Wikipedia:Reference desk/Archives/Mathematics/2008 February 3

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February 2

Isn't there an interesting question out there about February 2?

Anyone? (extending and editing my previous remarks about February 2 ... —hydnjo talk 06:47, 4 February 2008 (UTC))

How about $(2^{(}2^{2}))/2='08$ or something. --hydnjo talk 07:18, 4 February 2008 (UTC)

Oh, ok then, - guess that'll have to do :-( --hydnjo talk 07:18, 4 February 2008 (UTC)

Well done87.102.90.249 (talk) 12:54, 4 February 2008 (UTC)

2+2=2*2=2^2=2^^2=4

Feb 3

Distance

OK, so lets say I have points A,B,C on a plane forming the veritices of a triangle, with AB=1 and BC=1, both lines parallel to the axes. From the Pythagorean theorem, I know that AC=sqrt2. However, lets say in order to get from A to C, I instead started parallel to BC, walked, say 1/4 of the horizontal distance, then made a 90 degree turn and move forwards 1/4 of the horizontal distance, then turned 90 degrees again, etc. The distance of that path will be equal to AB+BC, because each of those little fragments in one direction corespond to a fragment of the line to which it is parallel. Using that logic, I could move, say, 1/32 of the distance before turning 90 degrees, etc, and the total distance would still be equal to AB+BC. In fact, any distance for each fragment will give AB+BC for the total distance! If I keep on making each incriment smaller and smaller, the total distance from A to C will always be 2. But, at the end of that sequence is the straight line, AC, which I know is not 2, but sqrt 2... An explanation? 70.156.60.236 (talk) 04:44, 3 February 2008 (UTC)

Wow, funny you should talk about that. We recently had a discussion about that in my analysis and my teacher gave us the same exact example. If you have a right triangle and one leg is A units long and the other leg is B units long. If you travel along the legs, the distance traveled is A+B and it will always be A+B unless you take a straight line along the hypotenuse in which case the distance will be sqrt{A^2+B^2}. So at which point, does the length change from A+B to ${\sqrt {(}}A^{2}+B^{2})$ and they are certainly not always equal? The answer has something to do with uniform convergence. I don't know how familiar with analysis you are but in a sequence of functions, if you don't have uniform convergence (as opposed to just convergence, which we shall now call pointwise convergence), then some of the most "obvious" results which we take for granted fall apart. For example, in calculus, if we do term by term integration or term by term differentiation of an infinite power series, we can only do that because that series converges uniformly. I will give you another example. Consider the sequence $\{x,x^{2},x^{3},x^{4},...,x^{n},...\}{\textrm {for}}x\in [0,1]$. This is a sequence of functions and all of them are continuous on the given interval. In fact, they are continuous everywhere. But if you take the limit as $n\rightarrow \infty$ then the limiting function becomes

$f(x)={\begin{cases}0&x\in [0,1)\\1&x=1\end{cases}}$

So, what happens? Each function in the sequence is continuous but the limit breaks up. The reason is uniform convergence. This sequence does NOT converge uniformly and that is why even if each terms is continuous, the limiting function is not guaranteed to be continuous.A Real Kaiser (talk) 05:27, 3 February 2008 (UTC)

While I agree with your description of uniform convergence, I struggle to see how it applies to this question. I can't see a sequence of functions, I can only see a sequence of numbers {2*n/n}, which ought to converge to 2, no sqrt(2) - there's no "x" or similar in there for the convergence to be uniform or not with respect to... I should probably have paid more attention in my Analysis in Many Variables lectures, but maybe you can point out what I'm missing? --Tango (talk) 16:36, 3 February 2008 (UTC)

Everything that you computed is right, you just need to go the last step and also believe it. You are currently confused by an assumption that you took for granted, but it is false. The limit of the lengths of a series of curves is not equal to the length of the limit of a series of curves. If you had made the analysis in the hope to find the length of AC, than your analysis is wrong, because you made this incorrect assumption. The length of a curve is defined by integrating along the curve in the direction of the curve, but the direction of the curve changes when you switch from the staircase like approximations to the the straight line limit. Another view is if you look at the definition in Arc length. Your approximation staircases have points which are not on the straight line, but the definition requires all points to be on the line.

btw.: Very good question. Clever people have reasoned about what would happen if length were defined by a metric that works like your analysis does. The result is called Manhattan distance. Thorbadil (talk) 18:32, 3 February 2008 (UTC)

It might provide perspective to imagine other examples. Say, for instance, that instead of walking parallel to the axes you walk in curliques that get smaller and smaller, or back and forth along the line itself, or in a hairpin-turn zigzag that covers dozens of times the length of the line while never going far from it or backtracking. In general, you can get these paths be of whatever length you want while never getting more than an arbitrarily small distance from the diagonal line. With one exception: you can never produce a path with less length than the actual diagonal distance, and you can never achieve that minimum except with a perfectly straight line. So, it seems reasonable to think of these other paths as extremely intricate detours. Another bit of perspective might be gained by studying more pathological examples, like the Koch curve, Peano curve, and Weierstrass function. Black Carrot (talk) 20:09, 3 February 2008 (UTC)

By the way, I just noticed that the introduction of Koch curve makes the exact assumption that started this. It's important to notice the difference: The Koch curve is the actual limit curve, and any description of its length has to keep track of that. It is the set of all points that remain part of the simpler curves from some step onwards, and the limit points thereof. You can justify that it has infinite length by letting curves of known length hug it more and more tightly. They must have length less than the Koch curve, since they're smoothed-out versions of it, but can be made arbitrarily long. These approximation curves can be most easily taken as the curves that defined the Koch curve in the first place, since they certainly hug it tightly. The curve you mention works in the opposite way. Each point of the diagonal line is a point that was fixed at some point in the process (where the stepped line meets it) and the limit points thereof. However, there's no reason to think the stepped line is shorter than the diagonal. Since it's rougher, it could only be longer, meaning that you could prove the length of the diagonal at most 2, but not at least 2. Black Carrot (talk) 20:20, 3 February 2008 (UTC)