Friday Science: Geometry of Logarithms

This is my post on the fifth chapter. Well, Roger Penrose is a smart dude. I followed some of the material in this chapter and knew of some of it, but a lot of it is a bit beyond me. Still more fundamentally, I'm not exactly sure why we are talking about the material in this chapter. It's relevant to quantum physics I know, but I can't quite catch the context.

1. So I know the chapter dances with graphs of the addition and multiplication of complex numbers. Penrose makes a connection between the way logarithms behave and the way the graphs of complex numbers work. Complex number graphs 1) have an x axis that is used for the "real" part of a complex number (see my last post for a + bi as the standard form of a complex number, with a as the real part and bi as the "imaginary" part). Then 2) instead of a y axis they have an "i" axis.

2. Then he shifts to treat these graphs in terms of polar coordinates. Polar coordinates treat a point in terms of an angle that is at zero and then moves in a circle in a counterclockwise direction. Then there is a distance "r" from the origin (0,0) to that point. So the point is described in terms of an (r, ϑ) instead of an (x, y). I believe he is doing this because these ways of thinking about imaginary numbers help us get some little grasp of why the equations work the way they do.

3. Logarithms are a way of conceptualizing how exponential functions work, one that was developed in the 1600s. For some reason, I've always had trouble conceptualizing them. I get exponents. So the following relationship makes sense to me: bⁿ=x. Thinking about this relationship logarithmically rearranges the relationship to say that the log_bx=n.

4. Now I understand all these ingredients. I just don't quite get what these things all cook. The next ingredient is the number e (2.718281828...). It is a curious number that is somehow basic to the universe. It can be derived from the following formula:

The logarithm base e is called the natural logarithm (often abbreviated "ln"). Apparently, the logarithms of imaginary numbers can be expressed and graphed in terms of polar coordinates: z=log_er+iϑ.

5. I'm afraid I'm still struggling to have some sense of perspective on a number of other relations that are in this home stretch. e^2πi=1. And imaginary numbers can be put into polar format using the form w=re^iϑ.

6. Another relation he mentions is iⁱ. I can't entirely follow why but this is equal to e^ilogi =0.207879576...

7. The final section implies that there is a direct application of this math to the quantum world. Although I'm still not clear how. :-)