What is the best way to tell people what Analysis is about? What is the best way to tell people what Analysis is about? I am currently taking Analysis course. However, I am really having a big difficulty explaining to people what Mathematical Analysis is about. Does anyone have any idea how to do it?
 A: In my experience, most people who ask these questions are not interested in any mathematical details. I suggest that you tell them that analysis is like calculus, but done more carefully and in more detail with more difficult problems. I think that will convey the idea clearly to people who already know calculus, or who know what calculus is.
People who don't know what calculus is won't really understand, but I don't think those people will understand a more specific or detailed explanation either. They will understand that analysis is a difficult branch of advanced mathematics, which is probably the best you can do.
If they ask for details, that is another matter. As an example, I would try to briefly explain the convergence of the geometric series $\sum \frac1{2^n}$ (no matter how many terms you take, the sum is always less than 2), point out that the series $\sum 1$ diverges, and then present the question of the convergence of the harmonic series $\sum \frac1n$. Because  $\frac1{2^n} < \frac1n < 1$, it could go either way, and your comrade may recognize that the answer is not obvious; if they do think the answer is obvious, there is a 50-50 chance they will guess wrong, and you can tell them so.  Then I would say that one of the central problems of analysis is recognizing whether a given series converges, and if so, to what value, and add that such questions are of enormous practical importance in the solution of problems in physics and engineering.  By that time they will probably have heard enough.
