Infinity

MathsIsFun · 2006-12-09 17:48:14

OK ... this is really just a draft because I feel I am over-simplifying the subject ("Everything should be made as simple as possible, but not simpler." said Einsten) ...

... but I have wanted to have a page like this for a while, so I thought I would "bite the bullet", write it, and we can discuss.

So ... here is a page called What is Infinity?

It does not address infinity as used in Sets (Aleph etc), but I would like to say as much about infinity as possible without getting too complicated, and leave that for another page one day. But this page should not be wrong, either.

I would be happy for a full-blown controversial debate on it.

Toast · 2006-12-09 17:56:48

Wow what an excellent read! Infinity is really a subject which needed to be addressed and I think you did it quite well. I got kinda confused with why ∞-∞=/=0 and ∞/∞=/=1, so maybe you could explain that a little? I dunno good work.

Just a question:
If real numbers lie between -∞ and ∞, then where to complex numbers lie?

Last edited by Toast (2006-12-09 18:00:47)

Ricky · 2006-12-09 20:34:11

If real numbers lie between -∞ and ∞, then where to complex numbers lie?

Complex numbers are, uhhh..., more complex. They can not be represented by a single real number. In fact, they are represented by two real numbers, typically a and b:

a + bi

Because of such, the complex numbers lie on a plane, and not a line. However, it is rather easy to see that the real line is merely just a subset of the complex plane, when b = 0.

MathsIsFun · 2006-12-09 20:50:35

No comments on the page, Ricky? Constructive criticism? Corrections?

justlookingforthemoment · 2006-12-09 21:12:17

Wow, this is great!

It was so good I couldn't actually find any mistakes, so I'm just going to have to invent some things which 'seem' wrong, I think.

Okay, so you wanted corrections ... well, they're not really corrections, just fussy things like missing commas and full stops and the word 'so', but I'm going to put them and you can decide ...

Possible insertions are in bold:

1) You will never reach it, so don't try.

2) But infinity does not do anything, it just is.

3) Even these faraway galaxies can't compete with infinity.

4) If it does have an end it is either a Ray (one end) or a line segment (two ends).

5) Which means they could be counted (it may be very hard, but they could be counted).

6) Here are some more proeprties:

All right. I think number six is probably the only real correction, the others were just me being silly and you can correct them if you can actually be bothered.

Devantè · 2006-12-09 21:18:25

Hehe, if we're bringing full stops in, I can recall many with no full stops in them.

I'm not sure they matter, however, since it isn't really a website for topologists.

MathsIsFun · 2006-12-09 21:36:25

Thanks!

I will wait for a few more opinions then make the corrections.

LQ · 2006-12-10 20:53:16

Awesome, And it doesn't say that little extra that makes it all crackpot! Never thought I would find that.

MathsIsFun · 2006-12-10 22:08:19

But it probably does require a last note hinting at the advanced study of infinity.

Ricky · 2006-12-11 02:45:04

I certainly see nothing wrong, and I think it gives a good first glance at what infinity is.

As for a last note hinting at more advanced study, probably the most interesting and non-intuitive part of infinity is that there are different sizes of infinity. If you can number everything, for example the set of even positive integers:

Even integer: 2 4 6 8 10 ...
Number: 1 2 3 4 5 ...

Then we say it is "countable". If we can't, for example, the reals, then we say it isn't countable. We prove countability by finding a way to number everything with a pattern, as above. We prove uncountability by assuming we have some, any, list of numbers, and we show that there exists a number not in this list. For example, assume we have a list of numbers:

Note that we don't say want list. It could be any conceivable or inconceivable list. Now lets create a number, x. If a_0 = 2, then let x_0 = 1, otherwise let x_0 = 2. Repeat this for every single digit. Since every number differs from at least one digit of new number x in our list, x is a number that is not in our list. Since we can do this however many times we want, we will always have an x which is not in our list, and so the real number are not countable.

But when we take rational numbers, we may list them out:

1/1, 1/2, 2/1, 3/1, 1/3, 3/2, 2/3, 4/1, 1/4, 4/2, 2/4, 4/3, 3/4, ... (continue the pattern)

And for any rational number n/m, we will eventually hit it. This means that the rationals are countable.

mathsyperson · 2006-12-11 06:10:56

I echo justlooking completely. I couldn't find anything wrong with the page, and it explains things very well indeed. Because I couldn't find anything wrong with the page at all, I'm going to have to resort to pointing out that you've written Einsten in that first post up there. Because that's the only slightly wrong thing I can find.
Edit: Yay I found one. 'All if these are "undefined":'

Superlative work, as always.

MathsIsFun · 2006-12-12 13:17:26

This is good.

I have revised the page, given an example of why infinity/infnity is not equal to 1 (someone please check).

And I added a "taster" of further study which says: "For example, there are infinitely many whole numbers {0,1,2,3,4,...}, but there are more real numbers (such as 12.308 or 1.111111) because any real number could have an infinite number of decimal places."

Here is the page again: What is Infinity?

Is that both correct and simple?

Devantè · 2006-12-13 04:39:29

Y'all missed out an important one;

Becasue if something has an end, you have to define where that end is.

MathsIsFun · 2008-05-26 12:52:28

I have just updated my What is Infinity? page.

This has become a mildly popular page and I don't want to mislead people, so constructive criticism please!

Maracuya · 2008-06-15 19:50:10

I liked your explanation of "Isn't ∞ / ∞ equal to 1?"

I was not as clear on why 0 × ∞ , ∞^0 , and ∞ - ∞ are undefined.

Questions/ clarifications:
1) Is ∞ = ∞ defined? I'm guessing that it is not, since ∞ -∞ is not defined.
2)You mentioned different magnitudes of infinity. Is the symbol ∞ only for the smallest (Aleph-null ) infinity, or is it generic for infinity of any size?
3) n^0 is defined to be equal to 1 for any n not equal to zero, because this makes the operations consistent for numbers raised to any power.
Is there a similar logic to why ∞^0 must not equal one?

Ricky · 2008-06-16 03:52:13

From the article:

For example, there are infinitely many whole numbers {0,1,2,3,4,...}, but there are more real numbers (such as 12.308 or 1.111111) because any real number could have an infinite number of decimal places.

It's not just that they could have an infinite number of decimal places. Certain rational numbers do as well. But there is a restriction on rational numbers that is not upon the reals. There is no easy way to explain this, so I would personally just leave it that there are more real numbers.

John E. Franklin · 2008-06-19 10:06:55

Infinity is weird, so I have no comment, since I can't distinguish between pure math and applied math, or the difference between what we perceive and what is purely thought processes and ideas. How do mathematicians decide what is right when it comes to things we cannot see?

MathsIsFun · 2008-06-19 11:44:47

Ricky: I changed the example to:

For example, there are infinitely many whole numbers {0,1,2,3,4,...}, but there are more real numbers (such as 12.308 or 1.1111115) because there are infinitely many possible variations after the decimal place as well.

It is tricky to express it simply. Any wording ideas appreciated.

John E. Franklin · 2008-06-20 09:32:46

One has a higher density, but they both go on forever, so
both are endless. I wouldn't say there is answer, just that
one is a subset of the other, but the subset is also infinite.

Ricky · 2008-06-22 14:08:38

I see the same problems with that wording as well. Perhaps the 2nd example should be pi or the square root of 2. The only way I see how to do it is to first have a small blurb about how the decimal digits in a rational number have to either repeat or terminate, then how real numbers don't have this restriction, and then finally conclude there are more reals than just rationals.

But I'm not certain if that's a good idea, I think perhaps it's a bit beyond the scope of the piece. No?

Last edited by Ricky (2008-06-22 14:09:01)

Math Is Fun Forum

#1 2006-12-09 17:48:14

Infinity

#2 2006-12-09 17:56:48

Re: Infinity

#3 2006-12-09 20:34:11

Re: Infinity

#4 2006-12-09 20:50:35

Re: Infinity

#5 2006-12-09 21:12:17

Re: Infinity

#6 2006-12-09 21:18:25

Re: Infinity

#7 2006-12-09 21:36:25

Re: Infinity

#8 2006-12-10 20:53:16

Re: Infinity

#9 2006-12-10 22:08:19

Re: Infinity

#10 2006-12-11 02:45:04

Re: Infinity

#11 2006-12-11 06:10:56

Re: Infinity

#12 2006-12-12 13:17:26

Re: Infinity

#13 2006-12-13 04:39:29

Re: Infinity

#14 2008-05-26 12:52:28

Re: Infinity

#15 2008-06-15 19:50:10

Re: Infinity

#16 2008-06-16 03:52:13

Re: Infinity

#17 2008-06-19 10:06:55

Re: Infinity

#18 2008-06-19 11:44:47

Re: Infinity

#19 2008-06-20 09:32:46

Re: Infinity

#20 2008-06-22 14:08:38

Re: Infinity

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