Interesting Math Column - VI

Yet another weekly column in the NY Times series by Cornell mathematician Steven Strogatz. This week it's Finding Your Roots and about complex numbers. This one gets a bit more challenging with its mention of Newton's method and fractals.

The images shown in this article are very similar to the better known Mandelbrot Set which is also plotted in the complex plane by continued iterations (long ago, in college, I wrote a Turbo Pascal program to generate the Mandelbrot set).  I love zooming into fractals and exploring these images of infinite complexity generated by such simple equations.  Fractals describe many diverse forms of geometric objects including branched trees, coastlines, cloud shapes, and fault surfaces to name only a few objects that are self-similar at different scales.

One interesting thing not mentioned in the article is Euler's Formula.  Leonhard Euler (1707-1783), pronounced "Oil-er", was one of the most prolific mathematicians who ever lived.  He discovered an amazing formula:

e^ix = cos(x) * i sin(x)
 
The constant e is the base of the natural logarithms and is equal to 2.71828183 (like p, it goes on forever).  The formula cos(x) + i sin(x) denotes a set of complex numbers (the number depends, of course, on what x is equal to).  What's so amazing about this formula?  Well, if you set x = p, it becomes:

e^ip = cos(p) + i sin(p)

e^ip = (-1) + i (0)

e^ip = -1

It's a formula that now relates two of the most fundamental constants in modern mathematics, e and p (both infinite non-repeating decimals), with i, the square root of -1.  The brilliant Nobel-prize-winning physicist Richard Feynman called this "one of the most remarkable, almost astounding, formulas in all of mathematics."

By the way, this formula also pops up in the book discussed in the previous post, The Housekeeper and the Professor.