Timeline for The staircase paradox, or why $\pi\ne4$
Current License: CC BY-SA 3.0
18 events
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| Jun 13, 2020 at 1:00 | comment | added | sasquires | @N.S. Perhaps you're right and the first-year calculus students wouldn't answer this correctly. But my point is basically that it's really easy to refute the paradox using the language of the calculus of infinitesimals. You just need to know how to define the arc length of a differentiable curve (i.e., $\int_C \sqrt{ dx^2 + dy^2}$), which is something they teach in first-year calculus. But I want to re-iterate that I also appreciate your point, which is that learning how to write a rigorous error bound is essential for avoiding stupid mistakes. | |
| Jun 12, 2020 at 4:29 | comment | added | N. S. | @sasquires I really dissagree that any calculus student should be able to figure this out, unless your calculus students are really smart. | |
| Jun 12, 2020 at 1:43 | comment | added | sasquires | @N.S. I agree with the spirit of your answer, but I don't think it supports the conclusion that it is wrong to teach "calculus" rather than "analysis." There is a perfectly good one-line response to this question that any calculus student should be able to give, which I will post now as a separate answer. The problem isn't calculus versus analysis, but thinking (in this case, defining the quantity you are trying to calculate) versus not thinking. | |
| Dec 2, 2019 at 5:45 | comment | added | tia | The problem is that it's a problem we assume that π < 4 at the first place or not. If we don't, we also can't say that the error is a non-zero constant toward infinite iteration. But if we assume that π < 4, so the proof is still a contradiction itself... | |
| Sep 25, 2018 at 17:48 | comment | added | N. S. | showing that regular partitions are sufficient is highly non-trivial :)To put it simple, yea we could show them how to calculate using properly Riemann sums in Calculus, but we could not explain them why it works.....From my experience, most students which take Calculus struggle to calculate $\int_0^3 x^2+1 dx$ using left end points, imagine asking them to do something where Darboux sums are really necessary. | |
| Sep 25, 2018 at 17:47 | comment | added | N. S. | @DavidK Yes and no... I think that the main problem is that most students and even some instructors don't really get all subtilities, and everything which could go wrong. Most of the times they only look as you said at left hand points vs right hand points, instead of looking at upper/lower (Darboux sums). Also, and this is part of the problem, I am not aware any simple way of proving that if the upper minus lower Darboux sum converges to 0 then the function is Riemann integrable. Last but not least, and this is another subtle problem, | |
| Sep 25, 2018 at 17:27 | comment | added | David K | Of course it is not. But this answer does mention Riemann sums. Using that topic as an example, I wonder if there is a useful level of rigor that may often be missing from some "calculus" courses that can be accomplished without turning the "calculus" course into a full-blown real analysis course. | |
| Sep 25, 2018 at 15:36 | comment | added | N. S. | @DavidK Nothing. But in this picture that is NOT a Riemann sum :) | |
| Sep 25, 2018 at 11:31 | comment | added | David K | There may be a tendency among students of calculus (and perhaps also among teachers) to consider only the lower Riemann sum; that is, to look only at rectangles inscribed under a curve. If you look at both the upper and lower Riemann sums, however, and can prove they converge to the same value, what's wrong with that? | |
| May 21, 2013 at 16:02 | history | rollback | robjohn♦ | Rollback to Revision 2 | |
| May 21, 2013 at 12:15 | history | edited | ABC | CC BY-SA 3.0 | edited body |
| Mar 20, 2013 at 12:44 | comment | added | Ray | N.S. isn't talking about numerical analysis; he's referring to en.wikipedia.org/wiki/Mathematical_analysis, which provides the foundations of calculus. | |
| Nov 13, 2012 at 8:14 | comment | added | Rory O'Kane | Wikipedia links explaining some concepts mentioned here: calculus and (numerical) analysis (classes). Riemann sum. limsup (limit superior). | |
| S Nov 13, 2012 at 8:11 | history | suggested | Rory O'Kane | CC BY-SA 3.0 | replaced \sim(ilar order of magnitude) with \approx(imately equal); improved spelling, grammar, typography |
| Nov 13, 2012 at 8:01 | review | Suggested edits | |||
| S Nov 13, 2012 at 8:11 | |||||
| May 2, 2012 at 10:55 | comment | added | Paul Slevin | @phv3773 Leibniz. | |
| Jun 29, 2011 at 19:51 | comment | added | phv3773 | To some extent, this puzzle illustrates the arc of mathematics from Archimedes to Newton. Archimedes (who would not have made this error) knew about approximation by tiny increments, but he did not have the formal theorems that are supposed to keep us out of trouble. That was the program that Newton and Leibnetz (or Leibnez?) finished. | |
| May 30, 2011 at 17:07 | history | answered | N. S. | CC BY-SA 3.0 |