Thursday, January 24, 2008

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 e2θi in two ways. First, it is ei*(2θ), which is cos(2θ)+i sin(2θ) (Euler's formula), and secondly it is (e)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.

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