How to define the exponential function without calculus? For fun, I would like to define the complex exponential function from these two properties:


*

*$\exp(0) = 1$

*$\exp(z + w) = \exp(z) \exp(w)$


From here, I would like to find a way to compute values of $\exp(z)$, or at least to compute $\exp(1)$. 
So far, I found only two ways:


*

*Noting that $\exp'(z) = \exp(z)$ and solving the differential equation, which leads to $\int \frac{\exp'(z)}{\exp(z)} dz = \log(\exp(z)) + C = z$.

*Noting that $\exp'(z) = \exp(z)$, computing its Taylor series and checking that what I get is an entire function.
The first approach is simply wrong because it involves logarithms, which I have not defined yet. The second approach looks much better. I haven't tried, but I guess I can find a way to manipulate the Taylor series to obtain the limit definition of $e$ and conclude that $\exp(1) = e$, which is my aim.
However, I'm struggling to find another way that does not involve differentiation or limits in general. I would be happy to find a way to say $\exp(1) = e$ without calculus. I think that the irrational nature of $e$ forces me to use limits -- am I right?

Not sure how helpful this is, but you might compare the situation with $\pi$.
There is a geometric "definition" of $\pi$ that relies on the intuitive ideas of the diameter and the circumference of a circle.
Actually "calculating" $\pi$ requires a method of successive approximation.
If we are willing to accept an intuitive idea of the (directed) area between the graph of the reciprocal function and the horizontal axis "from" $1$ to $r$, then we can give a geometric "definition" of the real exponential function.
$\exp{x}$ will be the value of $r$ needed to get an area of $x$.
Actually "calculating" values of this function will require a method of successive approximation.
