Skip to main content

[Community Question] Calculus: Why is Euler's number 2.718 and not any thing else?

One of our user asked:

Why is Euler's number $\mathtt 2.71828$ and not for example $\mathtt 3.7589$ ???

I know that e is the base of natural logarithms, I know about areas on hyperbola xy=1 and I know it's formula : $$e =\sum_{n=0}^\infty \frac{1}{n!}$$
And I also know it has many other characterizations.
But, why is e equal to that formula (which sum is approximately $\mathtt 2.71828$) ???
I googled that many times and every time it ends in having "e is the base of natural logarithms",
I don't want to work out any equations using e without understanding it perfectly.

Thanks in advance.


Labels:

Comments

Post a Comment

Popular posts from this blog

[Community Question] Calculus: Manifold with boundary - finding the boundary

One of our user asked: I have the manifold with boundary $M:= \lbrace (x_1,x_2,x_3) \in \mathbb R^3 : x_1\geq 0, x_1^2+x_2^2+x_3^2=1\rbrace \cup\lbrace (x_1,x_2,x_3) \in \mathbb R^3 : x_1= 0, x_1^2+x_2^2+x_3^2\leq1\rbrace$ and I need to find the boundary of this manifold. I think it is $\lbrace (x_1,x_2,x_3) \in \mathbb R^n : x_1= 0, x_2^2+x_3^2=1\rbrace$ , the other option is that the boundary is the empty set? I think the first is right? Am I wrong?
Post a Comment
Read more

[Community Question] Linear-algebra: Are linear transformations between infinite dimensional vector spaces always differentiable?

One of our user asked: In class we saw that every linear transformation is differentiable (since there's always a linear approximation for them) and we also saw that a differentiable function must be continuous, so it must be true that all linear operators are continuous, however, I just read that between infinite dimensional vector spaces this is not necessarily true. I would like to know where's the flaw in my reasoning (I suspect that linear transformations between infinite dimensional vector spaces are not always differentiable).
Post a Comment
Read more