Monday, July 05, 2004

A neat infinity paradox

As Brian Clegg says, "One of the great delights of infinity is its ability to throw up mind-bending paradoxes." Here's one I have not heard before.

The basic idea is to cover the line from 0 to infinity with a combination of line segments whose length adds up to 1.

The segments will be of length 1/2, 1/4, 1/8, etc. Attach these segments to the non-negative rational numbers in some order. Since there are the same number of segments as there are rational numbers, such an assignment can be done. (The number of each is the same as the number of integers.)

Each rational number r now has a line segment of length x associated with it. That line segment covers a portion of the line from r - x/2 to r + x/2. Since the end points of these line segments are also rational numbers, each of which also have line segments associated with them, the entire line is covered. In fact, many of the line segments overlap. But the sum of the lengths of the line segments adds up to only 1. How can that be?

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