STUDENT: You know that i to n power just keep repeating itself- which creates a circle, BUT THAT CIRCLE IS A UNIT CIRCLE.

Now, take that point on the complex plane that I gave you, which the angle is pi/4 or 45 degrees.

If you use Euler formula, and you put e^ipi/4= pi/square root of 2 + pi/ square root of 2 i

BUT, that is just the point on the complex plane. Why that happen knowing that I just put e^ i pi/4 ?????/ but why e to the i angle gives me the exact point on the plane???? why the base must be e??? It is just a simple plane, whyy base e to the i angle????

I hope you understand what I mean, because I do not even understand myself.

ME: I think I know what you mean.

A point on the unit circle is just (cos(theta), sin(theta)). If that unit circle is on the complex plane, we write that point as cos(theta) + i sin(theta). So far, that’s no big deal: it’s just how it is, because of what we mean by sin and cos and i.

But Euler’s formula says, that same point on the unit circle on the complex plane, cos(theta) + i sin(theta), is also e^(i theta)!!

I think you are asking, how the hell (excuse me) did e get into it?

That is a VERY good question.

STUDENT: YES THAT IS MY QUESTION- but you are not answering it??

ME: What fun would that be?

-Summer correspondence with my most inquisitive student.

I think “whyy” should be a new word.

this is enchanting.