Here is something really funky that popped up in my head as I was thinking about walking across my room, which is a rectangle. Let's assume that it has side lengths of a and b. Now, If I walk from one corner to the opposite corner by only walking along the side lengths, I end up going a+d. Now, if I walk like this, I still end up going a+d. Even if I walk like this, I still end up going a+d. My calculus intuition tells me that we should have ended up with sqrt(a^2+d^2) or should at least be approaching it. What is going on?
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2See this problem: http://www.askamathematician.com/2011/01/q-%CF%80-4/ – Mathematician 42 Aug 01 '18 at 07:20
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In short, the problem is that even though the path in some sense converges to the diagonal path, the lengths do not. The first issue is that you need a proper definition of 'length of a path'. This will involve a derivative. You need more conditions to get convergence of this derivative as well! – Mathematician 42 Aug 01 '18 at 07:23
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And version on this site. – user202729 Aug 01 '18 at 07:24
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Very related. – user202729 Aug 01 '18 at 07:25
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@user202729 : The drawings and cartoons in the first version are hilarious! :D – Mathematician 42 Aug 01 '18 at 07:28
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Can someone plz explain this in layman's terms? I only know derivatives, not integrals – Dude156 Aug 01 '18 at 07:34