Importance of Reasonable ApproximationDate: 08/07/99 at 13:38:28 From: David Ross Subject: Triangle geometry Imagine looking at a stairway in profile, such that it is essentially a right triangle with the "hypotenuse" being a series of stairs. Let's label the height a, the base b, and the staggered hypotenuse c. Now looking at the stairs, the sum of the vertical portions is a and the sum of the horizontal portions is b. An ant at the base and crawling up the stairs by sequentially going up the verticals and along the horizontals would at the top have covered the distance a + b. Now increase the number of stairs; i.e. decrease the vertical and horizontal lengths and add more stairs. The ant's new journey is, of course, still a + b. Ultimately the stairs can be shrunk to an infinitesimal size, and nothing has changed. The symbol c now truly represents a straight line instead of a wiggled line, and c = a + b. But it's a right triangle, and so that can't be correct. Where is the error? This is neither a riddle nor a trick question. I genuinely don't know where the error is. Thanks, David Ross Date: 08/07/99 at 14:13:14 From: Doctor Tom Subject: Re: Triangle geometry Hi David, You have to be careful when you wish to calculate the length of a "curve" by successive approximation. When I say "curve," I mean any sort of path which may be a straight line as it is in this case. Just because a series of "curves" (in this case the series of smaller and smaller steps) approaches the curve in question does not mean that the lengths of the approximations get close to the length of the real curve. Here is a method that will always work. Pick a series of points on the curve to be measured, draw straight lines between them in order, and add up the lengths of the straight segments. As the number of points gets large ("goes to infinity"), as long as the largest distance between pairs of points goes to zero, the sum of the lengths will approximate the length of the limiting curve. With your staircase approximations, you always have about half of the points not lying on the hypotenuse. In general, the calculation of arc length and curved surface area is tricky, and you have to be quite careful to use a "reasonable" approximation. It stumped mathematicians for quite a while, so don't feel bad. - Doctor Tom, The Math Forum http://mathforum.org/dr.math/ |
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