Hilbert Hotel

Oh and I forgot to credit that picture, it was from berto-meister.blogspot.com

Hilbert hotel (revisited)

Just wanted to show you an interesting picture to do with the hilbert hotel paradox.

This is just a trivial picture that demonstrates the paradox. A little present to all our visual learners out there.

I realize that in my last post I forgot to link my infinity paradox to geography, its very simple, in geography there are points on a map and in the previous infinity paradox there were points on a dart board. Consider that paradox linked.

Infinity Problems

           Infinity is a very interesting idea. It can confuse many people and as a concept has some very interesting properities. Some of these properties are demonstrated in infinity paradoxes. One such infinity paradox is the "Hilbert Hotel" Paradox. This was presented by David Hilbert in the 1920's and is a Veridical Paradox (as apose to a Falsidical Paradox). 

           Imagine a hypothetical hotel with an infinite number of rooms, all of which are occupied. We shall call this hotel the "Zambezi Sun Hotel" because the Zambezi Sun is an actual hotel in Livingstone, Zambia. I have many happy memories of the Zambezi Sun so it is a suitable hotel to link with such an interesting paradox. One would think that if a guest were to come to the Zambezi Sun, he would be turned away as the Zambezi Sun would be "full". If the manager of the Zambezi Sun moves guest one to room two, the guest in room two to room three, and keeps doing this then he can fit the new guest in. Similarly if an infinite number of guests came to the Zambezi Sun, by moving guest N into room N+1 the hotel manager can fit an infinite number of guests into his hotel. This is interesting as you could not do this with a finite number.

            This is linked to geography because hotels are tertiary econemic activities which is a geographical term. Also this hotel has a location(Mozi-oa-tunya road, Livingstone 20100, Zambia) which is quite geographical as I'm sure you'll agree.

               I have come up with some more infinity paradoxes, as well as an interesting video that fits in quite nicely. The video is by a group on you tube called Numberphile. http://www.youtube.com/watch?v=dDl7g_2x74Q is the link to the video. In this video he talks about four different infinity paradoxes and probably explains them better than I can on this blog.

               The paradox I'd like to share with you today is a dart board paradox, for all those dart enthusiasts out there. Imagine throwing a dart at a dart board and having a 100% guarantee that the dart is going to hit the dart board. Now think about the point of the end of the dart, and imagine a mathematical point on the dart board. What exactly is the probability that the dart is going to hit that point on the dart board?

                Turns out there isn't a sensible answer for this. If we say that there is a probability greater than one ( which seems like a reasonable deduction), we must consider that there is an infinite number of points on this dart board. All these points would also have the same probability of being hit. As there is an infinite number of point on the dart board, you would have to add up all the probabilities and come to the conclusion that there is an infinite probability of that point being hit. This is complete rubbish as you can't have a probability greater than one.

                On the other hand if we say that the chance of hitting that point is zero, we could say the same for any point on the dart board. This means that the probability of hitting any point on the dart board is zero, which again doesn't make sense because we know that there is a 100% probability of the dart hitting the board...

                  This again in another strange infinity paradox, if you're interested and not too bored by my non-nonsensical ranting then maybe you could look up these paradoxes for yourself and check them out. I think they're cool... 

               

Introduction

My name is Chris and in this blog I'm going to try and show you some interesting mathematical conumdrums/problems and relate them to Geography. Everything is relatable to geography and mathematical problems are no exception. I'm very interesed in mathematical problems, especially ones that haven't been solved yet.