K-theory. An elementary introduction

This survey paper is an expanded version of lectures given at the Clay Mathematics Academy ; see http://www.claymath.org/programs/outreach/academy/colloquium2005.php These lectures were intended to very young (and motivated) college students with little background. Therefore, they are accessible to a mathematician of any speciality willing to understand the subject. A much more complete introduction to K-theory may be found in the "Handbook of K-theory", recently edited by Springer.