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Understands that mathematics has been helpful in practical ways for many centuries.

Why Is There a Tidal Bulge Opposite the Moon?

Demonstrate and calculate the reasons for tidal bulges of water on both hemispheres of Earth, due to the Moon.

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Square Root of 2: Irrational, Yes! Impractical, No!

For the mathematically inclined person, irrational numbers such as a2 are fascinating, both from a historical perspective, and as a classic example of using the reductio ad absurdum proof, to prove the irrationality of a2. However, the majority of students, when introduced to the irrational number a2, might be excused for not being fascinated by it, and might believe that a2 has not much practical use.

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Logarithms: Taking the Curve Out

Logarithms are very handy when dealing with numbers at lots of different scales (see related PUMAS example: Just What is a Logarithm, Anyway?). But they also have another useful feature: they help us average measurements of physical phenomena that have nonlinear behavior. A common example in my field of study relates cloud "albedo" to cloud optical depth (Fig. 1); but similar examples may be found when examining many natural phenomena.

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