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### Finite vs. Infinite

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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?

Angie
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```
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

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/
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Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry
High School Number Theory

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