Associated Topics || Dr. Math Home || Search Dr. Math

### 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.

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

- Doctor Tom, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Conic Sections/Circles
High School Geometry
High School Triangles and Other Polygons

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search