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### Where Does Pi Come From?

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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.
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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/
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Associated Topics:
Elementary Math History/Biography
Middle School History/Biography