Friday, November 30, 2007

Should you go shopping?

This is a simple proof that you shouldn't. You cannot find anything useful in shops. And I am going to prove you this right now. The proof is by induction. I am going to prove that in any group of shops no matter how large, there is nothing interesting or useful.
Lets start with n=0 this is with the empty set. There are no shops in it so there is nothing interesting in them. Now lets suppose that we proved this for n=k. We just need to prove it now for n=k+1.
Lets therefore look at a random group of k+1 shops - {a1, a2, a3, a4 ....a(K+1)}. It can be viewed as a union of two groups - {a1,...,ak} and {a2,...,a(K+1)}. Since they both have the some number of shops in them - k, we know from our assumption that there is nothing interesting or useful in any shop in both of them. And since we don't get anything new when we unite the groups there is nothing interesting in the union also. And this completes the proof.

Now lets see another example. Lets suppose you have a random group of people. Is it true that all of them male or all of them female? In other words is it possible by choosing people randomly to get a group where there are both men and women? By now you probably think that I am crazy since I actually asked such a question, right?
Well lets see what math says about this. For n=1 we have a group with one human in it - either a a man or a woman obviously. Now lets suppose that we proved for n=k that any random group of size k has only men or only women. As in the previous example we can now look on a K+1 group as a union of two groups. And since we know that any group of the size k contains only men or only women they both contain only men or only women. Because a2 belongs to both groups and naturally has only one gender :) all the rest have to be the same gender - that is either male of female. And this completes the proof.

So now, after you saw this what do you think about mathematics?

Update: I guess I should have written this part in the first place but anyway: These proofs are wrong. It can be seen easily - just by simple example. In fact all the idea behind those proofs is that induction must be used with great care - or you might proof something that is plain wrong. A good way to check yourself is to try and prove your statement for n=3 and n=4. In the examples here there was no proof whatsoever. Induction is a very valuable and powerful tool but it is meant to be used for proving that some statement is correct for all n, when you know that it is correct for at least some n, and not for proving something you know that is not correct.

Wednesday, November 28, 2007

Evolution of numbers

We all know how numbers look. And sometimes it is even hard to imagine that they can have a different form that what is familiar to us. But while math is a rather ancient science our number system and our numbers are not very old. This photo shows their development in time:

As you can see even numbers in the middle ages were different from modern numbers. And I am sure that you would not even think that there is any connection between our numbers and Hindu.

It is a surprising fact however that there is just one number system. You will probably say it is not correct. But it is. Yes, there were a lot of different number system in history. Even in the present we have more then one. Take the binary system for example.
But if we want a number system that is good enough for both algebra and geometry, that is a system that lets us measure distance we discovery that there is only one such system - they are all isomorphic to each other. It means that you can give different names to numbers, you can have it in base 17 (for example) and not in base 10 but it will still be the same thing. In mathematical language - we have only one mathematical structure that is a complete ordered field.

Saturday, November 24, 2007

A bit of humor

Theorem: Every positive integer is interesting.
Proof: By contradiction, assume that there exists an uninteresting positive integer. Then there must be a smallest uninteresting positive integer. But that's pretty interesting! Therefore a contradiction!

(I really doubt that you will manage to proof this one in school..)

Top ten excuses for not doing homework:

1. I accidentally divided by zero and my paper burst into flames.
2. It was Isaac Newton's birthday.
3. I could only get arbitrarily close to my textbook. I couldn't actually reach it.
4. I have the proof, but there isn't room to write it in this margin.
5. I was watching the World Series and got tied up trying to prove that it converged.
6. I have a solar powered calculator and it was cloudy.
7. I locked the paper in my trunk but a four-dimensional dog got in and ate it.
8. I couldn't figure out whether I am the square root of negative one or i is the square root of negative one.
9. I took time out to snack on a doughnut and a cup of coffee. I spent the rest of the night trying to figure which one to drink.
10. I could have sworn I put the homework inside a Klein bottle, but this morning I couldn't find it

Q: What is the world's longest song?
A: "Aleph-nought Bottles of Beer on the Wall."

(Aleght-nought is infinity. It is from Group theory)

A story:
Some famous mathematician was to give a keynote speech at a conference. When he was asked for an advance summary, he said he would present a proof of Fermat's Last Theorem -- but they shouldn't announce it. Although when he arrived, he spoke on a much more prosaic topic. Afterwards the conference organizers asked why he said he'd talk about the theorem and then didn't. He replied that this was his standard practice, just in case he was killed on the way to the conference.

Another story:
I don't know if the previous story was real but this one is:
WHEN G. H. Hardy faced a stormy sea passage from Scandinavia to England, he took out an unusual insurance policy. Hardy scribbled a postcard to a friend with the words: "Have proved the Riemann hypothesis". God, Hardy reasoned, would not let him die in a shipwreck, because he would then be feted for solving the most famous problem in mathematics. He survived the trip.

A mathematician organizes a lottery in which the prize is an infinite amount of money. When the winning ticket is drawn, and the jubilant winner comes to claim his prize, the mathematician explains the mode of payment:
"1 dollar now, 1/2 dollar next week, 1/3 dollar the week after that..."

(This is a reason for not playing with mathematician in math games. Also as a interesting fact, you chance of winning in a real world lottery is less then the possibility that you will die this week.)