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/ |
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