Wednesday, February 12, 2014

How Thin Can You Slice Your Pi?

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That is the value of π from the calculator on my laptop. I cannot give the value off my graphic calculator since I do not own one.

Students should use the π symbol and its corresponding value when applicable on the Regents Exams in Algebra I, Geometry, and Algebra II (Common Core). The approximate values of π, such as 3.1416, 3.14 or 227 , are unacceptable unless otherwise specified.

What could possibly be "its corresponding value"?  Mathematicians throughout the world know that π is a transcendental number (worse than irrational, to the uninitiated). If the state's guidance wants to use an exact value in any place-value system, it is asking for the impossible. If it saying that the π button on a calculator must be used, it is just accepting the calculator's approximation as "better" than any others.

Before anyone passes judgment on my comments, I suggest that first read a column from the July 21, 2012 at Scientific American entitled "How Much Pi Do You Need?" (get it here).

Another good read is a page from BetterExplained.com  by entitled Learning Calculus: Overcoming Our Artificial Need for Precision (see it here) which says
...approximations are a part of Nature, yet everything works out. Why? We only need to be accurate within our scale. Uncertainty at the atomic level doesn’t matter when you’re dealing with human-sized objects.
The web site Ask a Mathematician/Ask a Physicist states (here)
Every major civilization has been aware of π and have relied on various approximate values.  Basically they’d find an approximation that was “good enough” to find the length around any circle given the length across it (the definition of π) and go with that. 
One question I pose is this: knowing that  π is by definition the circumference of a circle divided by twice its radius, how large a circle would we need to get 8-digit accuracy in our approximation  if our measurements of length were no better than the nearest millimeter?
On a simpler note: take any circle you wish, and measure its circumference and radius (I say measure, not calculate), then divide the circumference by twice the radius. Take the result of your calculation, and see how closely it agrees to the decimal at the top of this post.
I think you will be amazed at how different it is.
Do not force students to try to pin down a value that cannot be pinned down, rely solely on a calculator's value (not all calculators have the same approximation), or depend on "otherwise specified".  Instead, use this famous number as a tool in teaching the need for approximations, helping students learn to deal with accuracy and its colleague precision, and developing skill in answering questions such as "how accurate do I need to be?".

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