Math, Numbers,and Infinity

I love numbers and math, a lot of people don't and they should be warned that this post is almost exclusively about math.

I've been taking a Game theory course online through Cousera and it has been very interesting. Game theory is not actually about making board games or video games, it is a discipline that models strategic interactions of almost any kind. I'm not done with this course yet and don't want to start making posts about it until I have a better understanding of the subject. However the course has reminded me of a lot of the classes I took in college and a couple of the most interesting theories that I found there.



One of my favorite subjects was Statistics. The normal distribution is one of the most amazing concepts in my mind. Normal distribution has several interesting properties. The one that I remember most is the central limit theorem, which basically states the sum of a random process will approximate a normal distribution bell curve. So what does that mean? It means any time you are taking the sum of multiple random terms and plotting them on a graph the graph will be similar to a normal distribution. For example rolling multiple dice and taking the sum of the exposed sides repeatedly will produce a normal distribution. Which means your dice don't hate you, your just on the low end of the bell curve sometimes. Another nice thing about normal distribution is the 3-sigma rule, which tells us how much of a population exists with in 1,2, or 3 deviations of the mean. There are some very nice applications of the 3-sigma rule but they are more involved and require a lot of explanation.

The one concept that blows my mind every time I think about it is the idea of uncountable infinity. Before I get into why it blows my mind I'm going to go into a little bit of background information. An integer is any whole number, no decimals, no fractions, just whole numbers. A Real number is any non-imaginary number. When we talk about integers we call them an infinite set, because there is no "biggest" number there is always a number bigger than any number a person can name. Here's where it gets weird. There are an infinite number of Reals between 0-1, in fact there are an infinite number of Reals between all integers. Conceptually you could say that the Reals go to infinity much faster than the integers. The part that blows my mind is that if you took the entire infinite set of integers and assigned one of them to each Real number you wouldn't have enough numbers to go around. Which is why it's called uncountable infinity.

 If you are still interested in reading more about any of these ideas I would suggest using Google Scholar so that you are getting academic answers rather than yahoo questions or other less reliable sources. Thanks for reading everyone.
Tseb