Quick Book

In the past few days I read 'The Most Beautiful Mathematical Formulas', by L. Salem, F. Testard, and C. Salem. Which isn't to say it took a while to read. Probably about an hour in all. But it was a perfectly enjoyable hour. The formulas are fun and mostly simple. Each formula (~49ish) occupies a chapter, which generally corresponded to two pages: 1 of text, and 1 full-page drawing.

The one thing that stood out the most, for me, was a quick deduction of the double-angle identities for sin and cos, using complex numbers. You look at e^2θi in two ways. First, it is e^i*(2θ), which is cos(2θ)+i sin(2θ) (Euler's formula), and secondly it is (e^iθ)². Expanding this (again, Euler's formula on the inside, then square it out) and then equating real and imaginary parts for the two versions gives you the double-angle identities. Not bad.