# M9.3.1

**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.

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

For the mathematically inclined person, irrational numbers such as a^{2} are fascinating, both from a historical perspective, and as a classic example of using the reductio ad absurdum proof, to prove the irrationality of a^{2}. However, the majority of students, when introduced to the irrational number a^{2}, might be excused for not being fascinated by it, and might believe that a^{2} has not much practical use.

### 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.