Monday, 4 February 2008

Johnny Loxley and the Infinite Baffle

So I found out today that apparently you can get different sized infinities. My argument goes like this:

Say you flip a coin an infinite number of times. The probability of getting, say, four heads in a row is 1. It is definitely going to happen, given the number of times the coin is flipped. By the same argument, the probability is 1 that ten heads in a row will show up. And twenty. And a million. But also, an infinte amount of heads, since there's infinite time to do it in.

So, I argued, the chances of getting an infinite number of heads, with an inifinite number of flips is 1. Surely that would mean that the possibility of getting a tails at any time is 0 (which obviously isn't true).

My maths friends told me it's something to do with different sized infinities, so that even though there are two infinities involved, one is bigger than the other. Which is why the probability of getting a tails is not 0. What?

I don't know, it made me think is all.

In other news: a) Drinking tea and beer only staves off hunger for so long, b) Real food is expensive and c) Toast is a life saver.

Don't am stupid,
- Mike


Aled said...

I have another one-

There are an infinite number of numbers between 0 and 1. Each of these numbers will have an infinate number of decimal places, so if you put them all in a grid, you'd have an infinity by infinity square. this should theoretically contain every single number there is between 0 and 1 if every number has an infinite number of decimal places and the table has an infinate number of numbers.


If you make a new number by going through the table in a diagonal line (by taking the first digit after the decimal point of the first number, the second digit of the second number, third digit of the third number...and so on) and add 1 to each digit in this new number (so 0 becomes 1, 3 becomes 4, 9 becomes 0...) then that number will not be in your infinite table of numbers, because due to the way it is derived, it will have at least 1 digit different to every number in the original list (the first digit will be 1 higher than the first digit of the first number in the list, the second digit will be 1 higher than the second digit of the second muber in the list, the 143rd digit will be 1 higher than the 143rd digit of the 143rd number in the list...etc).

So even if you write an infinite list of numbers, that still won't have them all.

And that's how maths works.

Also, can you guys make it possible for annonymous comments on this page, because it's a right ballache signing in.

Smokey Bastard said...

I think it's a way of preventing spam, but I'm not sure. Kind of a shame, since I don't actually know anyone who has a blogger account anyway, so anyone wanting to comment would have to sign up etc...

We shall have to sort this out!

- Mike

Smokey Bastard said...

Also, don't different sized infinities imply different sized zeros?

- Mike

Aled said...

When you leave a comment you have to type in a security question thing fro a picture of rando letters to prevent spam, so i don't think allowing annonyous comments would lead to spam.

On other matters, I think zero has one specific value of absolutely nothing. with infinity, it is tending towards a high limit which can never be reached, whereas with zero, if you reach and surpass it you go into negative numbers so you can tend towards zero by oscilating round it which you can't do with infinity.

On the other hand, if you were to create a set of nubers (similar to negative nubers) which were beyond infity, so you could tend towards it by oscillating around it, and it could be "tied down" more easily.

Altough if you take the number "zero" as a limit derived from 1 over infinity, then there would be different zeros.