Subject: Re: Area of an infinite rectangle
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 {An}, n=1,2,... be such that rectangle An has width n2 and height 1/n2. 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 {Bn} and {Cn}, n=1,2,... Both Bn and Cn have width of n2. However, the height of Bn is 1/n while that of Cn is 1/n3. Now, the area of Bn is n and grows without limit as n grows. On the other hand, the area of Cn 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