TITLE: Motivation for algebraic K-theory?
QUESTION [92 upvotes]: I'm looking for a big-picture treatment of algebraic K-theory and why it's important.  I've seen various abstract definitions (Quillen's plus and Q constructions, some spectral constructions like Waldhausen's) and a lot of work devoted to calculation in special cases, e.g., extracting information about K-theory from Hochschild and cyclic homology.  As far as I can tell, K-theory is extremely difficult to compute, it yields deep information about a category, and in some cases, this produces highly nontrivial results in arithmetic or manifold topology.  I've been unable to piece these results into a coherent picture of why one would think K-theory is the right tool to use, or why someone would want to know that, e.g., K22(Z) has an element of order 691.  Explanations and pointers to readable literature would be greatly appreciated.

REPLY [5 votes]: Grothendieck's original motivation for K-theory was to give a natural setting for the intersection theory on algebraic varieties.