The importance of 401 and 405 to majors cannot be overstated. This course and 405 are the only two courses absolutely required for all majors. These two together provide an introduction to the ideas and the methods of modern abstract mathematics. In 401 and 405 you will learn algebra and analysis, but you will also learn how proofs are developed and used, and methods to construct your own proofs. This involves an important transition in mathematical maturity: Until this point you have focused on solving problems by applying methods, and the central unit of learning is the "new method." Once you reach these more abstract courses, you will focus on uncovering information by applying theorems and their proofs. The central unit of learning will be the theorem. This is somehow more natural and more straightforward, because "theorem" is just another word for "fact." Instead of learning to do things you will now learn facts, and infer how to do things. In the best case, this transition happens early, even in Calculus class. In any case, expect it to be complete after you do 401 and 405!