Division of Number Line

Date: 5/14/96 at 17:31:34
From: Anonymous
Subject: Division of number line

Using the number line, more specifically the interval from zero to 
one, how do I explain that when you find half the distance and then 
half of that distance and so on, you'll never settle on one number?

Date: 5/17/96 at 19:13:17
From: Doctor Ken
Subject: Re: Division of number line

Hello -

Here's one way you can do it.  You start out with distance 1.  At the 
next step, you have distance 1/2.  Then 1/4, 1/8, and so on.  Notice 
that at each step, if the distance from one step was x, then the 
distance at the next step is x/2.  Now, if x is positive, then x/2 is 
also positive and - here's the main point - x/2 isn't equal to x (you 
can show this with inequalities - start with x > 0).  

So if you know that at each successive step you get a different number 
than you had at the last step, that means you won't settle on one 
number.

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