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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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closed as too localized by t.b., Zev Chonoles Sep 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
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And maybe also mathoverflow.net/questions/18100/… . –  Qiaochu Yuan Aug 21 '10 at 21:21
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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. –  t.b. Sep 5 '11 at 22:09

91 Answers 91

that properties of recursive function are not always provable. For example, existence of an algorithm which non-terminates and whose non-termination cannot be proved.

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There exists $f\colon \mathbb{N}\times\mathbb{N}\to\mathbb{N}$ which is bijective.

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This is too similar to the fact that Rationals are countable –  botismarius Aug 24 '10 at 13:01

For me it would be the Green-Tao theorem, which states: For any natural number $k$, there exist $k$-term arithmetic progressions of primes.

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Bounded holomorphic function is constant; integration of a meromorphic function.

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The consequences of busy beaver problem are really suprising. For every conjecture/hypothesis about countable number of cases if we can write a program that can verify whether this conjecture holds for some case then we need to check only FINITE number of cases to prove that this conjecture is true for every case.

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Morley’s Miracle: The three points of intersection of the adjacent trisectors of the angles of any triangle always form an equilateral triangle.

This is a stunning gem that slipped through the fingers of the ancients.

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Skolem’s Paradox:

From the Wikipedia article: “Skolem’s paradox is that every countable axiomatisatin of set theory in first-order logic, if it is consistent, has a model that is countable.”

Here’s the link:

http://en.wikipedia.org/wiki/Skolem%27s_paradox

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The fact that any known first order property of $\mathbb{C}$ in the language of rings is valid in any algebraically closed field.

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Well, of characteristic 0... –  Harry Altman Jun 10 '11 at 8:29

Almost everything I've seen so far, but especially that the area under Gaussian curve converges to the square root of the ratio of circumference of the circle to its diameter. This result is old and well-known, but these two things seem so unrelated that I still find it amazing!

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The divergence of the Harmonic Series.

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Another example from the numerics front: it's surprising that despite the theoretical fact that Gaussian elimination can be unstable (even with pivoting!), examples that trigger this instability are in fact very rare in practice, and can be handled by a simple fix if they do arise.

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I got really struck by duality, when my professor lectured about it the first time. I think that even though the algebraic concept is easy to understand, to think that there exists a space such that all inclusions are switched always had a special place in my mind.

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Similar to Thomae's function, I was impressed by the Dirichlet function, which is not only discontinuous everywhere, but impossible to plot. The function is defined as:

$f(x)=\begin{cases} 1 \mbox{ if } x\in\mathbb{Q} \\ 0 \mbox{ if }x\notin\mathbb{Q} \end{cases}$

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I remember a homework question in elementary school. Something like this: Billy and Jane's house is x blocks east and y blocks north of school. Billy walks home by walking east for x blocks and then north for y blocks. Jane decides to take a short cut: she walks alternately a block north and a block east. (There is a picture: Jane's route is a step-like hypotenuse.) Is Jane's route really a short cut?

Of course it is exactly the same distance, but I found this really hard to digest. I knew that in the triangle the sum of the two sides would exceed the hypotenuse.

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Boy, are you gonna like this! –  Raskolnikov Nov 19 '10 at 10:39
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Thus illustrating the difference between the $L^\infty$ and $W^{1,\infty}$ metrics. –  Nate Eldredge Nov 19 '10 at 21:19

If $G$ is a (Hausdorff, locally compact) totally disconnected abelian group which is a filtered union of its compact open subgroups (e.g. the additive group of $\mathbb{Q}_p$), then the category of smooth complex representations of $G$ (smooth meaning the action map is continuous when the vector space has the discrete topology) is canonically equivalent to the category of sheaves of complex vector spaces on the Pontryagin dual of $G$.

This is a beautiful example of an algebro-geometric duality in representation theory and quite shocking to me. The situation is sort of analogous to the equivalence, for a commutative ring $R$, of $R$-modules with quasi-coherent sheaves on $\text{Spec} \ R$.

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All eigenvalues of a Hermitian matrix A are real. No immediate intuition as to why it must be true. If we think of the Riemann hypothesis "All non-trivial zeros of the Riemann zeta function are of the form $ \frac{1}{2}+zi $ where z is real" and try to think of Hermitian matrices having real eigenvalues in equal awe, the wonderment increases. The two ideas are also related.

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This is not a very specific answer but I was struck with awe when I saw the entanglement between partial differential equations and stochastic processes.

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The extension principle of fuzzy subset theory. More generally, the extension of several parts of classical mathematics to an infinite-valued context.

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The Feit-Thompson theorem, although I don't know yet how to prove it. It was impressive for me that only from the condition of having an odd order a group would be solvable.

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Aside from some results I found amazing that have already been mentionned, Lagrange's Theorem in group theory is one that amazed me for some time.

For those who don't know about it, it tells us that the order of any subgroup of a group $G$ divides the order of $G$.

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I have had two results in the last year that surprised me.

One appeals to my intuition in physics, although it may seem more obvious to those more versed mathematics.

The set of pauli matrices (with identity) when multiplied by $i$ form the group of unitary quaternions (under matrix multiplication).

The was surprising to me how you can connect such a physical concept as electron spin to an abstract algebraic concept. Its what led me to dive into group and representation theory as it applies to physics. Now I understand that we can consider spin symmetry as SU(2) , which is injective into SO(3) the group of symmetry of $R^3$ of which the quaternions can be thought of as a representation.

The second result, which I had to prove in an algebra assignment:

For a field $F$ of characteristic $p$ where $p$ is prime $(x+y)^p = x^p + y^p$ for $x,y \in F$. Lovely little result that spits in the face of everything that our grade school teachers taught us in algebra.

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Supersymmetry. From a mathematical standpoint it implies that every bosonic (resp. fermionic) particle in the universe has a fermionic (resp. bosonic) superpartner, i.e., the existence of two physical bijections.

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This one goes hand in hand with the enumerability of $\mathbb{Q}$

The fact that though most of the real numbers are transcendental, it is extremely difficult to find one. (excluding some slight modification of the already known ones)

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I wonder what is the exact meaning of it is extremely difficult to find one (excluding some slight modification of the already known ones). –  Did Jan 7 '13 at 10:34

Kind of simple, but I find it really counterintuitive:

A strictly increasing function can have zero derivative.

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A strictly increasing function can have zero derivative almost everywhere. –  Jonas Meyer Sep 5 '11 at 20:25

The fact that surface area is simply the derivative of volume!

Bear in mind that in mathematics, “volume” is a generic term (i.e., a convenient handle, no pun intended) for a “container” of arbitrary dimension. So, this holds not only for dimension 3 (i.e., a sphere), but also for dimension 2 (i.e., a circle, a circle being a 2-sphere, and perimeter of a 2-dimensional object corresponding to the surface area of a 3-dimensional object), and for dimension 1 (i.e., an interval, an interval being a 1-sphere, and the set of endpoints of a 1-dimensional object corresponding to the surface of a 3-dimensional object). So, we have:

Dimension 3: d(4/3)πr^3/dr = 4πr^2

Dimension 2: dπr^2/dr = 2πr

Dimension 1: d2r/dr = 2

Beautiful!

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$r/{\mathrm{d}r}=1$ does not make sense to me. –  Rasmus Jun 10 '11 at 9:11

Mamikon’s Theorem: The area of a tangent sweep is equal to the area of its tangent cluster, regardless of the original shape of the curve.

This theorem allows you to, among other things, easily obtain results that were obtained before only with difficulty, such as the area under one arch of a cycloid. This theorem is the basis of what has come to be known as Visual Calculus. Here’s the link to Tom Apostol’s account of this awesome insight:

http://eands.caltech.edu/articles/Apostol%20Feature.pdf

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The fact that the sum of the first n odd numbers is n squared. Also, not just this fact, but the nice visual “wrapping” proof of it (as opposed to the induction proof).

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Computational instability of the Quadratic Formula. Who would have thought?

Due to this computational stability an alternative formula is also employed. Here is the relevant quote from the Wikipedia article:

“The alternative formula can reduce loss of precision in the numerical evaluation of the roots, which may be a problem if one of the roots is much smaller than the other in absolute magnitude.”

And here is the link to the full article:

http://en.wikipedia.org/wiki/Quadratic_equation

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Dirichlet’s Theorem: Every proper arithmetic sequence contains a prime.

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In fact, an infinite number of them. –  I. J. Kennedy May 20 '11 at 19:05

How the "inverse" of area under a curve is the slope.

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

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