Locating Pi on a Number Line

Date: 08/23/2000 at 13:26:17
From: Jennifer Reed
Subject: Locating pi on a number line

I need to find out how to locate pi on a number line. I have to show 
it to the class. Can you tell me the best way to do this? What number 
for pi should I use? I mean, how many places should I go to? Should I 
just use 3.14, or should I go further?

Date: 08/23/2000 at 15:56:31
From: Doctor Ian
Subject: Re: Locating Pi on a number line

Hi Jennifer,

A common approximation for pi is 3.14, so you could just draw a number 
line and label pi like this:

     <---|---------|---------|---->
         2         3         4
                     ^
                     |
                     pi

But that shows where pi _is_, more or less. If you want to show how to 
_find_ it, you might do this: Take a circle whose diameter is one, and 
'set' it on a number line so that it rests on zero.

               .
           .       .

         .           .

           .       .
               .
     <---------|-------------------------->
               0

Cut the circle at this point, and roll it out flat, so that one end 
remains at zero.

               .  .  .  .  .  .  .  .  .
     <---------|------------------------------->
               0                       ^
                                       |
                                       ?

How far will the other end reach? 

On the other hand, it's possible to compute pi by summing various 
infinite series. For example, it's true that

     pi/4 = 1 - 1/3 + 1/5 - 1/7 + ...

which means that 

       pi = 4(1 - 1/3 + 1/5 - 1/7 + ...)

          = 4 - 4/3 + 4/5 - 4/7 + ...

So you could draw a number line, start at 0, then move 4 units to the 
right, then 4/3 units to the left, then 4/5 units to the right, then 
4/7 units to the left, then 4/9 units to the right, ... and so on, 
until you get tired or the class period ends.  

                        -4/7<----
                       --------->+4/5
                   -4/3<-----------------
               ------------------------->4
     <---------|------------------------------->
               0 

With each move, you jump past pi, so pi will always be between the 
last two moves that you made. The more jumps you make, the closer 
you'll get to the actual position.  

                      -4/7<----
                     --------->+4/5
                 -4/3<-----------------
             ------------------------->4
    ---------|-------------------------------
             0
             [                        ]   pi is somewhere in here
                     [                ]         somewhere in here
                     [        ]                 somewhere in here
                          [   ]                 somewhere in here

I hope this helps. Be sure to write back if you have more questions, 
about this or anything else. 

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