Why Pi?

Date: 07/19/2002 at 04:51:41
From: Maxx
Subject: Pi ratio

Hi, I'm Maxx and I'll be taking Analysis 2 for math when I start 
10th grade next year. I know that if a diameter of a circle is an 
integer then then the circumference cannot be an integer and vice 
versa, and that is why although pi is a ratio, it is an irrational. 
But then, why don't we just use the ratio (for example 2.4/4 - not 
really pi, just an example) when we calculate the area of a circle 
to get more accurate answers? Is it impossible to accurately measure 
circumferences?

Date: 07/19/2002 at 13:35:35
From: Doctor Ian
Subject: Re: Pi ratio

Hi Maxx, 

A subtle point, which a lot of people fail to appreciate, is that 
mathematical shapes are _idealizations_ of real shapes. Or to put that 
another way, real shapes are only _approximations_ to mathematical 
shapes. You can look at it either way (although philosophers would 
probably make a big deal out of which way you choose to phrase it).  

For example, you can't construct a perfect circle in the real world, 
if for no other reason than that a perfect circle contains an infinite 
number of points, and a real object contains a finite number of 
pieces.

The formula

       circumference    area
  pi = ------------- = --------
         diameter      radius^2

holds exactly _only_ for mathematical circles, not for real objects.  
If you try to apply it to real object, you run into several problems, 
which you're probably already aware of.  One is that you can't make 
exact measurements! Another is that if the object isn't a perfect 
circle, the formula doesn't exactly describe the shape of the object. 

A third problem is that in many cases, you can't really measure the 
quantities you want to measure, because they don't really exist. For 
example, in a mathematical circle, every point on the circle is the 
same distance from the radius. That is, there is just the one radius, 
and it's the same everywhere. 

But now suppose you have an object that is approximately circular, and 
you want to find the 'radius'.  Where should you measure it? Depending 
on how precisely you measure things, you'll get a slightly different 
value everywhere you look! So what _is_ the 'radius' of the object?  

(It might seem at first that 'circumference' would be immune to this 
problem, but it's not. Think about measuring the circumference of an 
island. If you use a yardstick, and measure points a yard apart, you 
get one value. If you use a foot-long ruler, you get another value.  
If you use an inch-long ruler, you get a third value. And so on. By 
determining the level of detail that you're willing to ignore, the 
length of your ruler determines the circumference of the object!) 

It's to get past problems like these that we agree to treat objects as 
if they corresponded to exact shapes like circles and squares. Then we 
can forget about the objects and just deal with the shapes to do our 
calculations. We give up a little accuracy this way, but we more than 
make up for it in convenience.

Does this answer your question?  Write back if you'd like to talk 
more about this, or anything else. 

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