Teaching Pi

Date: 03/04/97 at 12:52:22
From: Leneda Laing
Subject: Pi's exact value?

I am teaching my 7th grade students about the value and history of pi.  
I was always taught that 22/7 is an approximate value for pi.  But 
this is not pi's value (3.141592...).  22/7 is 3.1428571...  

How can I teach my students WHERE the 3.141592... comes from and then 
let them experiment with it?

They are also interested in the Guinness Book of World Records and the 
places that pi has been divided out to and the number of places that 
pi has been  memorized to - I just want to tell then how to find this.

Thanks for your information!

Leneda Laing 

Date: 03/04/97 at 15:39:09
From: Doctor Daniel
Subject: Re: Pi's exact value?

Hi Leneda,

One important fact you need to remember is that pi is an irrational 
number. That means that there exist no two integers a and b such that 
a/b = pi. That's why it's so hard in general to compute pi to an 
arbitrary number of places.  

For example, not only is 22/7 = 3.142857142857142857..., but this is a 
very predictable sequence; the 6th, 12th, 18th, etc decimal places are 
all 7, and so on.  Pi doesn't have this property, and in fact, it's 
not even a predictable sequence like .10110011100011110000..., which, 
though not rational, is very easily characterized.  

Evaluation of pi has an extremely interesting history; the Egyptians 
and Greeks had decent approximations to it (I believe 22/7 and 3.2), 
which were actually better than the one used by medieval Europeans, 3.  
Back in the 19th century, the Indiana legislature even tried to set 
the value of pi (I believe to 3.2).  

What is pi?  There are lots of answers to that question; it's the 
ratio of the circumference of a circle to its diameter, for starters.  
But it also has a much more formal definition that has nothing to do 
with circles, let alone geometry, in that it's the value we get by 
adding an infinite number of terms in an infinite series, which is 
because arctan (1) = pi/4.  (That is, the inverse tangent of 1 is a 45 
degree angle, which is pi/4 radians.)  

Another fun way of looking at the number pi is that if you throw n 
needles, each of which is length 1, at a floor with horizontal stripes 
every 1 unit, the ratio of needles which cross a stripe to the total 
number of needles will approach 2/pi as you have more and more 
needles. This can be a fun classroom experiment.

Usually, pi is approximated by some infinite series method, and it's 
almost always the case that it's done by supercomputers; it's sort of 
a way of testing to make sure that a given supercomputer is much more 
powerful than the previous incarnation, to show that it can compute 
very many more digits of pi than the previous one.  This is very 
similar to whenever messages come out that a new largest-known prime 
has been found; basically, this means that there's a new generation of 
supercomputers that is more spiffy than the previous.
  
These methods may be outside the interest range of your students' 
interest; then again they might not.  It's kind of like looking at 
1+1/2+1/4+1/8+1/16+ ... and showing that it gets closer and closer 
to 2, except that the series that converge to pi are much much slower 
at converging and hence more boring.  The needles experiment always 
strikes me as more fun.  

Good luck,

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