Sunday, December 11, 2011

Novel way of explaining binary counting to absolute beginners

Binary is explained very well on thousands of excellent web pages out there, so this is just a snippet of an idea that I think is novel.  I just thought of a cool way to teach someone to count in binary, by just swapping the digit "1" for "9" and back, as follows:

A) Replace all the "1"s with "9"s;
B) Add 1 as you would in familiar decimal counting
C) Then change all "9"s back to "1"s.

For example, starting with say 1011
After step A) we have 9099
After step B) we have 9100
After step C) we have 1100

The goal is to very quickly just skip the step of actually writing any "9"s at all, and just know to treat the "1" as the "9", i.e. in binary, the "1" is the last digit.  Works for any base.

Thursday, December 01, 2011

Perhaps nature does not use the square root of negative one


I've recently finished reading "Introduction to Smooth Manifolds" by John M. Lee, which I highly recommend if you want a solid and readable introduction to some aspects of differential geometry, Lie groups and Lie algebras, and a whole lot more. (I must be honest and admit that I think I would probably learn just as much as I did the first time if I read it a second time.)

Thoughts on quantum physics: The use of complex algebra in quantum physics is puzzling because sqrt(-1) does not make sense in our "classical" "real world".  However, it may be as simple as accepting that complex algebra is just isomorphic to whatever the algebra is of the quantities that quantum physics is modelling. In other words, nature doesn't take the square root of negative one, but whatever is going on works in the same way as if it did.