Importance of Reasonable Approximation

Date: 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/