Line or Ray Longer?

Date: 12/11/2001 at 21:01:57
From: Leslie Williamson
Subject: Line vs ray

Which is longer, a ray or a line? A ray has a definite starting point 
and goes for infinity, so therefore I think a line is longer.  A line 
goes both ways for infinity, so in my opinion, it should theoretically 
be twice as long as a ray could ever be.

Date: 12/12/2001 at 13:47:38
From: Doctor Ian
Subject: Re: Line vs ray

Hi Leslie, 

That's one way to look at it, but let's look at it from a different 
angle. 

Suppose you think of a line and a ray as collections of points.  One 
would be 'longer' than the other if it has more points, right?  So 
another way to ask 'which is longer' is to ask 'which has more 
points'.

Suppose we start listing all the points on the ray, and label them a, 
b, c, d, etc. 

               a  b  c  d
             o------------->

Now, suppose we map those points onto alternating positive and 
negative points on the line.

               a  b  c  d
             o------------->
   <---------o------------->

       d  b   a  c

That is, 

    a(ray) ->  a(line)
    b(ray) -> -a(line)
    c(ray) ->  b(line)
    d(ray) -> -b(line)

and so on.  

To determine whether two sets have the same size, we can pair up the 
elements.  If we run out of elements in one set before we run out of 
elements in the other, then the set that runs out is smaller.  Does 
that make sense? 

   {bob    ted    fred    barney}      <---
     |      |      |        |              |-- same size
   {carol  alice  wilma   betty}       <---


   {john   paul    george   ringo}     <--- smaller
     |      |        |        |
   {greg   marsha  peter    jan   bobby  cindy}

Okay, so the question is:  If we pair up the elements on the line and 
the ray in this way, which set will run out of elements first? 

You can use the same kind of reasoning to show that the set of 
positive integers is the same size as the set of all integers... which 
is the same size as the set of rational numbers.  Or that the set of 
points in the interval from 0 to 1 is the same size as the set of 
points in the interval from a to b, where a and b are any finite 
numbers.

Weird, isn't it? 

What's trickier is to show that the set of real numbers is larger than 
the set of integers. 

You can find a nice introduction to these ideas at

 Infinity - Michigan Math and Science Scholars
 http://www.math.lsa.umich.edu/~mathsch/courses/Infinity/index.shtml   

I hope this helps.  Write back if you'd like to talk about this some 
more, or if you have any other questions. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/   

Date: 12/12/2001 at 14:09:43
From: Doctor Peterson
Subject: Re: Line vs ray

Hi, Leslie.

I'd like to add something to what Dr. Ian said.

You may object, as I do, that the length of a line is not the same 
thing as the number of points in it; for example, a line segment has 
finite length but infinitely many points. So instead of talking about 
points on the line and the ray, just cut each of them up into one-
centimeter segments. Then we can match up SEGMENTS just the way he 
said to match up points:

                 a   b   c   d
               o---+---+---+--->
   <---+---+---o---+---+---+--->
     f   d   b   a   c   e   g

You can match every piece of the line to a piece of the ray, so they 
are the same length!

The point is, infinity doesn't behave the way numbers do; 2 times 
infinity is equal to infinity. So even though the line IS twice as 
long, the ray is the same length. In fact, one of the characteristics 
of infinite sets is just this: a subset (the ray) has the same size as 
the whole set (the line).

Here is a similar discussion in our archives:

   Infinite Sets
   http://mathforum.org/dr.math/problems/gibbs9.24.97.html   

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/