The number $e$ seems to have multiple definitions:
$$\lim \limits_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$$
The unique number $a$ such that $\int_1^a\frac{1}{x} \, dx = 1$
The unique number $a$ such that $\frac{d}{dx}a^x=a^x$
The base of the natural logarithm
This always seems strange to me. Why is that? Why is there no 'agreed upon' definition, and all the other definitions are theorems? Some of the 'definitions' of $e$ are theorems that have to be proved if you use another definition of $e$. This can make looking up the proof for something involving $e$ or $e^x$ confusing and possibly promote circular reasoning.
Best Answer
Analogously, here are several ways to define me:
I am the citizen of the US with social security number [XYZ]. This is of primary interest to the government.
I am the oldest son of [my mother's name]. This is of primary interest to my family.
I am the instructor of [particular course meeting at particular days/times] at [university]. This is of primary interest to students in that class.
I am the author of [a particular master's thesis]. This is (maybe) of primary interest to my thesis advisor.
Of the above list, which is "the right definition" of me?
As you can see, I am related to the world in a multitude of very specific ways. Though some are quite different in their nature, they all determine me uniquely, with different people and institutions thinking of me primarily in different ways.
Similarly, the constant $e$ is related to various pieces of mathematics in many different, but specific, ways. The definition used may vary depending on what role $e$ is fulfilling in a particular context, but they all uniquely determine the same constant and are all important for their own reasons.