Taking Notes

In this post I want to share some ways of working with course notes that I am currently using. As you all know taking notes is one of the most basic parts of studying. While it is possible to do well without it, it usually only means that you are borrowing somebody else notes (or downloading them). However, taking notes and using them are two totally different things. Firstly, handwritten notes tend to differ greatly in quality due to people handwriting and the lecturer. In my case my notes are close to being unreadable for anyone except for me (for some reason I can read what I wrote easily enough, but I have problems reading other people notes). Also, if the lecturer speaks in a disorganized way the notes become difficult to read and understand.
Obviously handwriting and organization of notes is not much of a problem - it is after all perfectly possible to take notes on a computer. Actually, if I were studying a subject that don't have formulas I would use a computer to take notes myself. Since I study math, I do not believe that I should try to take notes on a computer, although I know people who do just that.

However, the really difficult part comes when you need to go over your notes. In the first and second year the lecturers tend to give you all the material in a very detailed way, but with time they stop doing this. Instead you are now supposed to figure all the extra stuff yourself. As a result, you basically need to add to your notes on your own. So how do you do this, while still keeping the notes organized and in a format that allows you go over them easily?

At this point of time I cannot honestly say that I found a real solution to this question. But I managed to come to the conclusion that I need two things. The first thing is to make sure that I have the notes made in an organized way. To do this I make a second (also handwritten) copy of my course notes. In this copy I write all the definitions and theorems (with their proofs) with as much details as I need to understand them. This copy is later used when I need to prepare for the exams.
The second thing is basically a reference list. The idea is to make a list of all the definitions and theorems, as well as links to any useful source of extra information on the topic. It is obvious that such a list should be done on a computer. The end result is basically another version of your notes, but instead of being detailed and organized it is easily to search. This makes it easy to check any general fact you are unsure of. It is especially useful if you want to check some specific definition of the wording of a theorem. Obviously you can do the same thing just by searching on Wikipedia, but using such a reference list makes it much easier - you are able to see what you look for by just taking a glance on it, instead of searching a whole site. Also, making such a list on the computer allows (depending on what software you use) to add extra notes and to modify them easily. Since I do not want to type math formulas myself I ended using Google notebook to clip content (mainly from Wikipedia) and then edit and categorize it the way I want. Unfortunately, Google notebook is the only service I managed to find that had all the features that I wanted.

If you follow all this you will end with three different sets of notes (plus, if available, a textbook). Obviously this is a lot of work, but I feel that this approach allows me to understand the material as well as I can.

Complex numbers and roots

Complex numbers appeared initially as a way to solve equations that don't have real solutions. However, what are we getting from this? We can write that i is the solution to x^2=-1, but what exactly is this result? It is an imaginary number, so it is not something that can represent for example area. The obvious result from this line of thinking is - are complex numbers really needed? What problems do they solve?

Somewhat surprisingly complex numbers solve a lot of different problems or at least make them easier. In this post I want to introduce one particular problem that is solved by using complex numbers. The problem is to define a^b for all values of b and for all values of a except zero. We can partially solve this problem without complex numbers. For example, we can agree that the functions log and exp are defined as usual and then:

a^b=exp(b*Log(a))

This will work for all values of b but only as long as a>0. This is as far as we can get without using complex numbers. It is important to note, that any solution of this problem must agree with the partial solution that we have here. In other words, in order to solve this problem we must basically extend the functions exp and log in such a way that log will be defined for negative values and exp will be defined for all values of log.

As a start, lets try to define exp(z) and log(z):

exp(z)=exp(x+iy)=exp(x)exp(iy)
exp(iy)=cos(y)+isin(y)

We now need to check what we got from this definition. Firstly, if y=0 then exp(z)=exp(x) as we wanted. Secondly:

exp(z+w)=exp(x+iy+a+ib)=exp(x+a)exp(iy+ib)
exp(x+a)exp(iy+ib)=exp(x)exp(a)[cos(y+b)+isin(y+b)]
cos(y+b)+isin(y+b)=cos(y)cos(b)-sin(y)sin(b)+isin(y)cos(b)+icos(y)sin(b)
cos(y+b)+isin(y+b)=[cos(y)+isin(y)][cos(b)+isin(b)]
exp(z+w)=exp(x)exp(a)exp(iy)exp(ib)
exp(z+w)=exp(x)exp(iy)exp(a)exp(ib)
exp(z+w)=exp(x+iy)exp(a+ib)=exp(z)exp(w)

The last last thing we want to check is the derivative. This is done in a similar way, and is left as an exercise.
As you can see it is rather easy to extend exp to the complex numbers. Extending log is more difficult. Lets start with what we want the function to do. We want that for any complex number z:

exp(log(z))=z

So, we want to say that if exp(w)=z then log(z)=w.

Unfortunately, we cannot use this as a definition for log. The problem is that there are many different complex numbers that satisfy this equation. According to the definition of exp:

exp(w+2(pi)i)=exp(w)exp(2(pi)i)=exp(w)[cos(2pi)+isin(2pi)]=exp(w)=z

This is obviously a problem. To solve this problem we must remove part of the complex numbers. There are many different ways to do this removal. In this post I decided to remove the numbers {z|z=ik, k=0 or k>0}.
Now that we do not have these numbers we can define log. We know that any complex number z can be written as z=r*exp(it), where t represents the angle between the line that connects zero and z. Obviously exp(it)=exp(it+2(pi)i). However, since we have some numbers removed we cannot go in a circle around zero. Therefore, for numbers that have a negative real or complex part we will take t to be 2pi minus the angle. This solves our little problem because we now have exactly one value of t assigned to every z. And now we can define log exactly as we tried to do before. Since the negative numbers were not removed we have log defined for negative numbers as well. For example: log(-1)=i*pi.

Now that we have this we can indeed define that for all real a,b such that a is not zero:

a^b=exp(b*Log(a))

This is true for complex numbers as well, but this a different topic.