a new way of defining $$e$$ the Euler number


This is inspired by a post on Quora by Alon Amit. I found this approach of defining e, the Euler number more intuitive than the traditional way($$e=\lim_{n \to \infty} (1+\frac{1}{n})^n$$). In addition, it explains the importance of the Euler number very well.

First of all, we need a function that solves the elementary differential equation:

$$\frac{d f(x)}{d x} = f(x)$$

Of course, without specifying an initial value, this equation will have infinite number of solutions. Just for convenience, let’s specify:


By Picard–Lindelöf theorem, our equation:

$$\begin{gather*} \frac{d f(x)}{d x} = f(x)\\f(0)=1 \end{gather*} $$

has one and only one solution.

Now, let’s see what kind of property this function $$f(x)$$ has.

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