Date: Tue, 11 Feb 1997 18:14:05 -0500

From: Alex Bogomolny

Dear Colin:

I may risk being perceived as a pedant but what is an infinite rectangle? Rectangle is a quadrilateral (meaning having four sides) whose four internal angles equal 90°. How many sides had the rectangle you've been discussing with your friend? And what angles did it have at infinity?

Now, assume, on the other hand, I'd like to be a little more forthcoming, and attempt to impute some meaning into your question. Then I wouldn't consider a figment of imagination like an infinite rectangle. Instead I'd look into a family of rectangles, each very finite and well defined. Members of such a family might obey some kind of law with regard to their shape and dimensions.

For example, let a family of rectangles {A_{n}}, n=1,2,... be such that
rectangle A_{n} has width n^{2} and height 1/n^{2}. Then every rectangle in this
family has area 1. As n grows, rectangles in the family become
progressively more narrow and longer so that "at the limit" you
may imagine a monstrosity you referred to as an infinite rectangle.
Since the area of the rectangles is invariably 1, we might assign
your object the same area, viz., 1.

However, we as well may consider two other families {B_{n}} and {C_{n}},
n=1,2,... Both B_{n} and C_{n} have width of n^{2}. However, the height of
B_{n} is 1/n while that of C_{n} is 1/n^{3}. Now, the area of B_{n} is n and
grows without limit as n grows. On the other hand, the area of
C_{n} is 1/n and tends to 0 as n grows.

In all three case, you would likely dare talking about infinite rectangles and their areas. All three cases satisfy your Description of infinitely tall yet infinitely narrow shape (assuming, of course, you'll manage to keep my rectangles upright - being wiser, I prefer having them flat on the ground). However, it must be admitted that the notion of area is not well defined for such objects. All depends on how fast they become narrow relative to how fast they become tall. You'll be surprised how many different possibilities are out there.

You asked a good question. Limits is the most basic and fundamental tool of Calculus. In time, I hope, you'll take up this important subject more seriously.

Sincerely,

Alexander Bogomolny