# What is the constant $e$, fundamentally? [duplicate]

The number $e$ is important in many respects. If you ask anyone why it is important, you will get multiple answers. Some say the defining property is that the derivative of $e^x$ is $e^x$. Some say that it is due to it being the limit $(1+\frac{1}{n})^n$, some define it using the integral of $\frac{1}{x}$. So far my personal intuition has been that all of those explanations are somewhat secondary. That is, rather than describing some fundamental property of $e$, they just demonstrate the consequences.

I understand that this opinion is somewhat subjective and the question is vague, but I hope someone here might understand what I mean and, perhaps, provide a "better" answer to the simple question of "what is $e$"?

Intuitively, I feel there are several directions which might provide a "satisfying" response.

• Firstly, perhaps $e$ can be expressed as an elegant and very important general invariant? Something like "for each isomorphism $f$ between the additive and the multiplicative groups of $\mathbb{R}$, $e=f(f(x)/f'(x))$". If only it were clear why such (or a similar) invariant must play an important role and participate in all those multiple well-known equations, it would clear things up a lot.
• Secondly, perhaps there is a nice "information-theoretical" explanation for $e$ playing, in some sense, the same role in the multiplicative group as $1$ does in the additive group of $\mathbb{R}$. I.e. with some generalizations $1$ can be viewed as a generator of $(\mathbb{R},+)$ (because any $a = 1\cdot a$), and although any other nonzero number would also work, $1$ is the most "reasonable" choice. Similarly, it seems that $e$ is the best choice for a generator in the multiplicative group. If it can be shown to be the "most reasonable" choice, perhaps it would become obvious (or, at least, "natural") for $e$ to pop up everywhere where infinite multiplications are involved.
• Finally, maybe the question I am asking is better phrased as "what properties of the field of real numbers can be stripped away for $e$ to still remain a fundamentally important object"? For example, is there a similarly-important "$e$" in any continuous group?

Thanks.

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