Bit of maths

[Lazlo]

Of course another way is to have the hotel room contain an infinite number of floors. The ground floor has doors numbered with all rational numbers of the form 1/m. the second floor has doors numbered with all rational numbers of the form 2/m, and so on.

If you introduce imaginary numbers, I'll respond by creating another dimension again, say an infinite number of blocks, each with an infinite number of floors .

[uwila]

Originally Posted by Lazlo

Yes: Take every invitation note, look at the rational number n/m and convert that rational number to a positive integer by combining the two numbers into one: e.g. 1/3 becomes 13, 26/9216 becomes 269216 and so on. Now everyone has a unique room number, and everyone is accommodated.

What about the person who had 2/69216 - surely that room was taken by the person with 26/9216?

Originally Posted by Lazlo

No. There are more rational numbers between 0 and 1 than there are positive integers, therefore there is not enough rooms.

Yup, this is the bizarre but correct answer.

The fact is that the number of rooms is countable, but the number of guests is not and is a much bigger number. It is even bigger than N squared (N is the biggest number in the sequence 1, 2,3,4,...)

[dry suit diver]

Originally Posted by uwila

What about the person who had 2/69216 - surely that room was taken by the person with 26/9216?


Yup, this is the bizarre but correct answer.

The fact is that the number of rooms is countable, but the number of guests is not and is a much bigger number. It is even bigger than N squared (N is the biggest number in the sequence 1, 2,3,4,...)

wil you stop, its friday and its supposed to be the relaxing day with flaming and trolling not maths.

[GLOC]

Originally Posted by uwila

What about the person who had 2/69216 - surely that room was taken by the person with 26/9216?


Yup, this is the bizarre but correct answer.

The fact is that the number of rooms is countable, but the number of guests is not and is a much bigger number. It is even bigger than N squared (N is the biggest number in the sequence 1, 2,3,4,...)

This is quite amusing, only because there are people out there who understand this...

However, I have a question, how can N be the biggest number in the sequence 1,2,3,4....if N^2 is a possible number?

[Lazlo]

Originally Posted by uwila

What about the person who had 2/69216 - surely that room was taken by the person with 26/9216?

Ok, so my algorithm is rubbish. But suppose you used something like:

....

hmm, can't think of one off-hand. Are you saying that it would be possible to prove that it is impossible to come up with a method for generating a unique positive integer from arbitrary values of n and m where n and m are positive integers and n< m.

[uwila]

Originally Posted by GLOC

However, I have a question, how can N be the biggest number in the sequence 1,2,3,4....if N^2 is a possible number?

If you imagine a big, well a very big , chessboard with N squares on each side. If you count the individual squares, you have N squared

Admitedtly it's going to take a long time to actually make, but being an ex pure mathematician, I can safely leave the practical stuff to someone else

[uwila]

Originally Posted by Lazlo

hmm, can't think of one off-hand. Are you saying that it would be possible to prove that it is impossible to come up with a method for generating a unique positive integer from arbitrary values of n and m where n and m are positive integers and n< m.

Yes. The sequence 1,2,3,4... is countable and the rationals are uncountable. Uncountable simply means that there are more values in the set/sequence than there are whole numbers.

In fact my earlier post gives a proof of exactly what you ask. Just using n=1 and n=2 proves it and there are quite a few more n values to consider

[nigelH]

Originally Posted by uwila

Admitedtly it's going to take a long time to actually make, but being an ex pure mathematician, I can safely leave the practical stuff to someone else

I started down that route (University of Sussex 1968) but somehow ended up working on life support systems (like aircraft) where you have to be very practical or death support systems where you have to be very very practical and rather left things like that alone.

In answer to the original question the sum of the series
9/10^1+9/10^2+9/10^3....+9/10^N where N->infinite is 1 by any acceptable proof.
As mathematicians know infinity is just a number that is bigger than any number you care to name. That's how we prove something is infinite. It is just bigger than any arbitrary number.

Mathematics is only as complicated as you want it to be.

"Is the set of all sets that do not contain themselves a member of itself?"

[Lazlo]

Slightly off-topic



xkcd: the site for maths nerds everywhere!

Another one of my favourites

[Lazlo]

Originally Posted by uwila

Yes. The sequence 1,2,3,4... is countable and the rationals are uncountable. Uncountable simply means that there are more values in the set/sequence than there are whole numbers.

Ooh - apparently the set of rational numbers IS countable.

Therefore it must be possible to fit all rational numbers in the hotel .