Why can't we write complex numbers as $2^{i \theta} $ or $-40^{i \theta} $? Why does it have to be $e$?
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$\begingroup$$e$ is a special number. See here.
$\endgroup$ 1 $\begingroup$We know that complex numbers can be expressed as $$\cos { \theta } +i\sin { \theta } ,$$ and we want to express this form as $$a^{i\theta}=\cos { \theta } +i\sin { \theta } .$$ Take the second derivatives fo the both equation: $$\frac { { d }^{ 2 } }{ d{ \theta }^{ 2 } } \left( { a }^{ i\theta } \right) =\frac { { d }^{ 2 } }{ d{ \theta }^{ 2 } } (\cos { \theta } +i\sin { \theta } )\\ -{ a }^{ i\theta }{ \left( \ln { a } \right) }^{ 2 }=-(\cos { \theta } +i\sin { \theta } )=-{ a }^{ i\theta }\\ a=e$$
$\endgroup$ $\begingroup$Because of Euler's formula $e^{ix} = \cos{x}+i\sin{x}$.
$\endgroup$ 1 $\begingroup$We can, actually : $2^i$ means $\cos\ln2+i\sin\ln2$, for instance... :-) The only difference is that, in the case of e, its natural logarithm ‘disappears’, thus becoming ‘invisible’ in the expression’s final form.
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