$20$ years ago I did my Calculus $101$ "term paper" on this (Okay, what else would you call it? It was more of a story problem that we presented to the class, but as everyone else chose a practical applied story problem from the book, I was the only one to tackle a theoretical problem.... )
I remember yes, you can, e is just a fraction multiple of pi or something like that..., but I can't find out/remember exactly how to write it....
Thanks
I think you should be more precise as "fraction" is an imprecise term. Is $e$ a rational multiple of $\pi$, or, equivalently, can you write $\frac{e} {\pi} = \frac ab$ where $a, b$ are integers sharing no common factors other than $1$? This is an open question, answer unknown. It is strongly suspected to be "no" (that $\frac{e} {\pi}$ is irrational) but without a conclusive proof one way or the other, the question remains open.
However, there is a very similar question about whether the product $e\pi$ is rational. This is also an open question. But the two quantities cannot both be rational (can you see why)?
Just for interest, there are similar open questions about $e+\pi$ and $e-\pi$, but again, both of these can't be simultaneously rational.
In fact of these four: $e+\pi, e-\pi, e\pi, \frac e{\pi}$, it is quite easy to show that at most one can be rational. Again, this is left as an exercise for you (should it interest you).