2012 Holiday Special III: A Very Special Message from David Hilbert

2012 Holiday Special III: A Very Special Message from David Hilbert

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2012 Holiday III Dolby

Thanks, David! As a math teacher, I know that lots of people out there have been pretty substantially burned by their mathsperiences of the past. I like to think of mathematics as the game with the Longest Tutorial Ever - something like 13 years before you get to REALLY play it, at which point most have given up. That's a shame, because recreational math is quite something, and the typical high school education gives you enough to start getting into some neat stuff. And you can do it anywhere! Instead of listening to Gangam Style for the 37th time, why not give your mind the exquisite delicacy of chewing on one of the following bits of fun?

ALGEBRA: Use Completing the Square to prove that the Quadratic Formula does indeed find the roots of a quadratic equation.

GEOMETRY: Prove that, for a given perimeter, the square is the shape that contains the greatest area!

TRIGONOMETRY: Prove our old friend, the Pythagorean Identity (the square of an angle's sine plus the square of its cosine equals 1!), using just SOHCAHTOA!

CALCULUS: Why does the power rule work for whole number values of the power? Hint: Think about Pascal's Triangle and how it relates to binomial expansions!

CLASSICAL ANALYSIS: Prove that, in a complete metric space, all Cauchy Sequences are Convergent! (Remember, in a general metric space, all convergent sequences are Cauchy, but the reverse is only true if you add completeness!)

ABSTRACT ALGEBRA: Prove that the kernel of a homomorphism between the groups G and H is a subgroup of G!

Happy New Year!

- Count Dolby von Luckner

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