TITLE: How do I interpret Euler's formula?
QUESTION [34 upvotes]: I don't understand the formula at all:
$$e^{ix} = \cos(x) + i \sin(x)$$
I've tried reading all sorts of webpages and answers on the subject but it's just not clicking with me. I don't understand how we can define things when these are already known quantities. We've already got an exact definition for $e$ as $\lim_{n \to 0} (1+\frac{1}{n})^n$, and we've got $i = \sqrt{-1}$, and we've got $\cos$ and $\sin$ as the $x$ and $y$ coordinates of where right triangles meet the unit circle. 
So I don't understand how we get from those figures to Euler's formula. Then again I also don't understand why complex numbers are of the form $a + bi$ so that might have something to do with it. I have a hard time separating what identities are "emergent" from the previous mathematics and which pieces are "by definition" and why they are defined that way. I don't understand why we start talking about "rotations" as if it's obvious that's what's happening. I don't even know why $e$ is involved in any of this to begin with. 
Is $e^{ix}$ simply being written in the slightly rearranged complex form $a + ib$ where $a = \cos(x)$ and $b = \sin(x)$? Is this to be interpreted as $\cos(x)$ being the $x$-coordinate and somehow $i \sin(x)$ is the $y$ coordinate?

REPLY [2 votes]: I'll try to address these pieces 

I don't understand how we can define things when these are already known quantities. We've already got an exact definition for $e$ as $\lim_{n \to 0} (1+\frac{1}{n})^n$, and we've got $i = \sqrt{-1}$, and we've got $\cos$ and $\sin$ as the $x$ and $y$ coordinates of where right triangles meet the unit circle. 



I have a hard time separating what identities are "emergent" from the previous mathematics and which pieces are "by definition" and why they are defined that way.

You're right, we do have those definitions. And yes, someone can't come up to you and define $e$ as $3.7$, because $e$ already has a definition.
We've had a definition of what $e^x$ is when $x$ is a real number. What we didn't have was a definition of $e^x$ when $x$ is an imaginary number. Before we give the definition, $e^{bi}$ is some meaningless writing. You could define it to to be whatever you like, ex. $b$, or $5$, or $\log 10b$, etc. But people chose to define it as $\cos(x) + i \sin(x)$ because it would have some really cool propreties (ex. defining it this way preserves some of the propreties of 
 the real $e^x$ we are used to).
Another common example is infinite sums. We have defined $\sum_{i=1}^n a_i$ as $a_1+a_2+ \cdots + a_n$. But again this does not tell us anything about the meaning of the symbol $\sum_{i=1}^{\infty} a_i$. We could again assign anything we want to it, such as $7$, or $a_4$, or $\pi$. But we have decided to assign it $\lim_{n \to \infty} \sum_{i=1}^n a_i$