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Vito Prosciutto: Teaching community college math on the road to a PhD.

Tuesday, October 07, 2003

21:02
A case of misusing calculators in the classroom 

In a discussion about the golden mean in my math teaching class, it was brought up that the golden mean had the wonderful property that
phi=1+1/phi=1+1/(1+1/phi)
which the student instructor followed up by saying, "you can check this on your calculators".

And to which, I say, wrong. This isn't a good use of calculators. Instead, we should use substitution to get from phi=1+1/phi to phi=1+1/(1+1/phi) (and of course extending into infinity).

Especially, since we then went into the series of continued fractions:

1+1/1=2
1+1/(1+1/1)=3/2
1+(1+1/(1+1/1))=5/3
1+(1+(1+1/(1+1/1)))=8/5
etc.
(can anyone recommend a good free, anonymous site for in-line image hosting so that I can replace nasty in-text equations with better-quality gif files?)

Anyway, this sequence will eventually converge on phi, and the limit proof is easier to get to if you use substitution (as is the proof that each successive continued fraction simplifies to a ratio of adjacent Fibonacci numbers).

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