The Road to Reality II

The next two chapters of The Road to Reality (i.e., chapters 3 and 4) covered number systems: integers, real, and complex (aka imaginary) numbers. I'm comfortable with all of these though I haven't dealt with complex numbers since college. If these were new to me I'm not sure how well I would have followed his discriptions, which makes me nervous.

Penrose asks something about each of these number systems that I don't remember from my math classes. That is, do they have some real meaning in the real world? While -3 has some obvious meaning when balancing your checkbook, what does it mean to have -3 cows in a field? It turns out that electron charges are a physical thing that require integers to represent them.

For real numbers he lists a few things that we use real numbers for. While there's a limit to distances we can measure, when you consider volumes you cube them and the numbers get large. On the small scale, irrationals are useful, but when dealing with quanta there might be a limit to how small we have to measure. He promises that real numbers will be necessary for calculus when we get to it.

For complex numbers I remember manipulating them but never knowing why they were important (aside from being how to deal with the square root of -1). First off it seems arithmetic and arbitrary roots in complex numbers always yields solutions in complex numbers. This wasn't the case for say the rational numbers where the square root of 2 gets you into trouble quickly. Also all polynomial equations have one (or more) complex solutions.

I also don't remember graphing complex numbers with the x axis for the real part and the y axis for the imaginary part. And I certainly didn't remember using such graphs to know if infinite series converge or diverge. And I didn't realize that Mandelbrot sets were on a complex plane (I never studied them in school, I just saw pretty pictures). Apparently complex numbers come up in quantum mechanics. Next up, logarithms.