Circle in the Euclidean space using Euler's Number

[OrthoJacobian]

0 to 1 in Euclidean space.

(1 + 1/n)^n using Euler's Number.

1 to 0 with the circle.

How amazing is Euler's Number?!

 

[Mentallic]

0 to 1 in Euclidean space.

(1 + 1/n)^n using Euler's Number.

1 to 0 with the circle.

How amazing is Euler's Number?!

What...?

But welcome to PF!

 

[HallsofIvy]

What do you mean by "0 to 1 in Euclidean space"? What is changing from 0 to 1?

What do you mean by "(1+ 1/n)^n using Euler's number"? Yes, the limit, as n goes to infinity is Euler's number but I would not say "with" Euler's number.

And, finally, what do you mean by "1 to 0 with the circle"? What is changing from 1 to 0 and what does that have to do with the circle?

 

[jbstemp]

I'm so confused by this post. Are you talking about how $e^{i\theta}$ is a circle in the complex plane with radius $1$, or how the series expansion for $(1+\frac{1}{n})^n$ is $e-\frac{e}{2n}+O(\frac{1}{n^2})$, or something else?

Regardless, e certainly is an amazing number and pops up in tons of (un)expected places.