# Which one result in mathematics has surprised you the most? [closed]

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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## closed as too localized by t.b., Zev ChonolesSep 5 '11 at 22:18

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big-list usually means community wiki. For this question it applies. –  Aryabhata Aug 21 '10 at 19:01
And maybe also mathoverflow.net/questions/18100/… . –  Qiaochu Yuan Aug 21 '10 at 21:21
–  Qiaochu Yuan Aug 21 '10 at 21:28
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. –  t.b. Sep 5 '11 at 22:09

(I mean: $\frac{d}{dx} \int x {\mathrm dx} = x$)