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A large part of my fascination in mathematics is because of some very surprising results that I have seen there.

I remember one I found very hard to swallow when I first encountered it, was what is known as the Banach Tarski Paradox. It states that you can separate a ball $x^2+y^2+z^2 \le 1$ into finitely many disjoint parts, rotate and translate them and rejoin (by taking disjoint union), and you end up with exactly two complete balls of the same radius!

So I ask you which are your most surprising moments in maths?

  • Chances are you will have more than one. May I request post multiple answers in that case, so the voting system will bring the ones most people think as surprising up. Thanks!
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    big-list usually means community wiki. For this question it applies.2010-08-21
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    I agree - I think I just did, I edited and checked 'community wiki'. Let me know if that doesn't work.2010-08-21
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    another relevant question, while not the same is http://math.stackexchange.com/questions/1352/surprising-generalizations2010-08-21
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    http://mathoverflow.net/questions/14574/your-favorite-surprising-connections-in-mathematics2010-08-21
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    And maybe also http://mathoverflow.net/questions/18100/theorems-with-unexpected-conclusions .2010-08-21
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    Finally, http://math.stackexchange.com/questions/250?tab=votes#tab-top .2010-08-21
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    Thanks, bookmarked all of them!2010-08-23
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    I added 4 answers to this question. I guess that means I don't have enough mathematics intuition to not be surprised by obvious stuff. Still, each of my answers seriously blows my mind.2010-09-02
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    I'm getting tired of this question being bumped every once in a while. It seems to have served its purpose and there's no need to accumulate more than 100 answers. Therefore I voted to close it.2011-09-05
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    @Theo: I agree, I will close the question ("too localized" is the best option out of the available ones, it seems).2011-09-05
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    @Theo, Zev: It's also very subjective-you gave me a negative one for my suggestion of the Cayley Isomorphism theorum, which I don't really agree with.2011-09-06
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    @Mathemagician: I gave you a -1 for the "no reason" remark; if you remove it I would be happy to change it to a +1. Also, the content of your answer had no bearing on the decision to close the question.2011-09-06

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