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# Seeing Infinity (for Ignite Cardiff Apr 09)

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Talking through Cantor's proof in the Ignite format - a bit of an experiment.

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• omg man really i like pie

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• @guest0f480d Sounds like you got to grips with it very well. The reason it doesn't work with integers is: the new number you'd create would have an infinite number of digits... and that would mean it wasn't an integer.

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• Interesting PPT, you might consider pasting the transcipt into the note section of each slide.

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• All of that is a bit of a simplification - if anyone wants to read more about infinity, I'd recommend:
* Infinity and the Mind: The Science and Philosophy of the Infinite by Rudy Rucker (the more maths-y but probably the more fun of the two)
* Brief History of Infinity: The Quest to Think the Unthinkable by Brian Clegg (more of a history of ideas, but with lots of interesting details)

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• Transcript and video

1 - I'm going to talk about infinity and beyond. Thanks to Buzz's catchphrase he's featured in almost everything written on the subject since Toy Story came out.

2 - My talk's called Seeing Infinity. It's a bit of an experiment - trying to get acorss some university-level maths in the Ignite format.

3 - So what I love about maths is the elegance and beauty (and, in some ways, the simplicity) of mathematical proofs. Like the Channel 4 logo here...

4 - ...a mathematician twists the problem around so that, looked at from a different angle, the answer becomes apparent. Infinity gives us a great example of this.

5 - You might have been told that infinity isn't a number. Well, it's probably more helpful to think of it as a number, but a very oddly behaved one. An eccentric but lovable uncle of a number.

6 - For example, think about an infinite hotel with an infinite number of rooms. Each of room has a guest in it - it's the high season. A new guest arrives. Can we accommodate him?

7 - In fact we can because we can put him into Room One, and then ask everyone else to move along one. That works, because having infinite rooms we'll never get to the end.

8 - Our hotel shows that infinity plus one equals infinity. It also turns out that infinity times anything equals infinity. So could there be anything bigger than infinity?

9 - The answer to this lies with real numbers. By 'real' we just mean all the numbers, including the untidy ones. Some of which are so untidy they can't be written down.

10 - The most famous 'untidy' number is pi. Because of pi's relationship to circles, it's been known about for a long time, and people have looked for all sorts of things in its unending tail of digits.

11 - The real numbers give us our thing bigger than infinity, because it turns out there are more of them even between zero and one than there are whole numbers between one and infinity.

12 - The proof of this is my favourite mathematical proof (I don't get out much). It's very short - it'll only take a minute of this talk.

13 - Let's assume that in fact there are the same number of real numbers between zero and one as there are whole numbers. That means we can line the two things up next to each other.

14 - Having lined them up, we build a new number. We take the 1st decimal place of the 1st number in our list as the 1st digit, the 2nd as the 2nd digit, and so on.

15 - Then we make another new number: which has a 4 in every decimal place, except where our first number had a 4, in which case it has a 5. This new number can't have been in our original list.

16 - Why not? Well, for any number in the list, it's different in at least one decimal place. For the Nth number it's different in the N'th place.

17 - Which means we have a real number that wasn't already in our list. That's a contradiction. So our original list wasn't big enough. So we've found a bigger kind of infinity.

18 - That proof, by Georg Cantor, is short, elegant - almost pictorial. Cantor twists the problem round so that the solution just appears. And he tells us something very fundamental and unintuitive about what infinity means.

19 - So, at the last Ignite Cardiff, the organisers were asking people to speak about something you care about - something you're interested in. I was like - nope, can't think of anything. So one of the guys handed me a book on knitting and told me to get a hobby...

20 - ... And I thought, yeah, actually, that is a bit pathetic. So I put this together. I've had lots of fun working on it - and I'd thoroughly recommend to any of you speaking at the next one.

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• ### Seeing Infinity (for Ignite Cardiff Apr 09)

1. http://www.flickr.com/photos/thomashawk/373249714/
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3. http://www.flickr.com/photos/rooreynolds/2104149226/
4. http://www.flickr.com/photos/rooreynolds/2104150484/
5. http://www.flickr.com/photos/mollivan_jon/67850108/
6. http://www.flickr.com/photos/pagedooley/1577999575/
7.
8.  + 1 =  2   =     = 
9. Whole numbers Real numbers 1, 2, 3, 4, etc 3.5728 ⅜ √ 2
10. http://xkcd.com/10/
11. There are more real numbers between 0 and 1, than whole numbers between 1 and  . So there is something bigger than  .
12. My favourite proof…
13. 1 0.2904710830... 2 0.0448736678... 3 0.3382994002... 4 0.2345324344... 5 0.5433447814... . . . . . .
14. 1 0. 2 904710830... 2 0.0 4 48736678... 3 0.33 8 2994002... 4 0.234 5 324344... 5 0.5433 4 47814... 0.24854... 
15. 1 0. 2 904710830... 2 0.0 4 48736678... 3 0.33 8 2994002... 4 0.234 5 324344... 5 0.5433 4 47814... 0.24854...  0.45445...
16. 1 0. 2 904710830... 2 0.0 4 48736678... 3 0.33 8 2994002... 4 0.234 5 324344... 5 0.5433 4 47814... 0.24854...  0.45445... This new number can’t be the same as any of the numbers in the list.
17. 1 0. 2 904710830... 2 0.0 4 48736678... 3 0.33 8 2994002... 4 0.234 5 324344... 5 0.5433 4 47814... 0.24854...  0.45445... This new number can’t be the same as any of the numbers in the list. So the list didn’t contain all the real numbers: lining the numbers up was impossible .
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19. http://www.flickr.com/photos/lollyknit/407335817/
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