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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?

MJD
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user10024395
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1 Answers1

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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.

MJD
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    When it comes to infinite series, you might think that the example $$\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\frac{1}{256}...$$ suits the purpose well. This series has the very appealing illustration which is also given here: http://en.wikipedia.org/wiki/1/4_%2B_1/16_%2B_1/64_%2B_1/256_%2B_%E2%8B%AF where you can immediately see the $1:3$ proportion of the seqeunce of squares to the main square. Then you can add that most problems, like $1+\frac{1}{2}+\frac{1}{3}+...$, do not have intuitive solutions like this, so a method is required. – String May 07 '14 at 13:53
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    I think the $\frac1{2^n}$ example is simpler, and requires no diagram. You point out that at each step, you are adding a term that gets you only half of the remaining distance to 2, and it will certainly be clear that you can't reach 2 by doing this, and probably clear that you can get as close to 2 as you like. – MJD May 07 '14 at 13:55
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    The $\frac{1}{2^n}$-example has a diagram too: http://en.wikipedia.org/wiki/1/2_%2B_1/4_%2B_1/8_%2B_1/16_%2B_%E2%8B%AF which has less symmetry, but is very nice anyway. However, the $\frac{1}{4^n}$-example can be seen at a glance from the diagram, which I personally like about it. – String May 07 '14 at 14:08
  • I don't know if it is important, but this answer in a way suggests that analysis comprises solely of infinite series. Was that picked arbitrarily or deliberately? I think that the study of curves that are more complex than circles, triangles and rectangles with respect to both tangents and areas is of great importance too. But perhaps that is because that part of the subject was already "parked" at lower educational levels as calculus? I am only asking out of curiousity! – String May 07 '14 at 14:12
  • I don't think suggests that. – MJD May 07 '14 at 14:13