman ...<p>Take successive square roots of 10.  As you get further and further you spot a pattern, that when <i>e</i> is very small, 10^e ~ 1+c.e for some constant <i>c.</i><p>Now take 10^(i.e) where <i>e</i> is very small and <i>i=sqrt(-1).</i>  Then 10^(i.e) ~ 1+c.i.e.  Take repeated powers and you end up moving in a circle.  Following that we end up with the formula e^(i.t) = cos(t)+i.sin(t) where t is a real angle.<p>None of that works if you don't basically take the reals for granted.  Or at least, it's very hard.