Different interpretations of imaginary number I went through a linear algebra course and I'm a bit confused..
I think I understand the geometric interpretation of imaginary numbers - multiplying by $i$ results in rotation by $90$ degrees in so that $1$ becomes $i$ and so forth. And this is where that $i^2 = -1$ comes from.
And then there's the matrix representation of $i$, which I understand emerged from a later generalization of complex numbers. I interpret the matrix representation as transform function which basically projects the imaginary axis to the real axis. I've thought of it as something very similar to vectors, with the difference that with vectors I write:
$P = x\mathbf{\hat{i}} + y\mathbf{\hat{j}}$ where $\mathbf{\hat{i}} = (1, 0)$ and $\mathbf{\hat{j}} = (0, 1)$
..and with complex numbers I can write:
$C = a + bi$ where $i$ = $2\times 2$ matrix, which represents the same $90$ degree transform logic by transformation.
Correct? Or at least close?
Anyways, as I understand, both of these interpretations of $i$ are actually later than $i=\sqrt{-1}$ itself. Is there an earlier interpretation? How did those who invented imaginary number prove that $i = \sqrt{-1}$ in the first place?
Thanks!
 A: As Cameron Williams' helpful comment clarifies: It wasn't a matter of "proving" that $i =\sqrt{-1}$, but more a matter of defining $i$ to represent the solution to $i^2 = -1$, as a means, for example, for solving polynomials like $x^2 + 1 = 0$.
You might find the following post helpful for help understanding $i$ in different contexts: 


*

*What are imaginary numbers? where you'll find many alternative perspectives regarding how to understand $i$ in various contexts.

A: As already commented, $i^2=-1$ was never proven, but lies at the origin of the creation of the imaginary numbers. The polynomial equation $x^2+1=0$ has no real root, therefore $\mathbb{R}$ is not algebraically closed. In order to create an algebraically closed field extension of $\mathbb{R}$, you have to add a solution of this equation. If we call the solution $i$, it must of course have the property of $i^2=-1$.
In order to see that adding $i$ is indeed sufficient to get to an algebraic closure, you might want to read up on the Fundamental theorem of Algebra (http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra).
A use of the matrix representation of complex numbers is to see how multiplication with a complex numbers acts on vectors in $\mathbb{R}^2 \simeq \mathbb{C}$.
A: $i$ wasn't proved, as much as it was conceived - invented, if you will, to solve a set of problems that previously did not have solutions. The most famous such problem (albeit not the first that was treated with this approach) is a second degree polynomial equation with no real roots
$ax^2+bx+c=0$ such that $\left(\frac{b}{a}\right)^2 - \frac{c}{a} < 0$
The equation $x^2 + 1=0$ is just one example of one such equation but once one defines the quantity $i$ to solve it, one can also specify solutions to any second-degree equation, and not just those with real roots. And not only that - you can also find all three solutions to cubic equations were only one root is real. And much more, of course...
When mathematicians started doing this it was not a matter of proving that $i$ solves $x^2+1=0$, so much as defining it to do so.
