Finite vs. Infinite

Date: 07/10/97 at 11:39:06
From: Aab
Subject: Geometry

If a line segment is a measurable part of a line, why is the number 
of points that make up a line segment infinite?

Thanks for your help!
Angie

Date: 07/15/97 at 13:44:04
From: Doctor Chita
Subject: Re: Geometry

Dear Angie,

What a provocative question. I'm not sure I can give you the 
definitive answer, but I'll try to provide one way of thinking about 
this paradox.

One way to begin is to quickly review some ideas from geometry. Euclid 
defined a "point" as "that which has no part." This means it has no 
measurable properties, including size. Therefore, a point takes up no 
space.

A line, according to Euclid, is "length without breadth." Further, 
"The extremities of a line are points," and "A straight line is a line 
which lies evenly with the points on itself." This suggests that a 
line has the property of length and contains at least two points.

Using these two notions of a point and a line, a good model is the 
number line, a geometric way to represent points on a line. First, to 
represent numbers, we have to impose a coordinate system on the line. 
Start by marking off a point to represent 0, the origin. Then choose 
a convenient scale that divides the line into equal intervals on 
either side of 0; the positive numbers to the right (by convention) 
and the negative numbers to the left. Every point on this line can be 
matched to a number, and every number can be matched to a point on the 
line. Since there is an infinite number of numbers, there must be an 
infinite number of points on the line. But what about a segment - a 
finite portion of a line (and your question)?

The number line houses all real numbers - positive and negative 
rational numbers (integers, fractions, decimals) and irrational 
numbers (pi, e, the square root of 2, etc.). Consider the rational 
numbers for now. Let two rational numbers be the endpoints of a 
segment, say 0 and 5. The set of numbers between the endpoints can be 
matched to points that lie on the segment. 

For any two rational numbers, you can always find a number between 
them. For example, if a segment has endpoints at 0 and 5, then 
it has a length of 5 units. The point halfway between 0 and 5 is 
(0+5)/2 = 2.5. Therefore, you could graph this point on the segment.

Find a point one half the distance from 0 to 2.5. That corresponds to 
5/4 = 1.25 and can be matched to a point on the segment between 0 and 
5. Now find the number that corresponds to the point that is 1/2 the 
distance between 0 and 1.25. That gives you another number and point. 

Repeat this process: Bisecting these segments locates points closer 
and closer to zero. However, they are never at 0 and, therefore, they 
occupy a "measureless" position on the segment. 

As you continue this process to infinity (that is, bisecting the set 
of tiny segments), you'll find you'll never run out of numbers and 
points. Since points don't take up any "real" space, they all fit 
infinitely comfortably on this finite length of the line! 

By the way, we haven't even inserted numbers/points that represent the 
irrational numbers! Between two given points, there is an infinite 
number of them, too. The square root of 2 is approximately 
1.414213562. Therefore, it lies on the segment between 0 and 5. And so 
do numbers like sqrt(2)/2, square(2)/4, sqrt(2)/8, etc., where 
"sqrt(2)" means "find the square root of 2."

It's one of those mind-boggling ideas that makes mathematics so 
fascinating. Don't you agree? 

-Doctor Chita,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/