TITLE: What is the purpose of K-Theory?
QUESTION [17 upvotes]: I have recognized that there is a theory called K-Theory in mathematics is used also for applications in mathematical physics. There is existing algebraic   K-Theory and topological K-Theory. Are these theories very similar?
For algebraic K-Theory by Milnor I have seen that the K-Groups are given by
$K_n = T^n / a \otimes (1-a)$ (Wikipedia).
Here, $T^n$ is the $n$-fold Tensor product. For n=2 one obtains abelian matrices. I don't understand this theory in depht. What is the reason that K- theory was introduced? (Is the theoretical physics application topological or algebraic?) And is there material (lecture Video or good pdf script) where the algebraic K-theory is explained?
I would greatly appreciate an answer.

REPLY [9 votes]: In short, algebraic $K$-theory starts with the observation that the dimension of vector spaces over a field is a very useful thing! The start is the study of the $K_0$ group of a ring, which is «the best thing for $A$-modules that feels like the dimension of vector spaces».
The next player in $K$-theory is the $K_1$ of a ring $A$, which again measures how far we are from the nice situation of linear algebra: there one can take, by applying row and column operations, matrices into very simple forms. This cannot be done for general rings, and $K_1$ tells you how badly it fails.
Higher $K$-theory is considerably more difficult to motivate and to explain, but can be felt in the same way.
You should browse Jonathan Rosenberg's beautiful book on $K$-theory.