Saturday, November 29, 2008

I wonder if I am pushing myself too hard...

Despite my best efforts I barely manage to finish all of my homework in time. Partially it is due to the fact that I was ill for a couple of days which caused me to slightly fall behind, but it seems that getting ahead is more hard that I thought. I also spent half a day yesterday fixing things in the new apartment. As it looks now I will finish all the exercises (of this week) on time, but this will require studying all day long (for the next three days). I don't mind doing so, but after a certain number of hours my head just stop working (and I get a headache). I also think that I am not getting enough sleep...

I was told once that the workload for math students raises with every year (unlike for biology students, who on fourth year can work on two part time jobs). It certainly seems true.
I am also planning on tutoring a bit, it seems line I have an hour or two once a week that can be used for such a purpose. It probably would be better to study during this time, but I am rather sure that I wouldn't manage to do this anyway - my homework requires much more energy to do that I have at that time of week, tutoring somebody will surely be simpler so I hope to manage.

As of now, my main concern is to manage to get ahead in doing homework, so that I will be able to read my notes and to look the material ahead. I really hope to mange to d0 it this week (I hoped the same the last week,so wish me luck.. ).

Now to brighten the mood a bit - a little math fun fact ( the link goes to wikipedia, and the explanation there is not very good, it is too difficult to read. I will try to find something better or to write it myself).

Thursday, November 20, 2008

Geometrical representation of numbers

This post s a response to a comment I got today. I was asked if there are numbers that cannot be represented at all using geometry.
The answer to this question depends greatly on what you consider to be a number, and what you consider to be a geometrical representation.

What is a number?
One possible approach is to define that x is a number if and only if it belongs to R (the set of all real numbers). There are different ways to define this set, but it can be proven that all those definitions give the same set, so this definition of a number doesn't have any problems.
However, there are objects that don't belong to R but are considered numbers (at least by some people). The most obvious example is the complex numbers. It can be easily shown that the "number" sqrt(-1)=i doesn't belong to R, so if we want to consider it as a number we need to extend our definition to include all the complex numbers - in other words x is a number if and only if it belongs to C (the set of all complex numbers).
It turns out however that even this definition can be extended. In addition to i we have other imaginary "numbers" - the infinitesimals (a non negative number smaller than any positive number) and infinity. Neither of them belong to R or to C.
Surprisingly, even this is not the end. There are also cardinal numbers - a cardinal number is basically a size of a set, so for finite sets this is just a natural number. For infinite sets a cardinal is a "number" (and there are infinitely many different sizes of infinite sets, so there is an infinite os such cardinals) that doesn't belong to any of the sets mentioned above (except for the cardinal of the set of the natural numbers).
It is possible to bring more examples of objects that can be called numbers but, in my opinion the best definition is the first one - x is a number if and only if it belongs to R. But you are free to chose the one you like.
Geometrical representation
It is important to notice that there are two different definition of what a geometrical representation is. The definitions are:
1. A number has a geometrical representation if there is a point on the real line that corresponds to this number.
2. A number has a geometrical representation if a line segment of a corresponding length can be constructed geometrically (using compass and straightedge alone).

Since R can be viewed as the real line, it follows immediately that all the numbers in R has a geometrical representation according to the first definition. It turn out that if you extend this definition of the numbers to include all C, there is also a geometrical representation, because every complex number can represented by a point on a plane. Infinitesimals, infinity and cardinals don't have such a representation.

The second definition is much more strict. The ancient Creeks never asked if there are numbers that cannot be represented in such a way, but it turned out (at about the 16 century I think) that there are such numbers. To better understand this, lets first see some examples. Lets look on the following numbers - 2, 9, sqrt(2) , pi.
It is obvious that we can construct the first two. All we need to do is to decide what we call a line segment of length one, and we are done. We can now draw 2 such segment to get 2 and 9 to get 9. For sqrt(2) it is a bit more complex, we need to make a right triangle with sides 1 and then we will get that the third side is sqrt(2). Generally it has been proven that a number that is a root of a polynomial with rational coefficients can be somehow constructed under the restrictions we put on ourselves. It was also proven that a number that is not a root of any such polynomial cannot be constructed in such a way. Not so long ago (in the last century) it was shown by Cantor that most numbers (numbers according to the first definition) are not constructible. Such numbers are called transcendental.
It is important to note that numbers that belong to C also can be constructed (not all of them, but some of them) this happens because C and R are sets of the same cardinality so you can assign a number in R for any number in C.

Thursday, November 13, 2008

Math wallpapers and some other images

When I checked Google webmaster tools today to see on what searches my blog showed up lately, I was surprised to see that it some people got here while searching for "math wallpapers" and "photo of Protagoras". I don't remember posting anything like this (I did post a link to a collection of Firefox wallpapers, but it wasn't related to math...), so I decided to post a collection of links to some math wallpapers I found on the net. If you are enthusiastic about math, and don't know how to show it, you can start but putting one of those wallpapers on your desktop. :)

From Deviant art:
Fractal Pi - probably the best wallpaper I ever seen with Pi in it.
Integrating - for all those who love calculus..
Integration fun - somewhat less simplistic than the above.
Pi - A classic of its kind the letter pi on the background of the first "insert large number here" digits of pi.
Infinity - I feel like there is too much objects in this image, but your opinion might be different.
Einstein - Somewhat small, but looks good.
Lambda - Rather simplistic.
Life is math - The title is great, but it is hard to figure the connection to math by looking on the image. However, if you add a caption to it in photoshop it will surely look great...
Three number lost -for those who love weird graphs...

From other sources:
Mathematica -photos of some famous historical figures, on a background with different number systems.

Well, this is all I found. I would really like to add to this short list, but unfortunately I don't have time to hunt for photos now... :(. In the case the links above don't work, you can download the wallpapers from my picasa album - Math wallpapers.
Last but not least - a photo of the Statue of Pythagoras in Pythagorion:

Tuesday, November 11, 2008

Dividing gold in a rational way

I finally got internet connection in my new apartment, I am really lucky it didn't take more time. Also, as I expected the semester in the university started on time despite the threats to not open it unless the government pays. I am doing 8 courses this semester, so I am really busy most of the week. Yesterday I was in the university from 8:00 to 20:00... Gladly today the lectures start at 16:00 so I can stay at home in the morning.

Now, about the title. One of the courses I am doing this semester is game theory. When I got the exercise for this course, one of the questions was about dividing gold between pirates (surprisingly, the other questions were rather difficult questions in analysis). What is interesting in this question is that the solution seems unbelievable, so I decided to post both the question and the solution on this blog:

The situation is as follows - five pirates got 50 coins of gold. They must find a way to divide those. The pirates all have different rank from 1 to five. They decide on how to divide the gold in a very simple manner: the pirate with the highest rank offers a way to divide the gold, and then the rest vote for or against his proposal. If the absolute majority is against he is killed and the process starts over with 4 pirates. There are three further assumptions. Firstly, the pirate who offers how to divide the gold wants to get as much gold as possible. Secondly, all the pirates are rational. Thirdly, if a pirate have no reason to vote for or against the proposal, he will vote against it.
A tip: go from the end to the beginning.

The solution is that the first pirate (who has the highest rank) will get 48 coins and the pirates 3 and 5 will each get 1 coin. I don't want to post the full solution, if is obtained by repeating a few simple steps on the situation, so I will just write the beginning of the solution:
Suppose there is only one pirate, he then takes all the gold. If there are two pirates (number 4 and 5), no matter what way to divide the gold the forth will offer the fifth will be always against, because then the forth will die and he will get all the gold. If there are three (3, 4 and 5) then the fifth will be always against and the forth doesn't have a reason to vote for or against (we assume that his life is not important to him), so he will vote against unless he gets something - at least one coin. In this situation, if the third will give him one coin, the forth will vote for him, so the third can keep 49 coins.