Where Does Pi Come From?

Date: 11/04/96 at 22:49:05
From: Lynn Hughes
Subject: Where does pi come from?

My students have asked me this often:  if pi is derived from measuring 
a circle's circumference and diameter and then dividing to determine 
the relationship, doesn't the answer's accuracy, and therefore the 
value of pi, depend on making perfect measurements with no degree of 
approximation?  And isn't that impossible?  Or was the relationship 
established in some other way?  How do we know we are working with 
good numbers?  

Thanks,
Lynn Hughes, 6th grade teacher, The Miquon School, Miquon, PA.

Date: 11/05/96
From: Doctor Pete
Subject: Re: Where does pi come from?

Hello,

I would venture to say that the issue at hand is more about real-life
measurements as an approximation of the mathematical abstraction of 
length, rather than pi as a measured value.  That is, pi is not really 
"derived" (in the sense of a mathematical argument) by taking pieces 
of string to circles. Instead, the value of pi as a ratio is obtained 
by using geometric and algebraic methods, much as one calculates 
quantities like area, length, and angle.  The significance of pi lies 
in the fact that the ratio of a circle's circumference to its diameter 
is *independent* of the radius of the circle.

To put things a bit more concretely, the ancient Greek geometer 
Archimedes of Syracuse stated that pi is independent of radius, and 
that the area of a circle is equal to pi times the square of the 
radius (he actually phrased it as being equal to the area of a right 
triangle whose legs were the radius and circumference of the circle, 
respectively - a bit of math shows these formulations are equivalent).  
He did this by using clever geometric arguments, not by taking strings 
to circles, nor by cutting up paper circles to rearrange them into 
rectangles.  

The point is that the notion of length and ratio when applied to 
geometric objects is an *abstraction*.  Saying that a square of side 
length 2 has area 4 is exact - it does not matter if one cannot cut a 
square of paper whose length is exactly two inches; such squares exist 
only in the ideal universe of one's mind.  Likewise, a circle is 
perfectly round on an abstract level, and it is possible to find pi in 
this abstract sense.

I hope this is a sufficient explanation. Unfortunately, I think that 
it would take a considerable amount of work to convey the above 
distinction to your students.  In a nutshell, lengths, ratios, areas, 
volumes, etc. of abstract figures in mathematics are not measured the 
way one measures with a ruler; their dimensions are exact by the very 
fact of their abstractness.

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