work in "k"-theory from this period culminated in bass' book "algebraic "k"-theory". in addition to providing a coherent exposition of the results then known, bass improved many of the statements of the theorems. of particular note is that bass, building on his earlier work with murthy, provided the first proof of what is now known as the fundamental theorem of algebraic "k"-theory. this is a four-term exact sequence relating "k" of a ring "r" to "k" of "r", the polynomial ring "r"["t"], and the localization "r"["t", "t"]. bass recognized that this theorem provided a description of "k" entirely in terms of "k". by applying this description recursively, he produced negative "k"-groups "k"("r"). in independent work, max karoubi gave another definition of negative "k"-groups for certain categories and proved that his definitions yielded that same groups as those of bass.