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Can you calculate infinity?
By Mark Jordan (P656) on Friday, April 16, 1999 - 04:55 pm:

I've always thought about this. 'Can you calculate infinity?'.
You might say you could: but how?
You might say you can't.
You might even say infinity is just an infinitesimally large number!
Well whatever it is , I've always wonderd

Bye!

By Chris Jefferson (Caj30) on Wednesday, April 21, 1999 - 08:02 am:

In maths the sum from 0 to ¥ is defined to be the sum from 0 to X as X gets very large if as X gets larger the sum tends to a limit. Or: the sum from 0 to ¥ of a series an is l if

for all E there exists D such that if X>D then -E<(Sn=1 Xan-l)<E

You will probably need to read that through a few times... I did!

Anyway the point of these ramblings is that infinity is almost always used in definition rather than existing on it's own. To say something is "equal" to ¥ is to say that it is bigger than all other numbers you could name. For example, as X gets very small, 1/X tends to infinity. However, 1/0 IS NOT EQUAL TO INFINITY!!

There are a few areas of maths where we seem to allow infinity in, but even here it is simply a special definition and nothing more.

Hope that that didn't confuse you too much and maybe helped to answer your question, bye!

By Bill Higgins (P1415) on Wednesday, November 17, 1999 - 04:34 pm:

Just remember!!

Take the Highest number you know and times it by itself many many times.

You see, there is no way of counting infinity!!!

There are no equations.

By Andrew Rogers (Adr26) on Wednesday, November 17, 1999 - 11:50 pm:

Hi there !

I thought I'd say a little about infinity.

Infinty (¥) is rather hard to understand, as we just generally don't see too many infinite numbers around us (we have lots of very large ones, but they are really just very large finite numbers).

If we cound the grains of sand on a beach or the number of molecules in the universe, these are still numbers Physicists (and we) can calculate, ie they are finite.

To cope with infinity cropping up Mathematicians have invented a sort of way of "counting" infinite numbers. Now the idea is that the set of Natural Numbers, ie {1,2,3,...}, are a good way of starting out to tackle the problem. They're fairly simple, and we know there are infinitely many of them. Actually, we can define Natural Numbers in terms of sets, but that's another story...

Basically we use 2 things to generate the Natural numbers:

1) We define "1"
2) The next natural number is the last one plus "1"

(We have to do other things like defining addition, etc, but as I said, it's a different story)

So we have:

1 = 1
2 = 1 + 1
3 = 2 + 1
4 = 3 + 1
....

Now we can say that there are infinitely many natural numbers. For suppose I had found all of them (ie a finite number of them). Then I could take the largest number from my set and use rule (2) to get a bigger one. There we have it. There isn't a finite number of natural numbers. To now look at all sorts of infinity, we use the natural numbers ...

Basically if you can find a way of pairing up every member of an infinite set with one (and only one) distinct natural number (this is known as a bijective map), then you can say that your set is of the same "order" of infinity as the natural numbers.

Let me give you an example to help demonstrate what I'm saying...

The set of (positive) even integers is as large as the set of natural numbers:

Define a relationship q, such that

q: n ® 2n

(Here n represents every natural number, and 2n represents every positive even integer)

Clearly, for every n, we can find a corresponding 2n, and for every 2n we can find a corresponding n, so that we have set up one-to-one correspondence between positive even numbers and the natural numbers.

Now, lots of more clever and experienced mathematicans than myself have gone on to prove lots of interesting things about infinite sets, including "uncountable" sets, where you can't set up a relation like the one I described.

The Mathematician who first thought this up was so ahead of his time that everyone else in Europe thought he was crazy, and he was locked up in a mental institution (which is perhaps where they souhld send me !), but nonetheless, this approach to infinty is the one used throughout Maths today (as far as I'm aware !).

If you are interested in this more, just reply back, and I'll fill you in on what I learn in my first year Numbers & Sets lectures, as I've been looking at the notes, and infinite sets look just round the corner....


Bye for now !


Andrew Rogers