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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.

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    $\begingroup$ The definitions are all equivalent, so you can use any of them and then prove the rest as theorems. This isn't circular, although I agree that it's confusing until you've actually done all of the proofs. Anyway, it's usually a good sign when something has lots of equivalent definitions: that means there are lots of reasons it's important. $\endgroup$ – Qiaochu Yuan Sep 9 '15 at 4:33
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    $\begingroup$ There isn't even an "agreed upon" definition of the number 1. Why should $e$ be any different? Everything in mathematics can be approached in multiple ways. Which way is used by any treatise depends both on the preferences of the author, and the particular needs of the moment. For instance, the limit expression for $e$ is often given in early courses because it is the most easily accessible, but of the ones given, it is also the least illuminating. So usually $e$ is defined by other means in more advanced courses. $\endgroup$ – Paul Sinclair Sep 9 '15 at 4:46
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    $\begingroup$ @QiaochuYuan You mean the integer after $0$. $\endgroup$ – Jack Lam Sep 9 '15 at 4:50
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    $\begingroup$ @JackLam: You mean the integer immediately after $0$ in the usual ordering of the integers. $\endgroup$ – Will R Sep 9 '15 at 5:03
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    $\begingroup$ The very fact that so many ways to characterize $e$ exist makes it worthwhile to coin a special name $e$ for this number. On the other hand, numbers like "the order of the monster group" have no nice alternative characterizations, which is why it is not a well-known and interesting mathematical constant $\endgroup$ – Hagen von Eitzen Sep 9 '15 at 6:49
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Analogously, here are several ways to define me:

  1. I am the citizen of the US with social security number [XYZ]. This is of primary interest to the government.

  2. I am the oldest son of [my mother's name]. This is of primary interest to my family.

  3. I am the instructor of [particular course meeting at particular days/times] at [university]. This is of primary interest to students in that class.

  4. 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.

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    $\begingroup$ You are user "pjs36" on MathStackExchange as of the time I post this comment. Look, another definition. $\endgroup$ – wythagoras Sep 9 '15 at 17:26
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    $\begingroup$ And the user with ID number 120540. $\endgroup$ – immibis Sep 9 '15 at 23:58
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    $\begingroup$ And, wow, look, the top answerer of question 1427618 as of this moment $\endgroup$ – Pierre Arlaud Sep 10 '15 at 9:14
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The number $e$ has many different characterizations.

The word "characterization" has a precise meaning in mathematics. An exercise in a textbook may say:

(a) Prove that $X$ is enormously purple but not largely purple.

(b) Prove that the property of being enormously purple but not largely purple characterizes $X$.

The student is expected to understand the difference between (a) and (b). To say that the property characterizes $X$ means that $X$ is the only thing that has that property.

The number $e$ has many characterizations besides the ones you mention. So does the golden ratio. So does $\pi$. The concept of a "parabola" can be characterized as (1) a certain conic section; or (2) the locus of point equidistant from the focus and the directrix; or (3) the curve having a certain reflecting property; or (4) the graph of a quadratic polynomial function; or any of a number of other ways.

Which characterization should be taken to be the "definition" depends on context. And unfortunately mathematicians have never collected their thoughts on the precise nature of that dependence in the same way they have on matters of deductive logic.

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Answer 1: pedagogy.

A definition is chosen because it fits into the story the author wants to write. An author may find a particular choice of definition fits nicely into their story; e.g. one author may prefer start from the simplicity and elegance of manipulating integrals, another might want to talk about solving differential equations, and yet another author might want to start talking about $e^x$ early on and so prefers the one in terms of limit.


Answer 2: they're 'different' constants, and it's a neat theorem that they all turn out to be equal!

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Let me explain your predicament. Look at the following equivalent definitions of $1$:

  1. The unique real number $x$ that satisfies $\int\limits_{0}^{x} y \,\text{d}y = \frac{1}{2}$.

  2. The sum of the series $\sum\limits_{n=2}^{\infty} \frac{1}{n(n-1)}$.

  3. The smallest non-negative real number that satisfies the equation $\cos (\pi x) = -1$.

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    $\begingroup$ This would probably be better as a comment, but this is the right response to the question. $\endgroup$ – nathan.j.mcdougall Sep 9 '15 at 4:51
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    $\begingroup$ This is different from the question in that taking any of these as the definition of $1$ would indeed be circular. $\endgroup$ – Stefan Sep 9 '15 at 10:30
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    $\begingroup$ You're using 1 to define 1. Isn't that a problem? $\endgroup$ – Arturo Torres Sánchez Sep 9 '15 at 13:36
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    $\begingroup$ It seems to be a problem with definition 2, but not necessarily with definition 3. You could probably find a way to construct a number system where "$-1$" was defined as "the square of $i$" without any reference to $1$ or the inverse operation of addition. $\endgroup$ – alephzero Sep 9 '15 at 19:41
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    $\begingroup$ Additional confusion. For the definition of $e,$ take: $$\sum_{n=0}^{\infty} \frac{1}{n!}.$$ Tweak it by putting the multiplier $(n+2)$ in the denominator to get a definition for $1:$ $$\sum_{n=0}^{\infty} \frac{1}{(n+2)n!}.$$ $\endgroup$ – Fred Kline Sep 11 '15 at 4:00
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About the first part of your question:
There is no need for an 'agreed upon' definition, if you have the given scenario.
You can choose any definition to start from and then prove your way through to the theorems that state equality to the other ones.

For instance, take two persons, person 1 (p1) and person 2 (p2).
Furthermore assume that p1 defines $e$ via definition A, and that p2 defines $e$ via definition B.

Then p1 can prove a theorem that states that $e$, defined via definition A, is equal to $e$, defined via definition B/ has the defining and unique property used in definition B.
So p1 can start proving the same theorems about 'his' $e$ (defined as in definition A) as p2 about 'his' $e$ (defined via B) and vice versa.
So, while strictly formally speaking, p1 and p2 are talking about different $e$'s, from an epistemological point of view, it doesn't make a difference:
Any theorems using properties of (any) $e$ hold true in both "branches" of mathematics that p1 and p2 are building by using different definitions (assuming that up to chosing different definitions of $e$, p1 and p2 went through mathematics with the just the same doesn't harm my argument here).

About the second part of your question:
Strictly speaking, you can only define a term once on your way through mathematics. Defining is simply giving initial meaning to something:

In mathematics, a definition is used to give a precise meaning to a new term, instead of describing a pre-existing term.

(quoted from Wikipedia with the liberty of making the word 'new' bold)

So to put it simple... since there can be no other definition of $e$ besides the one you have opted for on your educational/epistemological path through mathematics, the other optional definitions you could initially chose from can at most become statements about 'your' $e$ that have to be proven - and if you have proven them, they have become theorems/corollaries etc. about 'your' $e$.

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I think the answer is that it depends on your point of view and what you intend to accomplish, for which is the best one to take as the definition. The limit definition is natural if you are concerned with compound interest formulas, because it leads to the quickest derivation of the formula $A = Pe^{rt}$. The second makes sense if you are in a somewhat rigorous calculus course and trying to prove things. In my mind, this is "the official definition", because I feel that I can most conveniently prove everything with this as a starting point. The third would be sensible in a differential equations class, where it makes sense to define a function as the unique solution to y = y'. The fourth isn't that much different from the second, both are just e^x is the inverse of the natural log function. One major other equivalent definition that is used in some textbooks is the power series definition. This is particularly natural if you're doing complex analysis or some such thing.

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There's often a multitude of ways to uniquely characterize something that consequently could serve as a definition for this something. Often these characteristics are of different practicality in different contexts, and in some contexts they're out of reach for the level the text is written.

At more basic level one might want to choose one that is within the reach at that level and then at a later stage show that the others are equivalent. At a basic level it would perhaps not be feasible to use the differential equation to define $e^x$ since that would require a proof that the solution exists and is unique (or at least know that $D a^x = ka^x$ for a constant $k$ that can be proven to be $1$ for a particular choice of $a$). The integral would do nice if you introduce integrals at an early point.

At more advanced level one could just put down a bunch of characteristics and then prove them to be equivalent and then just state that these equivalent characteristics defines the concept. Or you could choose one or a few of them that are particularily easy to use for proving things (here the mentioned theorem/proofs come handy if one book defines it in a way $A$ and another in a way $B$ you can by these theorem conclude that both books talks about the same thing).

Often definitions differ between books. For example in basic analysis there's two different approaches - one that starts of by introducing integrals and the other that starts of with derivates, then you have the choice of when to define the elementary functions. These choices influences which definitions you would use for variouos concepts.

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